feat(builtin): automatically generate Lean/C++ interface for builtin theories

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

src/builtin/CMakeLists.txt

@@ -67,6 +67,20 @@ function(add_theory FILE DEPS)
   endif()
 endfunction()
 
+# The following command invokes the lean binary to update .cpp/.h interface files
+# associated with a .lean file.
+function(update_interface FILE DEST ARG)
+  get_filename_component(BASENAME ${FILE} NAME_WE)
+  set(CPPFILE "${LEAN_SOURCE_DIR}/${DEST}/${BASENAME}_decls.cpp")
+  set(HFILE "${LEAN_SOURCE_DIR}/${DEST}/${BASENAME}_decls.h")
+  add_custom_command(OUTPUT ${CPPFILE} ${HFILE}
+    COMMAND ${SHELL_DIR}/lean ${ARG} -q ${CMAKE_CURRENT_SOURCE_DIR}/${FILE} ${CMAKE_CURRENT_SOURCE_DIR}/name_conv.lua ${CMAKE_CURRENT_SOURCE_DIR}/lean2cpp.lean > ${CPPFILE}
+    COMMAND ${SHELL_DIR}/lean ${ARG} -q ${CMAKE_CURRENT_SOURCE_DIR}/${FILE} ${CMAKE_CURRENT_SOURCE_DIR}/name_conv.lua ${CMAKE_CURRENT_SOURCE_DIR}/lean2h.lean > ${HFILE}
+    DEPENDS ${FILE})
+  add_custom_target("${BASENAME}_decls" DEPENDS ${CPPFILE})
+  add_dependencies(builtin ${BASENAME}_decls)
+endfunction()
+
 add_kernel_theory("kernel.lean"    "${CMAKE_CURRENT_BINARY_DIR}/macros.lua")
 add_kernel_theory("Nat.lean"       "${CMAKE_CURRENT_BINARY_DIR}/kernel.olean")
 
@@ -75,3 +89,8 @@ add_theory("Int.lean"          "${CMAKE_CURRENT_BINARY_DIR}/if_then_else.olean")
 add_theory("Real.lean"         "${CMAKE_CURRENT_BINARY_DIR}/Int.olean")
 add_theory("specialfn.lean"    "${CMAKE_CURRENT_BINARY_DIR}/Real.olean")
 add_theory("cast.lean"         "${CMAKE_CURRENT_BINARY_DIR}/Nat.olean")
+
+update_interface("kernel.lean" "kernel" "-n")
+update_interface("Nat.lean"    "library/arith" "-n")
+update_interface("Int.lean"    "library/arith" "")
+update_interface("Real.lean"   "library/arith" "")

src/builtin/kernel.lean

@@ -61,7 +61,7 @@ axiom funext {A : TypeU} {B : A → TypeU} {f g : ∀ x : A, B x} (H : ∀ x : A
 axiom allext {A : TypeU} {B C : A → TypeU} (H : ∀ x : A, B x == C x) : (∀ x : A, B x) == (∀ x : A, C x)
 
 -- Alias for subst where we can provide P explicitly, but keep A,a,b implicit
-definition substp {A : TypeU} {a b : A} (P : A → Bool) (H1 : P a) (H2 : a == b) : P b
+theorem substp {A : TypeU} {a b : A} (P : A → Bool) (H1 : P a) (H2 : a == b) : P b
 := subst H1 H2
 
 theorem eta {A : TypeU} {B : A → TypeU} (f : ∀ x : A, B x) : (λ x : A, f x) == f
@@ -267,12 +267,6 @@ theorem and_absurd (a : Bool) : (a ∧ ¬ a) == false
 := boolext (λ H, absurd (and_eliml H) (and_elimr H))
            (λ H, false_elim (a ∧ ¬ a) H)
 
-theorem not_true : (¬ true) == false
-:= trivial
-
-theorem not_false : (¬ false) == true
-:= trivial
-
 theorem not_and (a b : Bool) : (¬ (a ∧ b)) == (¬ a ∨ ¬ b)
 := case (λ x, (¬ (x ∧ b)) == (¬ x ∨ ¬ b))
         (case (λ y, (¬ (true ∧ y)) == (¬ true ∨ ¬ y))   trivial trivial b)

src/builtin/lean2cpp.lean

@@ -0,0 +1,32 @@
+(*
+ -- Auxiliary script for generating .cpp files that define
+ -- constants defined in Lean
+ local env = get_environment()
+ local num_imports = 0
+ print('/*')
+ print('Copyright (c) 2013 Microsoft Corporation. All rights reserved.')
+ print('Released under Apache 2.0 license as described in the file LICENSE.')
+ print('*/')
+ print("// Automatically generated file, DO NOT EDIT")
+ print('#include "kernel/environment.h"')
+ print('#include "kernel/decl_macros.h"')
+ print('namespace lean {')
+ for obj in env:objects() do
+    if obj:is_begin_import() or obj:is_begin_builtin_import() then
+       num_imports = num_imports + 1
+    elseif obj:is_end_import() then
+       num_imports = num_imports - 1
+    elseif num_imports == 0 and obj:has_name() and obj:has_type() and not is_explicit(env, obj:get_name()) and not obj:is_builtin() then
+       local is_fn = env:normalize(obj:get_type()):is_pi()
+       io.write('MK_CONSTANT(')
+       name_to_cpp_decl(obj:get_name())
+       if is_fn then
+          io.write('_fn')
+       end
+       io.write(', ')
+       name_to_cpp_expr(obj:get_name())
+       print(');')
+    end
+ end
+ print('}')
+*)

src/builtin/lean2h.lean

@@ -0,0 +1,71 @@
+(*
+ -- Auxiliary script for generating .h file that declare mk_* and is_* functions for
+ -- constants defined in Lean
+ local env = get_environment()
+ local num_imports = 0
+ print('/*')
+ print('Copyright (c) 2013 Microsoft Corporation. All rights reserved.')
+ print('Released under Apache 2.0 license as described in the file LICENSE.')
+ print('*/')
+ print("// Automatically generated file, DO NOT EDIT")
+ print('#include "kernel/expr.h"')
+ print('namespace lean {')
+ for obj in env:objects() do
+    if obj:is_begin_import() or obj:is_begin_builtin_import() then
+       num_imports = num_imports + 1
+    elseif obj:is_end_import() then
+       num_imports = num_imports - 1
+    elseif num_imports == 0 and obj:has_name() and obj:has_type() and not is_explicit(env, obj:get_name()) and not obj:is_builtin() then
+       local ty    = env:normalize(obj:get_type())
+       local is_fn = ty:is_pi()
+       local arity = 0
+       while ty:is_pi() do
+          n, d, ty = ty:fields()
+          arity = arity + 1
+       end
+       local is_th = obj:is_theorem() or obj:is_axiom()
+       io.write("expr mk_")
+       name_to_cpp_decl(obj:get_name())
+       if is_fn then
+          io.write("_fn");
+       end
+       print("();")
+       io.write("bool is_")
+       name_to_cpp_decl(obj:get_name())
+       if is_fn then
+          io.write("_fn");
+       end
+       print("(expr const & e);")
+       if is_fn and not is_th then
+          io.write("inline bool is_")
+          name_to_cpp_decl(obj:get_name())
+          io.write("(expr const & e) { return is_app(e) && is_");
+          name_to_cpp_decl(obj:get_name())
+          print("_fn(arg(e, 0)); }")
+       end
+       if is_fn then
+          io.write("inline expr mk_")
+          name_to_cpp_decl(obj:get_name())
+          if is_th then
+             io.write("_th");
+          end
+          io.write("(");
+          for i = 1, arity do
+             if i > 1 then
+                io.write(", ")
+             end
+             io.write("expr const & e" .. tostring(i))
+          end
+          io.write(") { return mk_app({mk_");
+          name_to_cpp_decl(obj:get_name())
+          io.write("_fn()")
+          for i = 1, arity do
+             io.write(", e" .. tostring(i))
+          end
+          print ("}); }")
+       end
+    end
+ end
+ print('}')
+
+*)

src/builtin/name_conv.lua

@@ -0,0 +1,38 @@
+-- Output a C++ statement that creates the given name
+function name_to_cpp_expr(n)
+   function rec(n)
+      if not n:is_atomic() then
+         rec(n:get_prefix())
+         io.write(", ")
+      end
+      if n:is_string() then
+         io.write("\"" .. n:get_string() .. "\"")
+      else
+         error("numeral hierarchical names are not supported in the C++ interface: " .. tostring(n))
+      end
+   end
+
+   io.write("name(")
+   if n:is_atomic() then
+      rec(n)
+   else
+      io.write("{")
+      rec(n)
+      io.write("}")
+   end
+   io.write(")")
+end
+
+-- Output a C++ constant name based on the given hierarchical name
+-- It uses '_' to glue the hierarchical name parts
+function name_to_cpp_decl(n)
+   if not n:is_atomic(n) then
+      name_to_cpp_decl(n:get_prefix())
+      io.write("_")
+   end
+   if n:is_string() then
+      io.write(n:get_string())
+   else
+      error("numeral hierarchical names are not supported in the C++ interface: " .. tostring(n))
+   end
+end

src/frontends/lean/parser_calc.cpp

@@ -44,34 +44,20 @@ void calc_proof_parser::add_trans_step(expr const & op1, expr const & op2, trans
     m_trans_ops.emplace_front(op1, op2, d);
 }
 
-static name g_eq_imp_trans("eq_imp_trans");
-static name g_imp_eq_trans("imp_eq_trans");
-static name g_imp_trans("imp_trans");
-static name g_eq_ne_trans("eq_ne_trans");
-static name g_ne_eq_trans("ne_eq_trans");
-static name g_neq("neq");
-
 calc_proof_parser::calc_proof_parser() {
     expr imp = mk_implies_fn();
-    expr eq  = mk_homo_eq_fn();
-    expr iff = mk_iff_fn();
-    expr neq = mk_constant(g_neq);
+    expr eq  = mk_eq_fn();
+    expr neq = mk_neq_fn();
 
     add_supported_operator(op_data(imp, 2));
     add_supported_operator(op_data(eq, 3));
-    add_supported_operator(op_data(iff, 2));
     add_supported_operator(op_data(neq, 3));
     add_trans_step(eq, eq,    trans_data(mk_trans_fn(), 6, eq));
-    add_trans_step(eq, imp,   trans_data(mk_constant(g_eq_imp_trans), 5, imp));
-    add_trans_step(imp, eq,   trans_data(mk_constant(g_imp_eq_trans), 5, imp));
-    add_trans_step(imp, imp,  trans_data(mk_constant(g_imp_trans), 5, imp));
-    add_trans_step(iff, iff,  trans_data(mk_trans_fn(), 6, iff));
-    add_trans_step(iff, imp,  trans_data(mk_constant(g_eq_imp_trans), 5, imp));
-    add_trans_step(imp, iff,  trans_data(mk_constant(g_imp_eq_trans), 5, imp));
-    add_trans_step(eq, iff,   trans_data(mk_trans_fn(), 6, iff));
-    add_trans_step(iff, eq,   trans_data(mk_trans_fn(), 6, iff));
-    add_trans_step(eq,  neq,  trans_data(mk_constant(g_eq_ne_trans), 6, neq));
-    add_trans_step(neq, eq,   trans_data(mk_constant(g_ne_eq_trans), 6, neq));
+    add_trans_step(eq, imp,   trans_data(mk_eq_imp_trans_fn(), 5, imp));
+    add_trans_step(imp, eq,   trans_data(mk_imp_eq_trans_fn(), 5, imp));
+    add_trans_step(imp, imp,  trans_data(mk_imp_trans_fn(), 5, imp));
+    add_trans_step(eq,  neq,  trans_data(mk_eq_ne_trans_fn(), 6, neq));
+    add_trans_step(neq, eq,   trans_data(mk_ne_eq_trans_fn(), 6, neq));
 }
 
 optional<expr> calc_proof_parser::find_op(operator_info const & op, pos_info const & p) const {
@@ -108,12 +94,12 @@ static expr parse_step_pr(parser_imp & imp, expr const & lhs) {
         expr eq_pr = imp.parse_expr();
         imp.check_rcurly_next("invalid calculational proof, '}' expected");
         // Using axiom Subst {A : TypeU} {a b : A} {P : A → Bool} (H1 : P a) (H2 : a == b) : P b.
-        return imp.save(Subst(imp.save(mk_placeholder(), p),
-                              imp.save(mk_placeholder(), p),
-                              imp.save(mk_placeholder(), p),
-                              imp.save(mk_placeholder(), p), // let elaborator compute the first four arguments
-                              imp.save(Refl(imp.save(mk_placeholder(), p), lhs), p),
-                              eq_pr), p);
+        return imp.save(mk_subst_th(imp.save(mk_placeholder(), p),
+                                    imp.save(mk_placeholder(), p),
+                                    imp.save(mk_placeholder(), p),
+                                    imp.save(mk_placeholder(), p), // let elaborator compute the first four arguments
+                                    imp.save(mk_refl_th(imp.save(mk_placeholder(), p), lhs), p),
+                                    eq_pr), p);
     } else {
         return imp.parse_expr();
     }

src/frontends/lean/parser_expr.cpp

@@ -561,11 +561,11 @@ expr parser_imp::parse_led_id(expr const & left) {
 }
 
 /** \brief Parse <tt>expr '=' expr</tt>. */
-expr parser_imp::parse_eq(expr const & left) {
+expr parser_imp::parse_heq(expr const & left) {
     auto p = pos();
     next();
     expr right = parse_expr(g_eq_precedence);
-    return save(mk_eq(left, right), p);
+    return save(mk_heq(left, right), p);
 }
 
 /** \brief Parse <tt>expr '->' expr</tt>. */
@@ -947,7 +947,7 @@ expr parser_imp::mk_app_left(expr const & left, expr const & arg) {
 expr parser_imp::parse_led(expr const & left) {
     switch (curr()) {
     case scanner::token::Id:          return parse_led_id(left);
-    case scanner::token::Eq:          return parse_eq(left);
+    case scanner::token::Eq:          return parse_heq(left);
     case scanner::token::Arrow:       return parse_arrow(left);
     case scanner::token::LeftParen:   return mk_app_left(left, parse_lparen());
     case scanner::token::IntVal:      return mk_app_left(left, parse_nat_int());

src/frontends/lean/parser_imp.h

@@ -291,7 +291,7 @@ private:
                              pos_info const & p);
     expr parse_expr_macro(name const & id, pos_info const & p);
     expr parse_led_id(expr const & left);
-    expr parse_eq(expr const & left);
+    expr parse_heq(expr const & left);
     expr parse_arrow(expr const & left);
     expr parse_lparen();
     void parse_names(buffer<std::pair<pos_info, name>> & result);

src/frontends/lean/pp.cpp

@@ -234,7 +234,7 @@ class pp_fn {
                 return is_atomic(arg(e, 1));
             else
                 return false;
-        case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::Eq: case expr_kind::Let:
+        case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::HEq: case expr_kind::Let:
             return false;
         }
         return false;
@@ -441,7 +441,7 @@ class pp_fn {
         operator_info op = get_operator(e);
         if (op) {
             return op.get_precedence();
-        } else if (is_eq(e)) {
+        } else if (is_heq(e)) {
             return g_eq_precedence;
         } else if (is_arrow(e)) {
             return g_arrow_precedence;
@@ -459,7 +459,7 @@ class pp_fn {
         operator_info op = get_operator(e);
         if (op) {
             return op.get_fixity() == fx;
-        } else if (is_eq(e)) {
+        } else if (is_heq(e)) {
             return fixity::Infix == fx;
         } else if (is_arrow(e)) {
             return fixity::Infixr == fx;
@@ -1022,7 +1022,7 @@ class pp_fn {
     }
 
     /** \brief Pretty print the child of an equality. */
-    result pp_eq_child(expr const & e, unsigned depth) {
+    result pp_heq_child(expr const & e, unsigned depth) {
         if (is_atomic(e)) {
             return pp(e, depth + 1);
         } else {
@@ -1034,11 +1034,11 @@ class pp_fn {
     }
 
     /** \brief Pretty print an equality */
-    result pp_eq(expr const & e, unsigned depth) {
+    result pp_heq(expr const & e, unsigned depth) {
         result p_arg1, p_arg2;
         format r_format;
-        p_arg1 = pp_eq_child(eq_lhs(e), depth);
-        p_arg2 = pp_eq_child(eq_rhs(e), depth);
+        p_arg1 = pp_heq_child(heq_lhs(e), depth);
+        p_arg2 = pp_heq_child(heq_rhs(e), depth);
         r_format = group(format{p_arg1.first, space(), g_eq_fmt, line(), p_arg2.first});
         return mk_result(r_format, p_arg1.second + p_arg2.second + 1);
     }
@@ -1121,7 +1121,7 @@ class pp_fn {
                 case expr_kind::Lambda:
                 case expr_kind::Pi:         r = pp_abstraction(e, depth); break;
                 case expr_kind::Type:       r = pp_type(e);               break;
-                case expr_kind::Eq:         r = pp_eq(e, depth);          break;
+                case expr_kind::HEq:        r = pp_heq(e, depth);         break;
                 case expr_kind::Let:        r = pp_let(e, depth);         break;
                 case expr_kind::MetaVar:    r = pp_metavar(e, depth);     break;
                 }

src/kernel/builtin.cpp

@@ -9,6 +9,7 @@ Author: Leonardo de Moura
 #include "kernel/abstract.h"
 #include "kernel/io_state.h"
 #include "kernel/decl_macros.h"
+#include "kernel/kernel_decls.cpp"
 
 namespace lean {
 // =======================================
@@ -19,7 +20,6 @@ expr const TypeU = Type(u_lvl);
 
 // =======================================
 // Boolean Type
-MK_CONSTANT(Bool, "Bool");
 expr const Bool = mk_Bool();
 expr mk_bool_type() { return mk_Bool(); }
 bool is_bool(expr const & t) { return is_Bool(t); }
@@ -116,57 +116,6 @@ static register_builtin_fn if_blt("ite", []() { return mk_if_fn(); });
 static value::register_deserializer_fn if_ds("ite", [](deserializer & ) { return mk_if_fn(); });
 // =======================================
 
-MK_CONSTANT(implies_fn, name("implies"));
-MK_CONSTANT(iff_fn,     name("iff"));
-MK_CONSTANT(and_fn,     name("and"));
-MK_CONSTANT(or_fn,      name("or"));
-MK_CONSTANT(not_fn,     name("not"));
-MK_CONSTANT(forall_fn,  name("forall"));
-MK_CONSTANT(exists_fn,  name("exists"));
-MK_CONSTANT(homo_eq_fn, name("eq"));
-// Axioms
-MK_CONSTANT(mp_fn,          name("mp"));
-MK_CONSTANT(discharge_fn,   name("discharge"));
-MK_CONSTANT(case_fn,        name("case"));
-MK_CONSTANT(refl_fn,        name("refl"));
-MK_CONSTANT(subst_fn,       name("subst"));
-MK_CONSTANT(abst_fn,        name("funext"));
-MK_CONSTANT(htrans_fn,      name("htrans"));
-MK_CONSTANT(hsymm_fn,       name("hsymm"));
-// Theorems
-MK_CONSTANT(eta_fn,             name("eta"));
-MK_CONSTANT(imp_antisym_fn,     name("iffintro"));
-MK_CONSTANT(trivial,            name("trivial"));
-MK_CONSTANT(false_elim_fn,      name("false_elim"));
-MK_CONSTANT(absurd_fn,          name("absurd"));
-MK_CONSTANT(em_fn,              name("em"));
-MK_CONSTANT(double_neg_fn,      name("doubleneg"));
-MK_CONSTANT(double_neg_elim_fn, name("doubleneg_elim"));
-MK_CONSTANT(mt_fn,              name("mt"));
-MK_CONSTANT(contrapos_fn,       name("contrapos"));
-MK_CONSTANT(absurd_elim_fn,     name("absurd_elim"));
-MK_CONSTANT(eq_mp_fn,           name("eqmp"));
-MK_CONSTANT(not_imp1_fn,        name("not_imp_eliml"));
-MK_CONSTANT(not_imp2_fn,        name("not_imp_elimr"));
-MK_CONSTANT(conj_fn,            name("and_intro"));
-MK_CONSTANT(conjunct1_fn,       name("and_eliml"));
-MK_CONSTANT(conjunct2_fn,       name("and_elimr"));
-MK_CONSTANT(disj1_fn,           name("or_introl"));
-MK_CONSTANT(disj2_fn,           name("or_intror"));
-MK_CONSTANT(disj_cases_fn,      name("or_elim"));
-MK_CONSTANT(refute_fn,          name("refute"));
-MK_CONSTANT(symm_fn,            name("symm"));
-MK_CONSTANT(trans_fn,           name("trans"));
-MK_CONSTANT(congr1_fn,          name("congr1"));
-MK_CONSTANT(congr2_fn,          name("congr2"));
-MK_CONSTANT(congr_fn,           name("congr"));
-MK_CONSTANT(eqt_elim_fn,        name("eqt_elim"));
-MK_CONSTANT(eqt_intro_fn,       name("eqt_intro"));
-MK_CONSTANT(forall_elim_fn,     name("forall_elim"));
-MK_CONSTANT(forall_intro_fn,    name("forall_intro"));
-MK_CONSTANT(exists_elim_fn,     name("exists_elim"));
-MK_CONSTANT(exists_intro_fn,    name("exists_intro"));
-
 void import_kernel(environment const & env, io_state const & ios) {
     env->import("kernel", ios);
 }

src/kernel/builtin.h

@@ -5,7 +5,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 */
 #pragma once
-#include "kernel/expr.h"
+#include "kernel/io_state.h"
+#include "kernel/kernel_decls.h"
 
 namespace lean {
 // See src/builtin/kernel.lean for signatures.
@@ -32,191 +33,12 @@ bool is_true(expr const & e);
 /** \brief Return true iff \c e is the Lean false value. */
 bool is_false(expr const & e);
 
-expr mk_if_fn();
-bool is_if_fn(expr const & e);
-inline bool is_if(expr const & e) { return is_app(e) && is_if_fn(arg(e, 0)); }
-inline expr mk_if(expr const & A, expr const & c, expr const & t, expr const & e) { return mk_app(mk_if_fn(), A, c, t, e); }
-inline expr If(expr const & A, expr const & c, expr const & t, expr const & e) { return mk_if(A, c, t, e); }
-inline expr mk_bool_if(expr const & c, expr const & t, expr const & e) { return mk_if(mk_bool_type(), c, t, e); }
-inline expr bIf(expr const & c, expr const & t, expr const & e) { return mk_bool_if(c, t, e); }
-
-expr mk_implies_fn();
-bool is_implies_fn(expr const & e);
-inline bool is_implies(expr const & e) { return is_app(e) && is_implies_fn(arg(e, 0)); }
-inline expr mk_implies(expr const & e1, expr const & e2) { return mk_app(mk_implies_fn(), e1, e2); }
 inline expr Implies(expr const & e1, expr const & e2) { return mk_implies(e1, e2); }
-
-expr mk_iff_fn();
-bool is_iff_fn(expr const & e);
-inline bool is_iff(expr const & e) { return is_app(e) && is_iff_fn(arg(e, 0)); }
-inline expr mk_iff(expr const & e1, expr const & e2) { return mk_app(mk_iff_fn(), e1, e2); }
-inline expr Iff(expr const & e1, expr const & e2) { return mk_iff(e1, e2); }
-
-expr mk_and_fn();
-bool is_and_fn(expr const & e);
-inline bool is_and(expr const & e) { return is_app(e) && is_and_fn(arg(e, 0)); }
-inline expr mk_and(expr const & e1, expr const & e2) { return mk_app(mk_and_fn(), e1, e2); }
 inline expr And(expr const & e1, expr const & e2) { return mk_and(e1, e2); }
-
-expr mk_or_fn();
-bool is_or_fn(expr const & e);
-inline bool is_or(expr const & e) { return is_app(e) && is_or_fn(arg(e, 0)); }
-inline expr mk_or(expr const & e1, expr const & e2) { return mk_app(mk_or_fn(), e1, e2); }
 inline expr Or(expr const & e1, expr const & e2) { return mk_or(e1, e2); }
-
-expr mk_not_fn();
-bool is_not_fn(expr const & e);
-inline bool is_not(expr const & e) { return is_app(e) && is_not_fn(arg(e, 0)); }
-inline expr mk_not(expr const & e) { return mk_app(mk_not_fn(), e); }
 inline expr Not(expr const & e) { return mk_not(e); }
-
-expr mk_forall_fn();
-bool is_forall_fn(expr const & e);
-inline bool is_forall(expr const & e) { return is_app(e) && is_forall_fn(arg(e, 0)); }
-inline expr mk_forall(expr const & A, expr const & P) { return mk_app(mk_forall_fn(), A, P); }
-inline expr Forall(expr const & A, expr const & P) { return mk_forall(A, P); }
-
-expr mk_exists_fn();
-bool is_exists_fn(expr const & e);
-inline bool is_exists(expr const & e) { return is_app(e) && is_exists_fn(arg(e, 0)); }
-inline expr mk_exists(expr const & A, expr const & P) { return mk_app(mk_exists_fn(), A, P); }
 inline expr Exists(expr const & A, expr const & P) { return mk_exists(A, P); }
+// inline expr hEq(expr const & A, expr const & l, expr const & r) { return mk_eq(A, l, r); }
 
-expr mk_homo_eq_fn();
-bool is_homo_eq_fn(expr const & e);
-inline bool is_homo_eq(expr const & e) { return is_app(e) && is_homo_eq_fn(arg(e, 0)); }
-inline expr mk_homo_eq(expr const & A, expr const & l, expr const & r) { return mk_app(mk_homo_eq_fn(), A, l, r); }
-inline expr hEq(expr const & A, expr const & l, expr const & r) { return mk_homo_eq(A, l, r); }
-
-expr mk_mp_fn();
-inline expr MP(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_mp_fn(), a, b, H1, H2); }
-
-expr mk_discharge_fn();
-inline expr Discharge(expr const & a, expr const & b, expr const & H) { return mk_app(mk_discharge_fn(), a, b, H); }
-
-expr mk_case_fn();
-inline expr Case(expr const & P, expr const & H1, expr const & H2, expr const & a) { return mk_app(mk_case_fn(), P, H1, H2, a); }
-
-expr mk_refl_fn();
-inline expr Refl(expr const & A, expr const & a) { return mk_app(mk_refl_fn(), A, a); }
-
-expr mk_subst_fn();
-inline expr Subst(expr const & A, expr const & a, expr const & b, expr const & P, expr const & H1, expr const & H2) { return mk_app({mk_subst_fn(), A, a, b, P, H1, H2}); }
-
-expr mk_eta_fn();
-inline expr Eta(expr const & A, expr const & B, expr const & f) { return mk_app(mk_eta_fn(), A, B, f); }
-
-expr mk_imp_antisym_fn();
-inline expr ImpAntisym(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_imp_antisym_fn(), a, b, H1, H2); }
-
-expr mk_abst_fn();
-inline expr Abst(expr const & A, expr const & B, expr const & f, expr const & g, expr const & H) { return mk_app({mk_abst_fn(), A, B, f, g, H}); }
-
-expr mk_hsymm_fn();
-inline expr HSymm(expr const & A, expr const & B, expr const & a, expr const & b, expr const & H1) { return mk_app({mk_hsymm_fn(), A, B, a, b, H1}); }
-
-expr mk_htrans_fn();
-inline expr HTrans(expr const & A, expr const & B, expr const & C, expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2) {
-    return mk_app({mk_htrans_fn(), A, B, C, a, b, c, H1, H2});
-}
-
-expr mk_trivial();
-#define Trivial mk_trivial()
-
-expr mk_true_ne_false();
-#define TrueNeFalse mk_true_ne_false();
-
-expr mk_em_fn();
-inline expr EM(expr const & a) { return mk_app(mk_em_fn(), a); }
-
-expr mk_false_elim_fn();
-inline expr FalseElim(expr const & a, expr const & H) { return mk_app(mk_false_elim_fn(), a, H); }
-
-expr mk_absurd_fn();
-inline expr Absurd(expr const & a, expr const & H1, expr const & H2) { return mk_app(mk_absurd_fn(), a, H1, H2); }
-
-expr mk_double_neg_fn();
-inline expr DoubleNeg(expr const & a) { return mk_app(mk_double_neg_fn(), a); }
-
-expr mk_double_neg_elim_fn();
-inline expr DoubleNegElim(expr const & P, expr const & H) { return mk_app(mk_double_neg_elim_fn(), P, H); }
-
-expr mk_mt_fn();
-inline expr MT(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_mt_fn(), a, b, H1, H2); }
-
-expr mk_contrapos_fn();
-inline expr Contrapos(expr const & a, expr const & b, expr const & H) { return mk_app(mk_contrapos_fn(), a, b, H); }
-
-expr mk_false_imp_any_fn();
-inline expr FalseImpAny(expr const & a) { return mk_app(mk_false_imp_any_fn(), a); }
-
-expr mk_absurd_elim_fn();
-inline expr AbsurdElim(expr const & a, expr const & c, expr const & H1, expr const & H2) {
-    return mk_app(mk_absurd_elim_fn(), a, c, H1, H2);
-}
-
-expr mk_eq_mp_fn();
-inline expr EqMP(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_eq_mp_fn(), a, b, H1, H2); }
-
-expr mk_not_imp1_fn();
-inline expr NotImp1(expr const & a, expr const & b, expr const & H) { return mk_app(mk_not_imp1_fn(), a, b, H); }
-
-expr mk_not_imp2_fn();
-inline expr NotImp2(expr const & a, expr const & b, expr const & H) { return mk_app(mk_not_imp2_fn(), a, b, H); }
-
-expr mk_conj_fn();
-inline expr Conj(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_conj_fn(), a, b, H1, H2); }
-
-expr mk_conjunct1_fn();
-inline expr Conjunct1(expr const & a, expr const & b, expr const & H) { return mk_app(mk_conjunct1_fn(), a, b, H); }
-
-expr mk_conjunct2_fn();
-inline expr Conjunct2(expr const & a, expr const & b, expr const & H) { return mk_app(mk_conjunct2_fn(), a, b, H); }
-
-expr mk_disj1_fn();
-inline expr Disj1(expr const & a, expr const & H, expr const & b) { return mk_app(mk_disj1_fn(), a, H, b); }
-
-expr mk_disj2_fn();
-inline expr Disj2(expr const & b, expr const & a, expr const & H) { return mk_app(mk_disj2_fn(), b, a, H); }
-
-expr mk_disj_cases_fn();
-inline expr DisjCases(expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2, expr const & H3) { return mk_app({mk_disj_cases_fn(), a, b, c, H1, H2, H3}); }
-
-expr mk_refute_fn();
-inline expr Refute(expr const & a, expr const & H) { return mk_app({mk_refute_fn(), a, H}); }
-
-expr mk_symm_fn();
-inline expr Symm(expr const & A, expr const & a, expr const & b, expr const & H) { return mk_app(mk_symm_fn(), A, a, b, H); }
-
-expr mk_trans_fn();
-inline expr Trans(expr const & A, expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2) { return mk_app({mk_trans_fn(), A, a, b, c, H1, H2}); }
-
-expr mk_eqt_elim_fn();
-inline expr EqTElim(expr const & a, expr const & H) { return mk_app(mk_eqt_elim_fn(), a, H); }
-
-expr mk_eqt_intro_fn();
-inline expr EqTIntro(expr const & a, expr const & H) { return mk_app(mk_eqt_intro_fn(), a, H); }
-
-expr mk_congr1_fn();
-inline expr Congr1(expr const & A, expr const & B, expr const & f, expr const & g, expr const & a, expr const & H) { return mk_app({mk_congr1_fn(), A, B, f, g, a, H}); }
-
-expr mk_congr2_fn();
-inline expr Congr2(expr const & A, expr const & B, expr const & a, expr const & b, expr const & f, expr const & H) { return mk_app({mk_congr2_fn(), A, B, a, b, f, H}); }
-
-expr mk_congr_fn();
-inline expr Congr(expr const & A, expr const & B, expr const & f, expr const & g, expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app({mk_congr_fn(), A, B, f, g, a, b, H1, H2}); }
-
-expr mk_forall_elim_fn();
-inline expr ForallElim(expr const & A, expr const & P, expr const & H, expr const & a) { return mk_app(mk_forall_elim_fn(), A, P, H, a); }
-
-expr mk_forall_intro_fn();
-inline expr ForallIntro(expr const & A, expr const & P, expr const & H) { return mk_app(mk_forall_intro_fn(), A, P, H); }
-
-expr mk_exists_elim_fn();
-inline expr ExistsElim(expr const & A, expr const & P, expr const & B, expr const & H1, expr const & H2) { return mk_app({mk_exists_elim_fn(), A, P, B, H1, H2}); }
-
-expr mk_exists_intro_fn();
-inline expr ExistsIntro(expr const & A, expr const & P, expr const & a, expr const & H) { return mk_app(mk_exists_intro_fn(), A, P, a, H); }
-class io_state;
 void import_kernel(environment const & env, io_state const & ios);
 }

src/kernel/expr.cpp

@@ -142,13 +142,13 @@ expr mk_app(unsigned n, expr const * as) {
     to_app(r)->m_hash = hash_args(new_n, m_args);
     return r;
 }
-expr_eq::expr_eq(expr const & lhs, expr const & rhs):
-    expr_cell(expr_kind::Eq, ::lean::hash(lhs.hash(), rhs.hash()), lhs.has_metavar() || rhs.has_metavar()),
+expr_heq::expr_heq(expr const & lhs, expr const & rhs):
+    expr_cell(expr_kind::HEq, ::lean::hash(lhs.hash(), rhs.hash()), lhs.has_metavar() || rhs.has_metavar()),
     m_lhs(lhs),
     m_rhs(rhs) {
 }
-expr_eq::~expr_eq() {}
-void expr_eq::dealloc(buffer<expr_cell*> & todelete) {
+expr_heq::~expr_heq() {}
+void expr_heq::dealloc(buffer<expr_cell*> & todelete) {
     dec_ref(m_rhs, todelete);
     dec_ref(m_lhs, todelete);
     delete(this);
@@ -246,7 +246,7 @@ void expr_cell::dealloc() {
             case expr_kind::MetaVar:    delete static_cast<expr_metavar*>(it); break;
             case expr_kind::Type:       delete static_cast<expr_type*>(it); break;
             case expr_kind::Constant:   static_cast<expr_const*>(it)->dealloc(todo); break;
-            case expr_kind::Eq:         static_cast<expr_eq*>(it)->dealloc(todo); break;
+            case expr_kind::HEq:        static_cast<expr_heq*>(it)->dealloc(todo); break;
             case expr_kind::App:        static_cast<expr_app*>(it)->dealloc(todo); break;
             case expr_kind::Lambda:     static_cast<expr_lambda*>(it)->dealloc(todo); break;
             case expr_kind::Pi:         static_cast<expr_pi*>(it)->dealloc(todo); break;
@@ -279,10 +279,10 @@ bool is_arrow(expr const & t) {
     }
 }
 
-bool is_eq(expr const & e, expr & lhs, expr & rhs) {
-    if (is_eq(e)) {
-        lhs = eq_lhs(e);
-        rhs = eq_rhs(e);
+bool is_heq(expr const & e, expr & lhs, expr & rhs) {
+    if (is_heq(e)) {
+        lhs = heq_lhs(e);
+        rhs = heq_rhs(e);
         return true;
     } else {
         return false;
@@ -296,7 +296,7 @@ expr copy(expr const & a) {
     case expr_kind::Type:     return mk_type(ty_level(a));
     case expr_kind::Value:    return mk_value(static_cast<expr_value*>(a.raw())->m_val);
     case expr_kind::App:      return mk_app(num_args(a), begin_args(a));
-    case expr_kind::Eq:       return mk_eq(eq_lhs(a), eq_rhs(a));
+    case expr_kind::HEq:      return mk_heq(heq_lhs(a), heq_rhs(a));
     case expr_kind::Lambda:   return mk_lambda(abst_name(a), abst_domain(a), abst_body(a));
     case expr_kind::Pi:       return mk_pi(abst_name(a), abst_domain(a), abst_body(a));
     case expr_kind::Let:      return mk_let(let_name(a), let_type(a), let_value(a), let_body(a));
@@ -341,7 +341,7 @@ constexpr char g_first_app_size_kind = 32;
 constexpr char g_small_app_num_args  = 32;
 constexpr bool is_small(expr_kind k) { return 0 <= static_cast<char>(k) && static_cast<char>(k) < g_first_app_size_kind; }
 static_assert(is_small(expr_kind::Var) && is_small(expr_kind::Constant) && is_small(expr_kind::Value) && is_small(expr_kind::App) &&
-              is_small(expr_kind::Lambda) && is_small(expr_kind::Pi) && is_small(expr_kind::Type) && is_small(expr_kind::Eq) &&
+              is_small(expr_kind::Lambda) && is_small(expr_kind::Pi) && is_small(expr_kind::Type) && is_small(expr_kind::HEq) &&
               is_small(expr_kind::Let) && is_small(expr_kind::MetaVar), "expr_kind is too big");
 
 class expr_serializer : public object_serializer<expr, expr_hash_alloc, expr_eqp> {
@@ -384,7 +384,7 @@ class expr_serializer : public object_serializer<expr, expr_hash_alloc, expr_eqp
                     for (unsigned i = 0; i < num_args(a); i++)
                         write_core(arg(a, i));
                     break;
-                case expr_kind::Eq:        write_core(eq_lhs(a)); write_core(eq_rhs(a)); break;
+                case expr_kind::HEq:       write_core(heq_lhs(a)); write_core(heq_rhs(a)); break;
                 case expr_kind::Lambda:
                 case expr_kind::Pi:        s << abst_name(a); write_core(abst_domain(a)); write_core(abst_body(a)); break;
                 case expr_kind::Let:       s << let_name(a); write_core(let_type(a)); write_core(let_value(a)); write_core(let_body(a)); break;
@@ -441,9 +441,9 @@ public:
                         args.push_back(read());
                     return mk_app(args);
                 }
-                case expr_kind::Eq: {
+                case expr_kind::HEq: {
                     expr lhs = read();
-                    return mk_eq(lhs, read());
+                    return mk_heq(lhs, read());
                 }
                 case expr_kind::Lambda: {
                     name n = read_name(d);

src/kernel/expr.h

@@ -52,7 +52,7 @@ The main API is divided in the following sections
 - Miscellaneous
 ======================================= */
 class expr;
-enum class expr_kind { Var, Constant, Value, App, Lambda, Pi, Type, Eq, Let, MetaVar };
+enum class expr_kind { Var, Constant, Value, App, Lambda, Pi, Type, HEq, Let, MetaVar };
 class local_entry;
 /**
    \brief A metavariable local context is just a list of local_entries.
@@ -142,7 +142,7 @@ public:
     friend expr mk_constant(name const & n, optional<expr> const & t);
     friend expr mk_value(value & v);
     friend expr mk_app(unsigned num_args, expr const * args);
-    friend expr mk_eq(expr const & l, expr const & r);
+    friend expr mk_heq(expr const & l, expr const & r);
     friend expr mk_lambda(name const & n, expr const & t, expr const & e);
     friend expr mk_pi(name const & n, expr const & t, expr const & e);
     friend expr mk_type(level const & l);
@@ -215,14 +215,14 @@ public:
     expr const * end_args() const            { return m_args + m_num_args; }
 };
 /** \brief Heterogeneous equality */
-class expr_eq : public expr_cell {
+class expr_heq : public expr_cell {
     expr m_lhs;
     expr m_rhs;
     friend class expr_cell;
     void dealloc(buffer<expr_cell*> & todelete);
 public:
-    expr_eq(expr const & lhs, expr const & rhs);
-    ~expr_eq();
+    expr_heq(expr const & lhs, expr const & rhs);
+    ~expr_heq();
     expr const & get_lhs() const { return m_lhs; }
     expr const & get_rhs() const { return m_rhs; }
 };
@@ -392,7 +392,7 @@ inline bool is_var(expr_cell * e)         { return e->kind() == expr_kind::Var;
 inline bool is_constant(expr_cell * e)    { return e->kind() == expr_kind::Constant; }
 inline bool is_value(expr_cell * e)       { return e->kind() == expr_kind::Value; }
 inline bool is_app(expr_cell * e)         { return e->kind() == expr_kind::App; }
-inline bool is_eq(expr_cell * e)          { return e->kind() == expr_kind::Eq; }
+inline bool is_heq(expr_cell * e)         { return e->kind() == expr_kind::HEq; }
 inline bool is_lambda(expr_cell * e)      { return e->kind() == expr_kind::Lambda; }
 inline bool is_pi(expr_cell * e)          { return e->kind() == expr_kind::Pi; }
 inline bool is_type(expr_cell * e)        { return e->kind() == expr_kind::Type; }
@@ -404,7 +404,7 @@ inline bool is_var(expr const & e)         { return e.kind() == expr_kind::Var;
 inline bool is_constant(expr const & e)    { return e.kind() == expr_kind::Constant; }
 inline bool is_value(expr const & e)       { return e.kind() == expr_kind::Value; }
 inline bool is_app(expr const & e)         { return e.kind() == expr_kind::App; }
-inline bool is_eq(expr const & e)          { return e.kind() == expr_kind::Eq; }
+inline bool is_heq(expr const & e)         { return e.kind() == expr_kind::HEq; }
 inline bool is_lambda(expr const & e)      { return e.kind() == expr_kind::Lambda; }
 inline bool is_pi(expr const & e)          { return e.kind() == expr_kind::Pi; }
        bool is_arrow(expr const & e);
@@ -431,8 +431,8 @@ inline expr mk_app(expr const & e1, expr const & e2) { return mk_app({e1, e2});
 inline expr mk_app(expr const & e1, expr const & e2, expr const & e3) { return mk_app({e1, e2, e3}); }
 inline expr mk_app(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({e1, e2, e3, e4}); }
 inline expr mk_app(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5) { return mk_app({e1, e2, e3, e4, e5}); }
-inline expr mk_eq(expr const & l, expr const & r) { return expr(new expr_eq(l, r)); }
-inline expr Eq(expr const & l, expr const & r) { return mk_eq(l, r); }
+inline expr mk_heq(expr const & l, expr const & r) { return expr(new expr_heq(l, r)); }
+inline expr HEq(expr const & l, expr const & r) { return mk_heq(l, r); }
 inline expr mk_lambda(name const & n, expr const & t, expr const & e) { return expr(new expr_lambda(n, t, e)); }
 inline expr mk_pi(name const & n, expr const & t, expr const & e) { return expr(new expr_pi(n, t, e)); }
 inline bool is_default_arrow_var_name(name const & n) { return n == "a"; }
@@ -464,7 +464,7 @@ inline expr expr::operator()(expr const & a1, expr const & a2, expr const & a3,
 inline expr_var *         to_var(expr_cell * e)         { lean_assert(is_var(e));         return static_cast<expr_var*>(e); }
 inline expr_const *       to_constant(expr_cell * e)    { lean_assert(is_constant(e));    return static_cast<expr_const*>(e); }
 inline expr_app *         to_app(expr_cell * e)         { lean_assert(is_app(e));         return static_cast<expr_app*>(e); }
-inline expr_eq *          to_eq(expr_cell * e)          { lean_assert(is_eq(e));          return static_cast<expr_eq*>(e); }
+inline expr_heq *         to_heq(expr_cell * e)         { lean_assert(is_heq(e));         return static_cast<expr_heq*>(e); }
 inline expr_abstraction * to_abstraction(expr_cell * e) { lean_assert(is_abstraction(e)); return static_cast<expr_abstraction*>(e); }
 inline expr_lambda *      to_lambda(expr_cell * e)      { lean_assert(is_lambda(e));      return static_cast<expr_lambda*>(e); }
 inline expr_pi *          to_pi(expr_cell * e)          { lean_assert(is_pi(e));          return static_cast<expr_pi*>(e); }
@@ -475,7 +475,7 @@ inline expr_metavar *     to_metavar(expr_cell * e)     { lean_assert(is_metavar
 inline expr_var *         to_var(expr const & e)         { return to_var(e.raw()); }
 inline expr_const *       to_constant(expr const & e)    { return to_constant(e.raw()); }
 inline expr_app *         to_app(expr const & e)         { return to_app(e.raw()); }
-inline expr_eq *          to_eq(expr const & e)          { return to_eq(e.raw()); }
+inline expr_heq *         to_heq(expr const & e)         { return to_heq(e.raw()); }
 inline expr_abstraction * to_abstraction(expr const & e) { return to_abstraction(e.raw()); }
 inline expr_lambda *      to_lambda(expr const & e)      { return to_lambda(e.raw()); }
 inline expr_pi *          to_pi(expr const & e)          { return to_pi(e.raw()); }
@@ -497,8 +497,8 @@ inline optional<expr> const &  const_type(expr_cell * e) { return to_constant(e)
 inline value const & to_value(expr_cell * e)             { lean_assert(is_value(e)); return static_cast<expr_value*>(e)->get_value(); }
 inline unsigned      num_args(expr_cell * e)             { return to_app(e)->get_num_args(); }
 inline expr const &  arg(expr_cell * e, unsigned idx)    { return to_app(e)->get_arg(idx); }
-inline expr const &  eq_lhs(expr_cell * e)               { return to_eq(e)->get_lhs(); }
-inline expr const &  eq_rhs(expr_cell * e)               { return to_eq(e)->get_rhs(); }
+inline expr const &  heq_lhs(expr_cell * e)              { return to_heq(e)->get_lhs(); }
+inline expr const &  heq_rhs(expr_cell * e)              { return to_heq(e)->get_rhs(); }
 inline name const &  abst_name(expr_cell * e)            { return to_abstraction(e)->get_name(); }
 inline expr const &  abst_domain(expr_cell * e)          { return to_abstraction(e)->get_domain(); }
 inline expr const &  abst_body(expr_cell * e)            { return to_abstraction(e)->get_body(); }
@@ -530,8 +530,8 @@ inline unsigned      num_args(expr const & e)             { return to_app(e)->ge
 inline expr const &  arg(expr const & e, unsigned idx)    { return to_app(e)->get_arg(idx); }
 inline expr const *  begin_args(expr const & e)           { return to_app(e)->begin_args(); }
 inline expr const *  end_args(expr const & e)             { return to_app(e)->end_args(); }
-inline expr const &  eq_lhs(expr const & e)               { return to_eq(e)->get_lhs(); }
-inline expr const &  eq_rhs(expr const & e)               { return to_eq(e)->get_rhs(); }
+inline expr const &  heq_lhs(expr const & e)              { return to_heq(e)->get_lhs(); }
+inline expr const &  heq_rhs(expr const & e)              { return to_heq(e)->get_rhs(); }
 inline name const &  abst_name(expr const & e)            { return to_abstraction(e)->get_name(); }
 inline expr const &  abst_domain(expr const & e)          { return to_abstraction(e)->get_domain(); }
 inline expr const &  abst_body(expr const & e)            { return to_abstraction(e)->get_body(); }
@@ -545,7 +545,7 @@ inline local_context const & metavar_lctx(expr const & e) { return to_metavar(e)
 
 inline bool has_metavar(expr const & e) { return e.has_metavar(); }
 
-bool is_eq(expr const & e, expr & lhs, expr & rhs);
+bool is_heq(expr const & e, expr & lhs, expr & rhs);
 // =======================================
 
 // =======================================
@@ -653,15 +653,15 @@ template<typename F> expr update_let(expr const & e, F f) {
     else
         return e;
 }
-template<typename F> expr update_eq(expr const & e, F f) {
+template<typename F> expr update_heq(expr const & e, F f) {
     static_assert(std::is_same<typename std::result_of<F(expr const &, expr const &)>::type,
                   std::pair<expr, expr>>::value,
-                  "update_eq: return type of f is not pair<expr, expr>");
-    expr const & old_l = eq_lhs(e);
-    expr const & old_r = eq_rhs(e);
+                  "update_heq: return type of f is not pair<expr, expr>");
+    expr const & old_l = heq_lhs(e);
+    expr const & old_r = heq_rhs(e);
     std::pair<expr, expr> p = f(old_l, old_r);
     if (!is_eqp(p.first, old_l) || !is_eqp(p.second, old_r))
-        return mk_eq(p.first, p.second);
+        return mk_heq(p.first, p.second);
     else
         return e;
 }

src/kernel/expr_eq.h

@@ -64,7 +64,7 @@ class expr_eq_fn {
                 if (!apply(arg(a, i), arg(b, i)))
                     return false;
             return true;
-        case expr_kind::Eq:       return apply(eq_lhs(a), eq_lhs(b)) && apply(eq_rhs(a), eq_rhs(b));
+        case expr_kind::HEq:      return apply(heq_lhs(a), heq_lhs(b)) && apply(heq_rhs(a), heq_rhs(b));
         case expr_kind::Lambda:   // Remark: we ignore get_abs_name because we want alpha-equivalence
         case expr_kind::Pi:       return apply(abst_domain(a), abst_domain(b)) && apply(abst_body(a), abst_body(b));
         case expr_kind::Type:     return ty_level(a) == ty_level(b);

src/kernel/for_each_fn.h

@@ -85,9 +85,9 @@ class for_each_fn {
                 }
                 goto begin_loop;
             }
-            case expr_kind::Eq:
-                todo.emplace_back(eq_rhs(e), offset);
-                todo.emplace_back(eq_lhs(e), offset);
+            case expr_kind::HEq:
+                todo.emplace_back(heq_rhs(e), offset);
+                todo.emplace_back(heq_lhs(e), offset);
                 goto begin_loop;
             case expr_kind::Lambda:
             case expr_kind::Pi:

src/kernel/free_vars.cpp

@@ -38,7 +38,7 @@ protected:
             return true;
         case expr_kind::Var:
             return var_idx(e) >= offset;
-        case expr_kind::App: case expr_kind::Eq: case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::Let:
+        case expr_kind::App: case expr_kind::HEq: case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::Let:
             break;
         }
 
@@ -76,8 +76,8 @@ protected:
         case expr_kind::App:
             result = std::any_of(begin_args(e), end_args(e), [=](expr const & arg){ return apply(arg, offset); });
             break;
-        case expr_kind::Eq:
-            result = apply(eq_lhs(e), offset) || apply(eq_rhs(e), offset);
+        case expr_kind::HEq:
+            result = apply(heq_lhs(e), offset) || apply(heq_rhs(e), offset);
             break;
         case expr_kind::Lambda:
         case expr_kind::Pi:
@@ -166,7 +166,7 @@ class free_var_range_fn {
             return 0;
         case expr_kind::Var:
             return var_idx(e) + 1;
-        case expr_kind::MetaVar: case expr_kind::App: case expr_kind::Eq:
+        case expr_kind::MetaVar: case expr_kind::App: case expr_kind::HEq:
         case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::Let:
             break;
         }
@@ -199,8 +199,8 @@ class free_var_range_fn {
             for (auto const & c : args(e))
                 result = std::max(result, apply(c));
             break;
-        case expr_kind::Eq:
-            result = std::max(apply(eq_lhs(e)), apply(eq_rhs(e)));
+        case expr_kind::HEq:
+            result = std::max(apply(heq_lhs(e)), apply(heq_rhs(e)));
             break;
         case expr_kind::Lambda:
         case expr_kind::Pi:
@@ -258,7 +258,7 @@ protected:
                 return true; // assume that any free variable can occur in the metavariable
         case expr_kind::Var:
             return var_idx(e) >= offset + m_low && var_idx(e) < offset + m_high;
-        case expr_kind::App: case expr_kind::Eq: case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::Let:
+        case expr_kind::App: case expr_kind::HEq: case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::Let:
             break;
         }
 
@@ -297,8 +297,8 @@ protected:
         case expr_kind::App:
             result = std::any_of(begin_args(e), end_args(e), [=](expr const & arg){ return apply(arg, offset); });
             break;
-        case expr_kind::Eq:
-            result = apply(eq_lhs(e), offset) || apply(eq_rhs(e), offset);
+        case expr_kind::HEq:
+            result = apply(heq_lhs(e), offset) || apply(heq_rhs(e), offset);
             break;
         case expr_kind::Lambda:
         case expr_kind::Pi:

src/kernel/kernel_decls.cpp

@@ -0,0 +1,96 @@
+/*
+Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+*/
+// Automatically generated file, DO NOT EDIT
+#include "kernel/environment.h"
+#include "kernel/decl_macros.h"
+namespace lean {
+MK_CONSTANT(Bool, name("Bool"));
+MK_CONSTANT(TypeU, name("TypeU"));
+MK_CONSTANT(not_fn, name("not"));
+MK_CONSTANT(or_fn, name("or"));
+MK_CONSTANT(and_fn, name("and"));
+MK_CONSTANT(implies_fn, name("implies"));
+MK_CONSTANT(exists_fn, name("exists"));
+MK_CONSTANT(eq_fn, name("eq"));
+MK_CONSTANT(neq_fn, name("neq"));
+MK_CONSTANT(em_fn, name("em"));
+MK_CONSTANT(case_fn, name("case"));
+MK_CONSTANT(refl_fn, name("refl"));
+MK_CONSTANT(subst_fn, name("subst"));
+MK_CONSTANT(funext_fn, name("funext"));
+MK_CONSTANT(allext_fn, name("allext"));
+MK_CONSTANT(substp_fn, name("substp"));
+MK_CONSTANT(eta_fn, name("eta"));
+MK_CONSTANT(trivial, name("trivial"));
+MK_CONSTANT(absurd_fn, name("absurd"));
+MK_CONSTANT(eqmp_fn, name("eqmp"));
+MK_CONSTANT(boolcomplete_fn, name("boolcomplete"));
+MK_CONSTANT(false_elim_fn, name("false_elim"));
+MK_CONSTANT(imp_trans_fn, name("imp_trans"));
+MK_CONSTANT(imp_eq_trans_fn, name("imp_eq_trans"));
+MK_CONSTANT(eq_imp_trans_fn, name("eq_imp_trans"));
+MK_CONSTANT(not_not_eq_fn, name("not_not_eq"));
+MK_CONSTANT(not_not_elim_fn, name("not_not_elim"));
+MK_CONSTANT(mt_fn, name("mt"));
+MK_CONSTANT(contrapos_fn, name("contrapos"));
+MK_CONSTANT(absurd_elim_fn, name("absurd_elim"));
+MK_CONSTANT(not_imp_eliml_fn, name("not_imp_eliml"));
+MK_CONSTANT(not_imp_elimr_fn, name("not_imp_elimr"));
+MK_CONSTANT(resolve1_fn, name("resolve1"));
+MK_CONSTANT(and_intro_fn, name("and_intro"));
+MK_CONSTANT(and_eliml_fn, name("and_eliml"));
+MK_CONSTANT(and_elimr_fn, name("and_elimr"));
+MK_CONSTANT(or_introl_fn, name("or_introl"));
+MK_CONSTANT(or_intror_fn, name("or_intror"));
+MK_CONSTANT(or_elim_fn, name("or_elim"));
+MK_CONSTANT(refute_fn, name("refute"));
+MK_CONSTANT(symm_fn, name("symm"));
+MK_CONSTANT(trans_fn, name("trans"));
+MK_CONSTANT(ne_symm_fn, name("ne_symm"));
+MK_CONSTANT(eq_ne_trans_fn, name("eq_ne_trans"));
+MK_CONSTANT(ne_eq_trans_fn, name("ne_eq_trans"));
+MK_CONSTANT(eqt_elim_fn, name("eqt_elim"));
+MK_CONSTANT(congr1_fn, name("congr1"));
+MK_CONSTANT(congr2_fn, name("congr2"));
+MK_CONSTANT(congr_fn, name("congr"));
+MK_CONSTANT(exists_elim_fn, name("exists_elim"));
+MK_CONSTANT(exists_intro_fn, name("exists_intro"));
+MK_CONSTANT(boolext_fn, name("boolext"));
+MK_CONSTANT(iff_intro_fn, name("iff_intro"));
+MK_CONSTANT(eqt_intro_fn, name("eqt_intro"));
+MK_CONSTANT(or_comm_fn, name("or_comm"));
+MK_CONSTANT(or_assoc_fn, name("or_assoc"));
+MK_CONSTANT(or_id_fn, name("or_id"));
+MK_CONSTANT(or_falsel_fn, name("or_falsel"));
+MK_CONSTANT(or_falser_fn, name("or_falser"));
+MK_CONSTANT(or_truel_fn, name("or_truel"));
+MK_CONSTANT(or_truer_fn, name("or_truer"));
+MK_CONSTANT(or_tauto_fn, name("or_tauto"));
+MK_CONSTANT(and_comm_fn, name("and_comm"));
+MK_CONSTANT(and_id_fn, name("and_id"));
+MK_CONSTANT(and_assoc_fn, name("and_assoc"));
+MK_CONSTANT(and_truer_fn, name("and_truer"));
+MK_CONSTANT(and_truel_fn, name("and_truel"));
+MK_CONSTANT(and_falsel_fn, name("and_falsel"));
+MK_CONSTANT(and_falser_fn, name("and_falser"));
+MK_CONSTANT(and_absurd_fn, name("and_absurd"));
+MK_CONSTANT(not_and_fn, name("not_and"));
+MK_CONSTANT(not_and_elim_fn, name("not_and_elim"));
+MK_CONSTANT(not_or_fn, name("not_or"));
+MK_CONSTANT(not_or_elim_fn, name("not_or_elim"));
+MK_CONSTANT(not_iff_fn, name("not_iff"));
+MK_CONSTANT(not_iff_elim_fn, name("not_iff_elim"));
+MK_CONSTANT(not_implies_fn, name("not_implies"));
+MK_CONSTANT(not_implies_elim_fn, name("not_implies_elim"));
+MK_CONSTANT(not_congr_fn, name("not_congr"));
+MK_CONSTANT(eq_exists_intro_fn, name("eq_exists_intro"));
+MK_CONSTANT(not_forall_fn, name("not_forall"));
+MK_CONSTANT(not_forall_elim_fn, name("not_forall_elim"));
+MK_CONSTANT(not_exists_fn, name("not_exists"));
+MK_CONSTANT(not_exists_elim_fn, name("not_exists_elim"));
+MK_CONSTANT(exists_unfold1_fn, name("exists_unfold1"));
+MK_CONSTANT(exists_unfold2_fn, name("exists_unfold2"));
+MK_CONSTANT(exists_unfold_fn, name("exists_unfold"));
+}

src/kernel/kernel_decls.h

@@ -0,0 +1,273 @@
+/*
+Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+*/
+// Automatically generated file, DO NOT EDIT
+#include "kernel/expr.h"
+namespace lean {
+expr mk_Bool();
+bool is_Bool(expr const & e);
+expr mk_TypeU();
+bool is_TypeU(expr const & e);
+expr mk_not_fn();
+bool is_not_fn(expr const & e);
+inline bool is_not(expr const & e) { return is_app(e) && is_not_fn(arg(e, 0)); }
+inline expr mk_not(expr const & e1) { return mk_app({mk_not_fn(), e1}); }
+expr mk_or_fn();
+bool is_or_fn(expr const & e);
+inline bool is_or(expr const & e) { return is_app(e) && is_or_fn(arg(e, 0)); }
+inline expr mk_or(expr const & e1, expr const & e2) { return mk_app({mk_or_fn(), e1, e2}); }
+expr mk_and_fn();
+bool is_and_fn(expr const & e);
+inline bool is_and(expr const & e) { return is_app(e) && is_and_fn(arg(e, 0)); }
+inline expr mk_and(expr const & e1, expr const & e2) { return mk_app({mk_and_fn(), e1, e2}); }
+expr mk_implies_fn();
+bool is_implies_fn(expr const & e);
+inline bool is_implies(expr const & e) { return is_app(e) && is_implies_fn(arg(e, 0)); }
+inline expr mk_implies(expr const & e1, expr const & e2) { return mk_app({mk_implies_fn(), e1, e2}); }
+expr mk_exists_fn();
+bool is_exists_fn(expr const & e);
+inline bool is_exists(expr const & e) { return is_app(e) && is_exists_fn(arg(e, 0)); }
+inline expr mk_exists(expr const & e1, expr const & e2) { return mk_app({mk_exists_fn(), e1, e2}); }
+expr mk_eq_fn();
+bool is_eq_fn(expr const & e);
+inline bool is_eq(expr const & e) { return is_app(e) && is_eq_fn(arg(e, 0)); }
+inline expr mk_eq(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_eq_fn(), e1, e2, e3}); }
+expr mk_neq_fn();
+bool is_neq_fn(expr const & e);
+inline bool is_neq(expr const & e) { return is_app(e) && is_neq_fn(arg(e, 0)); }
+inline expr mk_neq(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_neq_fn(), e1, e2, e3}); }
+expr mk_em_fn();
+bool is_em_fn(expr const & e);
+inline expr mk_em_th(expr const & e1) { return mk_app({mk_em_fn(), e1}); }
+expr mk_case_fn();
+bool is_case_fn(expr const & e);
+inline expr mk_case_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_case_fn(), e1, e2, e3, e4}); }
+expr mk_refl_fn();
+bool is_refl_fn(expr const & e);
+inline expr mk_refl_th(expr const & e1, expr const & e2) { return mk_app({mk_refl_fn(), e1, e2}); }
+expr mk_subst_fn();
+bool is_subst_fn(expr const & e);
+inline expr mk_subst_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_subst_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_funext_fn();
+bool is_funext_fn(expr const & e);
+inline expr mk_funext_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5) { return mk_app({mk_funext_fn(), e1, e2, e3, e4, e5}); }
+expr mk_allext_fn();
+bool is_allext_fn(expr const & e);
+inline expr mk_allext_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_allext_fn(), e1, e2, e3, e4}); }
+expr mk_substp_fn();
+bool is_substp_fn(expr const & e);
+inline expr mk_substp_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_substp_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_eta_fn();
+bool is_eta_fn(expr const & e);
+inline expr mk_eta_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_eta_fn(), e1, e2, e3}); }
+expr mk_trivial();
+bool is_trivial(expr const & e);
+expr mk_absurd_fn();
+bool is_absurd_fn(expr const & e);
+inline expr mk_absurd_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_absurd_fn(), e1, e2, e3}); }
+expr mk_eqmp_fn();
+bool is_eqmp_fn(expr const & e);
+inline expr mk_eqmp_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_eqmp_fn(), e1, e2, e3, e4}); }
+expr mk_boolcomplete_fn();
+bool is_boolcomplete_fn(expr const & e);
+inline expr mk_boolcomplete_th(expr const & e1) { return mk_app({mk_boolcomplete_fn(), e1}); }
+expr mk_false_elim_fn();
+bool is_false_elim_fn(expr const & e);
+inline expr mk_false_elim_th(expr const & e1, expr const & e2) { return mk_app({mk_false_elim_fn(), e1, e2}); }
+expr mk_imp_trans_fn();
+bool is_imp_trans_fn(expr const & e);
+inline expr mk_imp_trans_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_imp_trans_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_imp_eq_trans_fn();
+bool is_imp_eq_trans_fn(expr const & e);
+inline expr mk_imp_eq_trans_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_imp_eq_trans_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_eq_imp_trans_fn();
+bool is_eq_imp_trans_fn(expr const & e);
+inline expr mk_eq_imp_trans_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_eq_imp_trans_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_not_not_eq_fn();
+bool is_not_not_eq_fn(expr const & e);
+inline expr mk_not_not_eq_th(expr const & e1) { return mk_app({mk_not_not_eq_fn(), e1}); }
+expr mk_not_not_elim_fn();
+bool is_not_not_elim_fn(expr const & e);
+inline expr mk_not_not_elim_th(expr const & e1, expr const & e2) { return mk_app({mk_not_not_elim_fn(), e1, e2}); }
+expr mk_mt_fn();
+bool is_mt_fn(expr const & e);
+inline expr mk_mt_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_mt_fn(), e1, e2, e3, e4}); }
+expr mk_contrapos_fn();
+bool is_contrapos_fn(expr const & e);
+inline expr mk_contrapos_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_contrapos_fn(), e1, e2, e3, e4}); }
+expr mk_absurd_elim_fn();
+bool is_absurd_elim_fn(expr const & e);
+inline expr mk_absurd_elim_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_absurd_elim_fn(), e1, e2, e3, e4}); }
+expr mk_not_imp_eliml_fn();
+bool is_not_imp_eliml_fn(expr const & e);
+inline expr mk_not_imp_eliml_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_not_imp_eliml_fn(), e1, e2, e3}); }
+expr mk_not_imp_elimr_fn();
+bool is_not_imp_elimr_fn(expr const & e);
+inline expr mk_not_imp_elimr_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_not_imp_elimr_fn(), e1, e2, e3}); }
+expr mk_resolve1_fn();
+bool is_resolve1_fn(expr const & e);
+inline expr mk_resolve1_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_resolve1_fn(), e1, e2, e3, e4}); }
+expr mk_and_intro_fn();
+bool is_and_intro_fn(expr const & e);
+inline expr mk_and_intro_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_and_intro_fn(), e1, e2, e3, e4}); }
+expr mk_and_eliml_fn();
+bool is_and_eliml_fn(expr const & e);
+inline expr mk_and_eliml_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_and_eliml_fn(), e1, e2, e3}); }
+expr mk_and_elimr_fn();
+bool is_and_elimr_fn(expr const & e);
+inline expr mk_and_elimr_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_and_elimr_fn(), e1, e2, e3}); }
+expr mk_or_introl_fn();
+bool is_or_introl_fn(expr const & e);
+inline expr mk_or_introl_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_or_introl_fn(), e1, e2, e3}); }
+expr mk_or_intror_fn();
+bool is_or_intror_fn(expr const & e);
+inline expr mk_or_intror_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_or_intror_fn(), e1, e2, e3}); }
+expr mk_or_elim_fn();
+bool is_or_elim_fn(expr const & e);
+inline expr mk_or_elim_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_or_elim_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_refute_fn();
+bool is_refute_fn(expr const & e);
+inline expr mk_refute_th(expr const & e1, expr const & e2) { return mk_app({mk_refute_fn(), e1, e2}); }
+expr mk_symm_fn();
+bool is_symm_fn(expr const & e);
+inline expr mk_symm_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_symm_fn(), e1, e2, e3, e4}); }
+expr mk_trans_fn();
+bool is_trans_fn(expr const & e);
+inline expr mk_trans_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_trans_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_ne_symm_fn();
+bool is_ne_symm_fn(expr const & e);
+inline expr mk_ne_symm_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_ne_symm_fn(), e1, e2, e3, e4}); }
+expr mk_eq_ne_trans_fn();
+bool is_eq_ne_trans_fn(expr const & e);
+inline expr mk_eq_ne_trans_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_eq_ne_trans_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_ne_eq_trans_fn();
+bool is_ne_eq_trans_fn(expr const & e);
+inline expr mk_ne_eq_trans_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_ne_eq_trans_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_eqt_elim_fn();
+bool is_eqt_elim_fn(expr const & e);
+inline expr mk_eqt_elim_th(expr const & e1, expr const & e2) { return mk_app({mk_eqt_elim_fn(), e1, e2}); }
+expr mk_congr1_fn();
+bool is_congr1_fn(expr const & e);
+inline expr mk_congr1_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_congr1_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_congr2_fn();
+bool is_congr2_fn(expr const & e);
+inline expr mk_congr2_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_congr2_fn(), e1, e2, e3, e4, e5, e6}); }
+expr mk_congr_fn();
+bool is_congr_fn(expr const & e);
+inline expr mk_congr_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6, expr const & e7, expr const & e8) { return mk_app({mk_congr_fn(), e1, e2, e3, e4, e5, e6, e7, e8}); }
+expr mk_exists_elim_fn();
+bool is_exists_elim_fn(expr const & e);
+inline expr mk_exists_elim_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5) { return mk_app({mk_exists_elim_fn(), e1, e2, e3, e4, e5}); }
+expr mk_exists_intro_fn();
+bool is_exists_intro_fn(expr const & e);
+inline expr mk_exists_intro_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_exists_intro_fn(), e1, e2, e3, e4}); }
+expr mk_boolext_fn();
+bool is_boolext_fn(expr const & e);
+inline expr mk_boolext_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_boolext_fn(), e1, e2, e3, e4}); }
+expr mk_iff_intro_fn();
+bool is_iff_intro_fn(expr const & e);
+inline expr mk_iff_intro_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_iff_intro_fn(), e1, e2, e3, e4}); }
+expr mk_eqt_intro_fn();
+bool is_eqt_intro_fn(expr const & e);
+inline expr mk_eqt_intro_th(expr const & e1, expr const & e2) { return mk_app({mk_eqt_intro_fn(), e1, e2}); }
+expr mk_or_comm_fn();
+bool is_or_comm_fn(expr const & e);
+inline expr mk_or_comm_th(expr const & e1, expr const & e2) { return mk_app({mk_or_comm_fn(), e1, e2}); }
+expr mk_or_assoc_fn();
+bool is_or_assoc_fn(expr const & e);
+inline expr mk_or_assoc_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_or_assoc_fn(), e1, e2, e3}); }
+expr mk_or_id_fn();
+bool is_or_id_fn(expr const & e);
+inline expr mk_or_id_th(expr const & e1) { return mk_app({mk_or_id_fn(), e1}); }
+expr mk_or_falsel_fn();
+bool is_or_falsel_fn(expr const & e);
+inline expr mk_or_falsel_th(expr const & e1) { return mk_app({mk_or_falsel_fn(), e1}); }
+expr mk_or_falser_fn();
+bool is_or_falser_fn(expr const & e);
+inline expr mk_or_falser_th(expr const & e1) { return mk_app({mk_or_falser_fn(), e1}); }
+expr mk_or_truel_fn();
+bool is_or_truel_fn(expr const & e);
+inline expr mk_or_truel_th(expr const & e1) { return mk_app({mk_or_truel_fn(), e1}); }
+expr mk_or_truer_fn();
+bool is_or_truer_fn(expr const & e);
+inline expr mk_or_truer_th(expr const & e1) { return mk_app({mk_or_truer_fn(), e1}); }
+expr mk_or_tauto_fn();
+bool is_or_tauto_fn(expr const & e);
+inline expr mk_or_tauto_th(expr const & e1) { return mk_app({mk_or_tauto_fn(), e1}); }
+expr mk_and_comm_fn();
+bool is_and_comm_fn(expr const & e);
+inline expr mk_and_comm_th(expr const & e1, expr const & e2) { return mk_app({mk_and_comm_fn(), e1, e2}); }
+expr mk_and_id_fn();
+bool is_and_id_fn(expr const & e);
+inline expr mk_and_id_th(expr const & e1) { return mk_app({mk_and_id_fn(), e1}); }
+expr mk_and_assoc_fn();
+bool is_and_assoc_fn(expr const & e);
+inline expr mk_and_assoc_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_and_assoc_fn(), e1, e2, e3}); }
+expr mk_and_truer_fn();
+bool is_and_truer_fn(expr const & e);
+inline expr mk_and_truer_th(expr const & e1) { return mk_app({mk_and_truer_fn(), e1}); }
+expr mk_and_truel_fn();
+bool is_and_truel_fn(expr const & e);
+inline expr mk_and_truel_th(expr const & e1) { return mk_app({mk_and_truel_fn(), e1}); }
+expr mk_and_falsel_fn();
+bool is_and_falsel_fn(expr const & e);
+inline expr mk_and_falsel_th(expr const & e1) { return mk_app({mk_and_falsel_fn(), e1}); }
+expr mk_and_falser_fn();
+bool is_and_falser_fn(expr const & e);
+inline expr mk_and_falser_th(expr const & e1) { return mk_app({mk_and_falser_fn(), e1}); }
+expr mk_and_absurd_fn();
+bool is_and_absurd_fn(expr const & e);
+inline expr mk_and_absurd_th(expr const & e1) { return mk_app({mk_and_absurd_fn(), e1}); }
+expr mk_not_and_fn();
+bool is_not_and_fn(expr const & e);
+inline expr mk_not_and_th(expr const & e1, expr const & e2) { return mk_app({mk_not_and_fn(), e1, e2}); }
+expr mk_not_and_elim_fn();
+bool is_not_and_elim_fn(expr const & e);
+inline expr mk_not_and_elim_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_not_and_elim_fn(), e1, e2, e3}); }
+expr mk_not_or_fn();
+bool is_not_or_fn(expr const & e);
+inline expr mk_not_or_th(expr const & e1, expr const & e2) { return mk_app({mk_not_or_fn(), e1, e2}); }
+expr mk_not_or_elim_fn();
+bool is_not_or_elim_fn(expr const & e);
+inline expr mk_not_or_elim_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_not_or_elim_fn(), e1, e2, e3}); }
+expr mk_not_iff_fn();
+bool is_not_iff_fn(expr const & e);
+inline expr mk_not_iff_th(expr const & e1, expr const & e2) { return mk_app({mk_not_iff_fn(), e1, e2}); }
+expr mk_not_iff_elim_fn();
+bool is_not_iff_elim_fn(expr const & e);
+inline expr mk_not_iff_elim_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_not_iff_elim_fn(), e1, e2, e3}); }
+expr mk_not_implies_fn();
+bool is_not_implies_fn(expr const & e);
+inline expr mk_not_implies_th(expr const & e1, expr const & e2) { return mk_app({mk_not_implies_fn(), e1, e2}); }
+expr mk_not_implies_elim_fn();
+bool is_not_implies_elim_fn(expr const & e);
+inline expr mk_not_implies_elim_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_not_implies_elim_fn(), e1, e2, e3}); }
+expr mk_not_congr_fn();
+bool is_not_congr_fn(expr const & e);
+inline expr mk_not_congr_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_not_congr_fn(), e1, e2, e3}); }
+expr mk_eq_exists_intro_fn();
+bool is_eq_exists_intro_fn(expr const & e);
+inline expr mk_eq_exists_intro_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_eq_exists_intro_fn(), e1, e2, e3, e4}); }
+expr mk_not_forall_fn();
+bool is_not_forall_fn(expr const & e);
+inline expr mk_not_forall_th(expr const & e1, expr const & e2) { return mk_app({mk_not_forall_fn(), e1, e2}); }
+expr mk_not_forall_elim_fn();
+bool is_not_forall_elim_fn(expr const & e);
+inline expr mk_not_forall_elim_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_not_forall_elim_fn(), e1, e2, e3}); }
+expr mk_not_exists_fn();
+bool is_not_exists_fn(expr const & e);
+inline expr mk_not_exists_th(expr const & e1, expr const & e2) { return mk_app({mk_not_exists_fn(), e1, e2}); }
+expr mk_not_exists_elim_fn();
+bool is_not_exists_elim_fn(expr const & e);
+inline expr mk_not_exists_elim_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_not_exists_elim_fn(), e1, e2, e3, e4}); }
+expr mk_exists_unfold1_fn();
+bool is_exists_unfold1_fn(expr const & e);
+inline expr mk_exists_unfold1_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_exists_unfold1_fn(), e1, e2, e3, e4}); }
+expr mk_exists_unfold2_fn();
+bool is_exists_unfold2_fn(expr const & e);
+inline expr mk_exists_unfold2_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_exists_unfold2_fn(), e1, e2, e3, e4}); }
+expr mk_exists_unfold_fn();
+bool is_exists_unfold_fn(expr const & e);
+inline expr mk_exists_unfold_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_exists_unfold_fn(), e1, e2, e3}); }
+}

src/kernel/max_sharing.cpp

@@ -56,8 +56,8 @@ struct max_sharing_fn::imp {
             cache(r);
             return r;
         }
-        case expr_kind::Eq : {
-            expr r = update_eq(a, [=](expr const & l, expr const & r) { return std::make_pair(apply(l), apply(r)); });
+        case expr_kind::HEq : {
+            expr r = update_heq(a, [=](expr const & l, expr const & r) { return std::make_pair(apply(l), apply(r)); });
             cache(r);
             return r;
         }

src/kernel/normalizer.cpp

@@ -309,13 +309,13 @@ class normalizer::imp {
             }
             break;
         }
-        case expr_kind::Eq: {
-            expr new_lhs = normalize(eq_lhs(a), s, k);
-            expr new_rhs = normalize(eq_rhs(a), s, k);
+        case expr_kind::HEq: {
+            expr new_lhs = normalize(heq_lhs(a), s, k);
+            expr new_rhs = normalize(heq_rhs(a), s, k);
             if (is_value(new_lhs) && is_value(new_rhs) && !is_closure(new_lhs) && !is_closure(new_rhs) && typeid(to_value(new_lhs)) == typeid(to_value(new_rhs))) {
                 r = mk_bool_value(new_lhs == new_rhs);
             } else {
-                r = mk_eq(new_lhs, new_rhs);
+                r = mk_heq(new_lhs, new_rhs);
             }
             break;
         }

src/kernel/replace_fn.h

@@ -156,12 +156,12 @@ public:
                 pop_rs(num);
                 break;
             }
-            case expr_kind::Eq:
-                if (check_index(f, 0) && !visit(eq_lhs(e), offset))
+            case expr_kind::HEq:
+                if (check_index(f, 0) && !visit(heq_lhs(e), offset))
                     goto begin_loop;
-                if (check_index(f, 1) && !visit(eq_rhs(e), offset))
+                if (check_index(f, 1) && !visit(heq_rhs(e), offset))
                     goto begin_loop;
-                r = update_eq(e, rs(-2), rs(-1));
+                r = update_heq(e, rs(-2), rs(-1));
                 pop_rs(2);
                 break;
             case expr_kind::Pi: case expr_kind::Lambda:

src/kernel/replace_visitor.cpp

@@ -22,9 +22,9 @@ expr replace_visitor::visit_app(expr const & e, context const & ctx) {
     lean_assert(is_app(e));
     return update_app(e, [&](expr const & c) { return visit(c, ctx); });
 }
-expr replace_visitor::visit_eq(expr const & e, context const & ctx) {
-    lean_assert(is_eq(e));
-    return update_eq(e, [&](expr const & l, expr const & r) { return std::make_pair(visit(l, ctx), visit(r, ctx)); });
+expr replace_visitor::visit_heq(expr const & e, context const & ctx) {
+    lean_assert(is_heq(e));
+    return update_heq(e, [&](expr const & l, expr const & r) { return std::make_pair(visit(l, ctx), visit(r, ctx)); });
 }
 expr replace_visitor::visit_abst(expr const & e, context const & ctx) {
     lean_assert(is_abstraction(e));
@@ -80,7 +80,7 @@ expr replace_visitor::visit(expr const & e, context const & ctx) {
     case expr_kind::Var:       return save_result(e, visit_var(e, ctx), shared);
     case expr_kind::MetaVar:   return save_result(e, visit_metavar(e, ctx), shared);
     case expr_kind::App:       return save_result(e, visit_app(e, ctx), shared);
-    case expr_kind::Eq:        return save_result(e, visit_eq(e, ctx), shared);
+    case expr_kind::HEq:       return save_result(e, visit_heq(e, ctx), shared);
     case expr_kind::Lambda:    return save_result(e, visit_lambda(e, ctx), shared);
     case expr_kind::Pi:        return save_result(e, visit_pi(e, ctx), shared);
     case expr_kind::Let:       return save_result(e, visit_let(e, ctx), shared);

src/kernel/replace_visitor.h

@@ -31,7 +31,7 @@ protected:
     virtual expr visit_var(expr const &, context const &);
     virtual expr visit_metavar(expr const &, context const &);
     virtual expr visit_app(expr const &, context const &);
-    virtual expr visit_eq(expr const &, context const &);
+    virtual expr visit_heq(expr const &, context const &);
     virtual expr visit_abst(expr const &, context const &);
     virtual expr visit_lambda(expr const &, context const &);
     virtual expr visit_pi(expr const &, context const &);

src/kernel/type_checker.cpp

@@ -176,7 +176,7 @@ class type_checker::imp {
             }
         case expr_kind::Type:
             return mk_type(ty_level(e) + 1);
-        case expr_kind::Eq:
+        case expr_kind::HEq:
             // cheap when we are just inferring types
             if (m_infer_only)
                 return mk_bool_type();
@@ -241,10 +241,10 @@ class type_checker::imp {
                     f_t = check_pi(f_t, e, ctx);
                 }
             }
-        case expr_kind::Eq:
+        case expr_kind::HEq:
             lean_assert(!m_infer_only);
-            infer_type_core(eq_lhs(e), ctx);
-            infer_type_core(eq_rhs(e), ctx);
+            infer_type_core(heq_lhs(e), ctx);
+            infer_type_core(heq_rhs(e), ctx);
             return save_result(e, mk_bool_type(), shared);
         case expr_kind::Lambda:
             if (!m_infer_only) {
@@ -422,7 +422,7 @@ public:
         // Catch easy cases
         switch (e.kind()) {
         case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::Type: return false;
-        case expr_kind::Eq: return true;
+        case expr_kind::HEq: return true;
         default: break;
         }
         expr t = infer_type(e, ctx, menv, nullptr);

src/kernel/update_expr.cpp

@@ -67,10 +67,10 @@ expr update_let(expr const & let, optional<expr> const & t, expr const & v, expr
         return mk_let(let_name(let), t, v, b);
 }
 
-expr update_eq(expr const & eq, expr const & l, expr const & r) {
-    if (is_eqp(eq_lhs(eq), l) && is_eqp(eq_rhs(eq), r))
+expr update_heq(expr const & eq, expr const & l, expr const & r) {
+    if (is_eqp(heq_lhs(eq), l) && is_eqp(heq_rhs(eq), r))
         return eq;
     else
-        return mk_eq(l, r);
+        return mk_heq(l, r);
 }
 }

src/kernel/update_expr.h

@@ -43,5 +43,5 @@ expr update_let(expr const & let, optional<expr> const & t, expr const & v, expr
 
    \remark Return \c eq if the given lhs and rhs are (pointer) equal to the ones in \c eq.
 */
-expr update_eq(expr const & eq, expr const & l, expr const & r);
+expr update_heq(expr const & eq, expr const & l, expr const & r);
 }

src/library/arith/Int_decls.cpp

@@ -0,0 +1,20 @@
+/*
+Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+*/
+// Automatically generated file, DO NOT EDIT
+#include "kernel/environment.h"
+#include "kernel/decl_macros.h"
+namespace lean {
+MK_CONSTANT(Int, name("Int"));
+MK_CONSTANT(Int_ge_fn, name({"Int", "ge"}));
+MK_CONSTANT(Int_lt_fn, name({"Int", "lt"}));
+MK_CONSTANT(Int_gt_fn, name({"Int", "gt"}));
+MK_CONSTANT(Int_sub_fn, name({"Int", "sub"}));
+MK_CONSTANT(Int_neg_fn, name({"Int", "neg"}));
+MK_CONSTANT(Int_mod_fn, name({"Int", "mod"}));
+MK_CONSTANT(Int_divides_fn, name({"Int", "divides"}));
+MK_CONSTANT(Int_abs_fn, name({"Int", "abs"}));
+MK_CONSTANT(Nat_sub_fn, name({"Nat", "sub"}));
+MK_CONSTANT(Nat_neg_fn, name({"Nat", "neg"}));
+}

src/library/arith/Int_decls.h

@@ -0,0 +1,50 @@
+/*
+Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+*/
+// Automatically generated file, DO NOT EDIT
+#include "kernel/expr.h"
+namespace lean {
+expr mk_Int();
+bool is_Int(expr const & e);
+expr mk_Int_ge_fn();
+bool is_Int_ge_fn(expr const & e);
+inline bool is_Int_ge(expr const & e) { return is_app(e) && is_Int_ge_fn(arg(e, 0)); }
+inline expr mk_Int_ge(expr const & e1, expr const & e2) { return mk_app({mk_Int_ge_fn(), e1, e2}); }
+expr mk_Int_lt_fn();
+bool is_Int_lt_fn(expr const & e);
+inline bool is_Int_lt(expr const & e) { return is_app(e) && is_Int_lt_fn(arg(e, 0)); }
+inline expr mk_Int_lt(expr const & e1, expr const & e2) { return mk_app({mk_Int_lt_fn(), e1, e2}); }
+expr mk_Int_gt_fn();
+bool is_Int_gt_fn(expr const & e);
+inline bool is_Int_gt(expr const & e) { return is_app(e) && is_Int_gt_fn(arg(e, 0)); }
+inline expr mk_Int_gt(expr const & e1, expr const & e2) { return mk_app({mk_Int_gt_fn(), e1, e2}); }
+expr mk_Int_sub_fn();
+bool is_Int_sub_fn(expr const & e);
+inline bool is_Int_sub(expr const & e) { return is_app(e) && is_Int_sub_fn(arg(e, 0)); }
+inline expr mk_Int_sub(expr const & e1, expr const & e2) { return mk_app({mk_Int_sub_fn(), e1, e2}); }
+expr mk_Int_neg_fn();
+bool is_Int_neg_fn(expr const & e);
+inline bool is_Int_neg(expr const & e) { return is_app(e) && is_Int_neg_fn(arg(e, 0)); }
+inline expr mk_Int_neg(expr const & e1) { return mk_app({mk_Int_neg_fn(), e1}); }
+expr mk_Int_mod_fn();
+bool is_Int_mod_fn(expr const & e);
+inline bool is_Int_mod(expr const & e) { return is_app(e) && is_Int_mod_fn(arg(e, 0)); }
+inline expr mk_Int_mod(expr const & e1, expr const & e2) { return mk_app({mk_Int_mod_fn(), e1, e2}); }
+expr mk_Int_divides_fn();
+bool is_Int_divides_fn(expr const & e);
+inline bool is_Int_divides(expr const & e) { return is_app(e) && is_Int_divides_fn(arg(e, 0)); }
+inline expr mk_Int_divides(expr const & e1, expr const & e2) { return mk_app({mk_Int_divides_fn(), e1, e2}); }
+expr mk_Int_abs_fn();
+bool is_Int_abs_fn(expr const & e);
+inline bool is_Int_abs(expr const & e) { return is_app(e) && is_Int_abs_fn(arg(e, 0)); }
+inline expr mk_Int_abs(expr const & e1) { return mk_app({mk_Int_abs_fn(), e1}); }
+expr mk_Nat_sub_fn();
+bool is_Nat_sub_fn(expr const & e);
+inline bool is_Nat_sub(expr const & e) { return is_app(e) && is_Nat_sub_fn(arg(e, 0)); }
+inline expr mk_Nat_sub(expr const & e1, expr const & e2) { return mk_app({mk_Nat_sub_fn(), e1, e2}); }
+expr mk_Nat_neg_fn();
+bool is_Nat_neg_fn(expr const & e);
+inline bool is_Nat_neg(expr const & e) { return is_app(e) && is_Nat_neg_fn(arg(e, 0)); }
+inline expr mk_Nat_neg(expr const & e1) { return mk_app({mk_Nat_neg_fn(), e1}); }
+}

src/library/arith/Nat_decls.cpp

@@ -0,0 +1,45 @@
+/*
+Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+*/
+// Automatically generated file, DO NOT EDIT
+#include "kernel/environment.h"
+#include "kernel/decl_macros.h"
+namespace lean {
+MK_CONSTANT(Nat, name("Nat"));
+MK_CONSTANT(Nat_ge_fn, name({"Nat", "ge"}));
+MK_CONSTANT(Nat_lt_fn, name({"Nat", "lt"}));
+MK_CONSTANT(Nat_gt_fn, name({"Nat", "gt"}));
+MK_CONSTANT(Nat_id_fn, name({"Nat", "id"}));
+MK_CONSTANT(Nat_succ_nz_fn, name({"Nat", "succ_nz"}));
+MK_CONSTANT(Nat_succ_inj_fn, name({"Nat", "succ_inj"}));
+MK_CONSTANT(Nat_add_zeror_fn, name({"Nat", "add_zeror"}));
+MK_CONSTANT(Nat_add_succr_fn, name({"Nat", "add_succr"}));
+MK_CONSTANT(Nat_mul_zeror_fn, name({"Nat", "mul_zeror"}));
+MK_CONSTANT(Nat_mul_succr_fn, name({"Nat", "mul_succr"}));
+MK_CONSTANT(Nat_le_def_fn, name({"Nat", "le_def"}));
+MK_CONSTANT(Nat_induction_fn, name({"Nat", "induction"}));
+MK_CONSTANT(Nat_pred_nz_fn, name({"Nat", "pred_nz"}));
+MK_CONSTANT(Nat_discriminate_fn, name({"Nat", "discriminate"}));
+MK_CONSTANT(Nat_add_zerol_fn, name({"Nat", "add_zerol"}));
+MK_CONSTANT(Nat_add_succl_fn, name({"Nat", "add_succl"}));
+MK_CONSTANT(Nat_add_comm_fn, name({"Nat", "add_comm"}));
+MK_CONSTANT(Nat_add_assoc_fn, name({"Nat", "add_assoc"}));
+MK_CONSTANT(Nat_mul_zerol_fn, name({"Nat", "mul_zerol"}));
+MK_CONSTANT(Nat_mul_succl_fn, name({"Nat", "mul_succl"}));
+MK_CONSTANT(Nat_mul_onel_fn, name({"Nat", "mul_onel"}));
+MK_CONSTANT(Nat_mul_oner_fn, name({"Nat", "mul_oner"}));
+MK_CONSTANT(Nat_mul_comm_fn, name({"Nat", "mul_comm"}));
+MK_CONSTANT(Nat_distributer_fn, name({"Nat", "distributer"}));
+MK_CONSTANT(Nat_distributel_fn, name({"Nat", "distributel"}));
+MK_CONSTANT(Nat_mul_assoc_fn, name({"Nat", "mul_assoc"}));
+MK_CONSTANT(Nat_add_inj_fn, name({"Nat", "add_inj"}));
+MK_CONSTANT(Nat_add_eqz_fn, name({"Nat", "add_eqz"}));
+MK_CONSTANT(Nat_le_intro_fn, name({"Nat", "le_intro"}));
+MK_CONSTANT(Nat_le_elim_fn, name({"Nat", "le_elim"}));
+MK_CONSTANT(Nat_le_refl_fn, name({"Nat", "le_refl"}));
+MK_CONSTANT(Nat_le_zero_fn, name({"Nat", "le_zero"}));
+MK_CONSTANT(Nat_le_trans_fn, name({"Nat", "le_trans"}));
+MK_CONSTANT(Nat_le_add_fn, name({"Nat", "le_add"}));
+MK_CONSTANT(Nat_le_antisym_fn, name({"Nat", "le_antisym"}));
+}

src/library/arith/Nat_decls.h

@@ -0,0 +1,119 @@
+/*
+Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+*/
+// Automatically generated file, DO NOT EDIT
+#include "kernel/expr.h"
+namespace lean {
+expr mk_Nat();
+bool is_Nat(expr const & e);
+expr mk_Nat_ge_fn();
+bool is_Nat_ge_fn(expr const & e);
+inline bool is_Nat_ge(expr const & e) { return is_app(e) && is_Nat_ge_fn(arg(e, 0)); }
+inline expr mk_Nat_ge(expr const & e1, expr const & e2) { return mk_app({mk_Nat_ge_fn(), e1, e2}); }
+expr mk_Nat_lt_fn();
+bool is_Nat_lt_fn(expr const & e);
+inline bool is_Nat_lt(expr const & e) { return is_app(e) && is_Nat_lt_fn(arg(e, 0)); }
+inline expr mk_Nat_lt(expr const & e1, expr const & e2) { return mk_app({mk_Nat_lt_fn(), e1, e2}); }
+expr mk_Nat_gt_fn();
+bool is_Nat_gt_fn(expr const & e);
+inline bool is_Nat_gt(expr const & e) { return is_app(e) && is_Nat_gt_fn(arg(e, 0)); }
+inline expr mk_Nat_gt(expr const & e1, expr const & e2) { return mk_app({mk_Nat_gt_fn(), e1, e2}); }
+expr mk_Nat_id_fn();
+bool is_Nat_id_fn(expr const & e);
+inline bool is_Nat_id(expr const & e) { return is_app(e) && is_Nat_id_fn(arg(e, 0)); }
+inline expr mk_Nat_id(expr const & e1) { return mk_app({mk_Nat_id_fn(), e1}); }
+expr mk_Nat_succ_nz_fn();
+bool is_Nat_succ_nz_fn(expr const & e);
+inline expr mk_Nat_succ_nz_th(expr const & e1) { return mk_app({mk_Nat_succ_nz_fn(), e1}); }
+expr mk_Nat_succ_inj_fn();
+bool is_Nat_succ_inj_fn(expr const & e);
+inline expr mk_Nat_succ_inj_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_succ_inj_fn(), e1, e2, e3}); }
+expr mk_Nat_add_zeror_fn();
+bool is_Nat_add_zeror_fn(expr const & e);
+inline expr mk_Nat_add_zeror_th(expr const & e1) { return mk_app({mk_Nat_add_zeror_fn(), e1}); }
+expr mk_Nat_add_succr_fn();
+bool is_Nat_add_succr_fn(expr const & e);
+inline expr mk_Nat_add_succr_th(expr const & e1, expr const & e2) { return mk_app({mk_Nat_add_succr_fn(), e1, e2}); }
+expr mk_Nat_mul_zeror_fn();
+bool is_Nat_mul_zeror_fn(expr const & e);
+inline expr mk_Nat_mul_zeror_th(expr const & e1) { return mk_app({mk_Nat_mul_zeror_fn(), e1}); }
+expr mk_Nat_mul_succr_fn();
+bool is_Nat_mul_succr_fn(expr const & e);
+inline expr mk_Nat_mul_succr_th(expr const & e1, expr const & e2) { return mk_app({mk_Nat_mul_succr_fn(), e1, e2}); }
+expr mk_Nat_le_def_fn();
+bool is_Nat_le_def_fn(expr const & e);
+inline expr mk_Nat_le_def_th(expr const & e1, expr const & e2) { return mk_app({mk_Nat_le_def_fn(), e1, e2}); }
+expr mk_Nat_induction_fn();
+bool is_Nat_induction_fn(expr const & e);
+inline expr mk_Nat_induction_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_Nat_induction_fn(), e1, e2, e3, e4}); }
+expr mk_Nat_pred_nz_fn();
+bool is_Nat_pred_nz_fn(expr const & e);
+inline expr mk_Nat_pred_nz_th(expr const & e1, expr const & e2) { return mk_app({mk_Nat_pred_nz_fn(), e1, e2}); }
+expr mk_Nat_discriminate_fn();
+bool is_Nat_discriminate_fn(expr const & e);
+inline expr mk_Nat_discriminate_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_Nat_discriminate_fn(), e1, e2, e3, e4}); }
+expr mk_Nat_add_zerol_fn();
+bool is_Nat_add_zerol_fn(expr const & e);
+inline expr mk_Nat_add_zerol_th(expr const & e1) { return mk_app({mk_Nat_add_zerol_fn(), e1}); }
+expr mk_Nat_add_succl_fn();
+bool is_Nat_add_succl_fn(expr const & e);
+inline expr mk_Nat_add_succl_th(expr const & e1, expr const & e2) { return mk_app({mk_Nat_add_succl_fn(), e1, e2}); }
+expr mk_Nat_add_comm_fn();
+bool is_Nat_add_comm_fn(expr const & e);
+inline expr mk_Nat_add_comm_th(expr const & e1, expr const & e2) { return mk_app({mk_Nat_add_comm_fn(), e1, e2}); }
+expr mk_Nat_add_assoc_fn();
+bool is_Nat_add_assoc_fn(expr const & e);
+inline expr mk_Nat_add_assoc_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_add_assoc_fn(), e1, e2, e3}); }
+expr mk_Nat_mul_zerol_fn();
+bool is_Nat_mul_zerol_fn(expr const & e);
+inline expr mk_Nat_mul_zerol_th(expr const & e1) { return mk_app({mk_Nat_mul_zerol_fn(), e1}); }
+expr mk_Nat_mul_succl_fn();
+bool is_Nat_mul_succl_fn(expr const & e);
+inline expr mk_Nat_mul_succl_th(expr const & e1, expr const & e2) { return mk_app({mk_Nat_mul_succl_fn(), e1, e2}); }
+expr mk_Nat_mul_onel_fn();
+bool is_Nat_mul_onel_fn(expr const & e);
+inline expr mk_Nat_mul_onel_th(expr const & e1) { return mk_app({mk_Nat_mul_onel_fn(), e1}); }
+expr mk_Nat_mul_oner_fn();
+bool is_Nat_mul_oner_fn(expr const & e);
+inline expr mk_Nat_mul_oner_th(expr const & e1) { return mk_app({mk_Nat_mul_oner_fn(), e1}); }
+expr mk_Nat_mul_comm_fn();
+bool is_Nat_mul_comm_fn(expr const & e);
+inline expr mk_Nat_mul_comm_th(expr const & e1, expr const & e2) { return mk_app({mk_Nat_mul_comm_fn(), e1, e2}); }
+expr mk_Nat_distributer_fn();
+bool is_Nat_distributer_fn(expr const & e);
+inline expr mk_Nat_distributer_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_distributer_fn(), e1, e2, e3}); }
+expr mk_Nat_distributel_fn();
+bool is_Nat_distributel_fn(expr const & e);
+inline expr mk_Nat_distributel_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_distributel_fn(), e1, e2, e3}); }
+expr mk_Nat_mul_assoc_fn();
+bool is_Nat_mul_assoc_fn(expr const & e);
+inline expr mk_Nat_mul_assoc_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_mul_assoc_fn(), e1, e2, e3}); }
+expr mk_Nat_add_inj_fn();
+bool is_Nat_add_inj_fn(expr const & e);
+inline expr mk_Nat_add_inj_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_Nat_add_inj_fn(), e1, e2, e3, e4}); }
+expr mk_Nat_add_eqz_fn();
+bool is_Nat_add_eqz_fn(expr const & e);
+inline expr mk_Nat_add_eqz_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_add_eqz_fn(), e1, e2, e3}); }
+expr mk_Nat_le_intro_fn();
+bool is_Nat_le_intro_fn(expr const & e);
+inline expr mk_Nat_le_intro_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_Nat_le_intro_fn(), e1, e2, e3, e4}); }
+expr mk_Nat_le_elim_fn();
+bool is_Nat_le_elim_fn(expr const & e);
+inline expr mk_Nat_le_elim_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_le_elim_fn(), e1, e2, e3}); }
+expr mk_Nat_le_refl_fn();
+bool is_Nat_le_refl_fn(expr const & e);
+inline expr mk_Nat_le_refl_th(expr const & e1) { return mk_app({mk_Nat_le_refl_fn(), e1}); }
+expr mk_Nat_le_zero_fn();
+bool is_Nat_le_zero_fn(expr const & e);
+inline expr mk_Nat_le_zero_th(expr const & e1) { return mk_app({mk_Nat_le_zero_fn(), e1}); }
+expr mk_Nat_le_trans_fn();
+bool is_Nat_le_trans_fn(expr const & e);
+inline expr mk_Nat_le_trans_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5) { return mk_app({mk_Nat_le_trans_fn(), e1, e2, e3, e4, e5}); }
+expr mk_Nat_le_add_fn();
+bool is_Nat_le_add_fn(expr const & e);
+inline expr mk_Nat_le_add_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_Nat_le_add_fn(), e1, e2, e3, e4}); }
+expr mk_Nat_le_antisym_fn();
+bool is_Nat_le_antisym_fn(expr const & e);
+inline expr mk_Nat_le_antisym_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_Nat_le_antisym_fn(), e1, e2, e3, e4}); }
+}

src/library/arith/Real_decls.cpp

@@ -0,0 +1,17 @@
+/*
+Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+*/
+// Automatically generated file, DO NOT EDIT
+#include "kernel/environment.h"
+#include "kernel/decl_macros.h"
+namespace lean {
+MK_CONSTANT(Real, name("Real"));
+MK_CONSTANT(nat_to_real_fn, name("nat_to_real"));
+MK_CONSTANT(Real_ge_fn, name({"Real", "ge"}));
+MK_CONSTANT(Real_lt_fn, name({"Real", "lt"}));
+MK_CONSTANT(Real_gt_fn, name({"Real", "gt"}));
+MK_CONSTANT(Real_sub_fn, name({"Real", "sub"}));
+MK_CONSTANT(Real_neg_fn, name({"Real", "neg"}));
+MK_CONSTANT(Real_abs_fn, name({"Real", "abs"}));
+}

src/library/arith/Real_decls.h

@@ -0,0 +1,38 @@
+/*
+Copyright (c) 2013 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+*/
+// Automatically generated file, DO NOT EDIT
+#include "kernel/expr.h"
+namespace lean {
+expr mk_Real();
+bool is_Real(expr const & e);
+expr mk_nat_to_real_fn();
+bool is_nat_to_real_fn(expr const & e);
+inline bool is_nat_to_real(expr const & e) { return is_app(e) && is_nat_to_real_fn(arg(e, 0)); }
+inline expr mk_nat_to_real(expr const & e1) { return mk_app({mk_nat_to_real_fn(), e1}); }
+expr mk_Real_ge_fn();
+bool is_Real_ge_fn(expr const & e);
+inline bool is_Real_ge(expr const & e) { return is_app(e) && is_Real_ge_fn(arg(e, 0)); }
+inline expr mk_Real_ge(expr const & e1, expr const & e2) { return mk_app({mk_Real_ge_fn(), e1, e2}); }
+expr mk_Real_lt_fn();
+bool is_Real_lt_fn(expr const & e);
+inline bool is_Real_lt(expr const & e) { return is_app(e) && is_Real_lt_fn(arg(e, 0)); }
+inline expr mk_Real_lt(expr const & e1, expr const & e2) { return mk_app({mk_Real_lt_fn(), e1, e2}); }
+expr mk_Real_gt_fn();
+bool is_Real_gt_fn(expr const & e);
+inline bool is_Real_gt(expr const & e) { return is_app(e) && is_Real_gt_fn(arg(e, 0)); }
+inline expr mk_Real_gt(expr const & e1, expr const & e2) { return mk_app({mk_Real_gt_fn(), e1, e2}); }
+expr mk_Real_sub_fn();
+bool is_Real_sub_fn(expr const & e);
+inline bool is_Real_sub(expr const & e) { return is_app(e) && is_Real_sub_fn(arg(e, 0)); }
+inline expr mk_Real_sub(expr const & e1, expr const & e2) { return mk_app({mk_Real_sub_fn(), e1, e2}); }
+expr mk_Real_neg_fn();
+bool is_Real_neg_fn(expr const & e);
+inline bool is_Real_neg(expr const & e) { return is_app(e) && is_Real_neg_fn(arg(e, 0)); }
+inline expr mk_Real_neg(expr const & e1) { return mk_app({mk_Real_neg_fn(), e1}); }
+expr mk_Real_abs_fn();
+bool is_Real_abs_fn(expr const & e);
+inline bool is_Real_abs(expr const & e) { return is_app(e) && is_Real_abs_fn(arg(e, 0)); }
+inline expr mk_Real_abs(expr const & e1) { return mk_app({mk_Real_abs_fn(), e1}); }
+}

src/library/arith/int.cpp

@@ -13,9 +13,11 @@ Author: Leonardo de Moura
 #include "library/kernel_bindings.h"
 #include "library/arith/int.h"
 #include "library/arith/nat.h"
+#include "library/arith/Int_decls.cpp"
 
 namespace lean {
-MK_CONSTANT(Int, "Int");
+expr mk_nat_to_int_fn();
+
 expr const Int = mk_Int();
 expr mk_int_type() { return mk_Int(); }
 
@@ -84,16 +86,16 @@ public:
 constexpr char int_add_name[] = "add";
 struct int_add_eval { mpz operator()(mpz const & v1, mpz const & v2) { return v1 + v2; }; };
 typedef int_bin_op<int_add_name, int_add_eval> int_add_value;
-MK_BUILTIN(int_add_fn, int_add_value);
-static value::register_deserializer_fn int_add_ds("int_add", [](deserializer & ) { return mk_int_add_fn(); });
-static register_builtin_fn g_int_add_value(name({"Int", "add"}), []() { return mk_int_add_fn(); });
+MK_BUILTIN(Int_add_fn, int_add_value);
+static value::register_deserializer_fn int_add_ds("int_add", [](deserializer & ) { return mk_Int_add_fn(); });
+static register_builtin_fn g_int_add_value(name({"Int", "add"}), []() { return mk_Int_add_fn(); });
 
 constexpr char int_mul_name[] = "mul";
 struct int_mul_eval { mpz operator()(mpz const & v1, mpz const & v2) { return v1 * v2; }; };
 typedef int_bin_op<int_mul_name, int_mul_eval> int_mul_value;
-MK_BUILTIN(int_mul_fn, int_mul_value);
-static value::register_deserializer_fn int_mul_ds("int_mul", [](deserializer & ) { return mk_int_mul_fn(); });
-static register_builtin_fn int_mul_blt(name({"Int", "mul"}), []() { return mk_int_mul_fn(); });
+MK_BUILTIN(Int_mul_fn, int_mul_value);
+static value::register_deserializer_fn int_mul_ds("int_mul", [](deserializer & ) { return mk_Int_mul_fn(); });
+static register_builtin_fn int_mul_blt(name({"Int", "mul"}), []() { return mk_Int_mul_fn(); });
 
 constexpr char int_div_name[] = "div";
 struct int_div_eval {
@@ -105,9 +107,9 @@ struct int_div_eval {
     };
 };
 typedef int_bin_op<int_div_name, int_div_eval> int_div_value;
-MK_BUILTIN(int_div_fn, int_div_value);
-static value::register_deserializer_fn int_div_ds("int_div", [](deserializer & ) { return mk_int_div_fn(); });
-static register_builtin_fn int_div_blt(name({"Int", "div"}), []() { return mk_int_div_fn(); });
+MK_BUILTIN(Int_div_fn, int_div_value);
+static value::register_deserializer_fn int_div_ds("int_div", [](deserializer & ) { return mk_Int_div_fn(); });
+static register_builtin_fn int_div_blt(name({"Int", "div"}), []() { return mk_Int_div_fn(); });
 
 class int_le_value : public const_value {
 public:
@@ -121,18 +123,9 @@ public:
     }
     virtual void write(serializer & s) const { s << "int_le"; }
 };
-MK_BUILTIN(int_le_fn, int_le_value);
-static value::register_deserializer_fn int_le_ds("int_le", [](deserializer & ) { return mk_int_le_fn(); });
-static register_builtin_fn int_le_blt(name({"Int", "le"}), []() { return mk_int_le_fn(); });
-
-MK_CONSTANT(int_sub_fn, name({"Int", "sub"}));
-MK_CONSTANT(int_neg_fn, name({"Int", "neg"}));
-MK_CONSTANT(int_mod_fn, name({"Int", "mod"}));
-MK_CONSTANT(int_divides_fn, name({"Int", "divides"}));
-MK_CONSTANT(int_abs_fn, name({"Int", "abs"}));
-MK_CONSTANT(int_ge_fn, name({"Int", "ge"}));
-MK_CONSTANT(int_lt_fn, name({"Int", "lt"}));
-MK_CONSTANT(int_gt_fn, name({"Int", "gt"}));
+MK_BUILTIN(Int_le_fn, int_le_value);
+static value::register_deserializer_fn int_le_ds("int_le", [](deserializer & ) { return mk_Int_le_fn(); });
+static register_builtin_fn int_le_blt(name({"Int", "le"}), []() { return mk_Int_le_fn(); });
 
 /**
    \brief Semantic attachment for the Nat to Int coercion.
@@ -153,9 +146,6 @@ MK_BUILTIN(nat_to_int_fn, nat_to_int_value);
 static value::register_deserializer_fn nat_to_int_ds("nat_to_int", [](deserializer & ) { return mk_nat_to_int_fn(); });
 static register_builtin_fn nat_to_int_blt("nat_to_int", []() { return mk_nat_to_int_fn(); });
 
-MK_CONSTANT(nat_sub_fn, name({"Nat", "sub"}));
-MK_CONSTANT(nat_neg_fn, name({"Nat", "neg"}));
-
 void import_int(environment const & env, io_state const & ios) {
     env->import("Int", ios);
 }

src/library/arith/int.h

@@ -9,6 +9,7 @@ Author: Leonardo de Moura
 #include "util/numerics/mpz.h"
 #include "kernel/expr.h"
 #include "kernel/builtin.h"
+#include "library/arith/Int_decls.h"
 
 namespace lean {
 /** \brief Integer numbers type */
@@ -22,68 +23,16 @@ inline expr iVal(int v) { return mk_int_value(v); }
 bool is_int_value(expr const & e);
 mpz const & int_value_numeral(expr const & e);
 
-/** \brief Addition, Int::add : Int -> Int -> Int */
-expr mk_int_add_fn();
-inline expr iAdd(expr const & e1, expr const & e2) { return mk_app(mk_int_add_fn(), e1, e2); }
-
-/** \brief Subtraction, Int::sub : Int -> Int -> Int */
-expr mk_int_sub_fn();
-inline expr iSub(expr const & e1, expr const & e2) { return mk_app(mk_int_sub_fn(), e1, e2); }
-
-/** \brief Unary minus, Int::neg : Int -> Int */
-expr mk_int_neg_fn();
-inline expr iNeg(expr const & e) { return mk_app(mk_int_neg_fn(), e); }
-
-/** \brief Multiplication, Int::mul : Int -> Int -> Int */
-expr mk_int_mul_fn();
-inline expr iMul(expr const & e1, expr const & e2) { return mk_app(mk_int_mul_fn(), e1, e2); }
-
-/** \brief Integer division, Int::mul : Int -> Int -> Int */
-expr mk_int_div_fn();
-inline expr iDiv(expr const & e1, expr const & e2) { return mk_app(mk_int_div_fn(), e1, e2); }
-
-/** \brief Modulus, Int::mul : Int -> Int -> Int */
-expr mk_int_mod_fn();
-inline expr iMod(expr const & e1, expr const & e2) { return mk_app(mk_int_mod_fn(), e1, e2); }
-
-/** \brief Divides predicate, Int::mul : Int -> Int -> Bool */
-expr mk_int_divides_fn();
-inline expr iDivides(expr const & e1, expr const & e2) { return mk_app(mk_int_divides_fn(), e1, e2); }
-
-/** \brief Absolute value function, Int::abs : Int -> Int */
-expr mk_int_abs_fn();
-inline expr iAbs(expr const & e) { return mk_app(mk_int_abs_fn(), e); }
-
-/** \brief Less than or equal to, Int::le : Int -> Int -> Bool */
-expr mk_int_le_fn();
-inline expr iLe(expr const & e1, expr const & e2) { return mk_app(mk_int_le_fn(), e1, e2); }
-
-/** \brief Greater than or equal to, Int::ge : Int -> Int -> Bool */
-expr mk_int_ge_fn();
-inline expr iGe(expr const & e1, expr const & e2) { return mk_app(mk_int_ge_fn(), e1, e2); }
-
-/** \brief Less than, Int::lt : Int -> Int -> Bool */
-expr mk_int_lt_fn();
-inline expr iLt(expr const & e1, expr const & e2) { return mk_app(mk_int_lt_fn(), e1, e2); }
-
-/** \brief Greater than, Int::gt : Int -> Int -> Bool */
-expr mk_int_gt_fn();
-inline expr iGt(expr const & e1, expr const & e2) { return mk_app(mk_int_gt_fn(), e1, e2); }
-
-/** \brief If-then-else for integers */
-inline expr iIf(expr const & c, expr const & t, expr const & e) { return mk_if(Int, c, t, e); }
-
-/** \brief Coercion from natural to integer */
+expr mk_Int_add_fn();
+inline expr mk_Int_add(expr const & e1, expr const & e2) { return mk_app(mk_Int_add_fn(), e1, e2); }
+expr mk_Int_mul_fn();
+inline expr mk_Int_mul(expr const & e1, expr const & e2) { return mk_app(mk_Int_mul_fn(), e1, e2); }
+expr mk_Int_div_fn();
+inline expr mk_Int_div(expr const & e1, expr const & e2) { return mk_app(mk_Int_div_fn(), e1, e2); }
+expr mk_Int_le_fn();
+inline expr mk_Int_le(expr const & e1, expr const & e2) { return mk_app(mk_Int_le_fn(), e1, e2); }
 expr mk_nat_to_int_fn();
-inline expr n2i(expr const & e) { return mk_app(mk_nat_to_int_fn(), e); }
-
-/** \brief Subtraction (for naturals), Nat::sub : Nat -> Nat -> Int */
-expr mk_nat_sub_fn();
-inline expr nSub(expr const & e1, expr const & e2) { return mk_app(mk_nat_sub_fn(), e1, e2); }
-
-/** \brief Unary minus (for naturals), Nat::neg : Nat -> Int */
-expr mk_nat_neg_fn();
-inline expr nNeg(expr const & e1, expr const & e2) { return mk_app(mk_nat_neg_fn(), e1, e2); }
+inline expr mk_nat_to_int(expr const & e) { return mk_app(mk_nat_to_int_fn(), e); }
 
 class environment;
 /**

src/library/arith/nat.cpp

@@ -12,9 +12,9 @@ Author: Leonardo de Moura
 #include "kernel/decl_macros.h"
 #include "library/kernel_bindings.h"
 #include "library/arith/nat.h"
+#include "library/arith/Nat_decls.cpp"
 
 namespace lean {
-MK_CONSTANT(Nat, "Nat");
 expr const Nat = mk_Nat();
 expr mk_nat_type() { return mk_Nat(); }
 
@@ -78,17 +78,17 @@ constexpr char nat_add_name[] = "add";
 /** \brief Evaluator for + : Nat -> Nat -> Nat */
 struct nat_add_eval { mpz operator()(mpz const & v1, mpz const & v2) { return v1 + v2; }; };
 typedef nat_bin_op<nat_add_name, nat_add_eval> nat_add_value;
-MK_BUILTIN(nat_add_fn, nat_add_value);
-static value::register_deserializer_fn nat_add_ds("nat_add", [](deserializer & ) { return mk_nat_add_fn(); });
-static register_builtin_fn nat_add_blt(name({"Nat", "add"}), []() { return mk_nat_add_fn(); });
+MK_BUILTIN(Nat_add_fn, nat_add_value);
+static value::register_deserializer_fn nat_add_ds("nat_add", [](deserializer & ) { return mk_Nat_add_fn(); });
+static register_builtin_fn nat_add_blt(name({"Nat", "add"}), []() { return mk_Nat_add_fn(); });
 
 constexpr char nat_mul_name[] = "mul";
 /** \brief Evaluator for * : Nat -> Nat -> Nat */
 struct nat_mul_eval { mpz operator()(mpz const & v1, mpz const & v2) { return v1 * v2; }; };
 typedef nat_bin_op<nat_mul_name, nat_mul_eval> nat_mul_value;
-MK_BUILTIN(nat_mul_fn, nat_mul_value);
-static value::register_deserializer_fn nat_mul_ds("nat_mul", [](deserializer & ) { return mk_nat_mul_fn(); });
-static register_builtin_fn nat_mul_blt(name({"Nat", "mul"}), []() { return mk_nat_mul_fn(); });
+MK_BUILTIN(Nat_mul_fn, nat_mul_value);
+static value::register_deserializer_fn nat_mul_ds("nat_mul", [](deserializer & ) { return mk_Nat_mul_fn(); });
+static register_builtin_fn nat_mul_blt(name({"Nat", "mul"}), []() { return mk_Nat_mul_fn(); });
 
 /**
    \brief Semantic attachment for less than or equal to operator with type
@@ -106,14 +106,9 @@ public:
     }
     virtual void write(serializer & s) const { s << "nat_le"; }
 };
-MK_BUILTIN(nat_le_fn, nat_le_value);
-static value::register_deserializer_fn nat_le_ds("nat_le", [](deserializer & ) { return mk_nat_le_fn(); });
-static register_builtin_fn nat_le_blt(name({"Nat", "le"}), []() { return mk_nat_le_fn(); });
-
-MK_CONSTANT(nat_ge_fn,  name({"Nat", "ge"}));
-MK_CONSTANT(nat_lt_fn,  name({"Nat", "lt"}));
-MK_CONSTANT(nat_gt_fn,  name({"Nat", "gt"}));
-MK_CONSTANT(nat_id_fn,  name({"Nat", "id"}));
+MK_BUILTIN(Nat_le_fn, nat_le_value);
+static value::register_deserializer_fn nat_le_ds("nat_le", [](deserializer & ) { return mk_Nat_le_fn(); });
+static register_builtin_fn nat_le_blt(name({"Nat", "le"}), []() { return mk_Nat_le_fn(); });
 
 void import_nat(environment const & env, io_state const & ios) {
     env->import("Nat", ios);

src/library/arith/nat.h

@@ -9,6 +9,7 @@ Author: Leonardo de Moura
 #include "kernel/expr.h"
 #include "kernel/builtin.h"
 #include "util/numerics/mpz.h"
+#include "library/arith/Nat_decls.h"
 
 namespace lean {
 /** \brief Natural numbers type */
@@ -22,36 +23,12 @@ inline expr nVal(unsigned v) { return mk_nat_value(v); }
 bool is_nat_value(expr const & e);
 mpz const & nat_value_numeral(expr const & e);
 
-/** \brief Addition, Nat::add : Nat -> Nat -> Nat */
-expr mk_nat_add_fn();
-inline expr nAdd(expr const & e1, expr const & e2) { return mk_app(mk_nat_add_fn(), e1, e2); }
-
-/** \brief Multiplication, Nat::mul : Nat -> Nat -> Nat */
-expr mk_nat_mul_fn();
-inline expr nMul(expr const & e1, expr const & e2) { return mk_app(mk_nat_mul_fn(), e1, e2); }
-
-/** \brief Less than or equal to, Nat::le : Nat -> Nat -> Bool */
-expr mk_nat_le_fn();
-inline expr nLe(expr const & e1, expr const & e2) { return mk_app(mk_nat_le_fn(), e1, e2); }
-
-/** \brief Greater than or equal to, Nat::ge : Nat -> Nat -> Bool */
-expr mk_nat_ge_fn();
-inline expr nGe(expr const & e1, expr const & e2) { return mk_app(mk_nat_ge_fn(), e1, e2); }
-
-/** \brief Less than, Nat::lt : Nat -> Nat -> Bool */
-expr mk_nat_lt_fn();
-inline expr nLt(expr const & e1, expr const & e2) { return mk_app(mk_nat_lt_fn(), e1, e2); }
-
-/** \brief Greater than, Nat::gt : Nat -> Nat -> Bool */
-expr mk_nat_gt_fn();
-inline expr nGt(expr const & e1, expr const & e2) { return mk_app(mk_nat_gt_fn(), e1, e2); }
-
-/** \brief Identity function for natural numbers, Nat::id : Nat -> Nat */
-expr mk_nat_id_fn();
-inline expr nId(expr const & e) { return mk_app(mk_nat_id_fn(), e); }
-
-/** \brief If-then-else for natural numbers */
-inline expr nIf(expr const & c, expr const & t, expr const & e) { return mk_if(Nat, c, t, e); }
+expr mk_Nat_add_fn();
+inline expr mk_Nat_add(expr const & e1, expr const & e2) { return mk_app(mk_Nat_add_fn(), e1, e2); }
+expr mk_Nat_mul_fn();
+inline expr mk_Nat_mul(expr const & e1, expr const & e2) { return mk_app(mk_Nat_mul_fn(), e1, e2); }
+expr mk_Nat_le_fn();
+inline expr mk_Nat_le(expr const & e1, expr const & e2) { return mk_app(mk_Nat_le_fn(), e1, e2); }
 
 class environment;
 /** \brief Import Natural number library in the given environment (if it has not been imported already). */

src/library/arith/real.cpp

@@ -14,9 +14,9 @@ Author: Leonardo de Moura
 #include "library/arith/real.h"
 #include "library/arith/int.h"
 #include "library/arith/nat.h"
+#include "library/arith/Real_decls.cpp"
 
 namespace lean {
-MK_CONSTANT(Real, "Real");
 expr const Real = mk_Real();
 expr mk_real_type() { return mk_Real(); }
 
@@ -85,18 +85,18 @@ constexpr char real_add_name[] = "add";
 /** \brief Evaluator for + : Real -> Real -> Real */
 struct real_add_eval { mpq operator()(mpq const & v1, mpq const & v2) { return v1 + v2; }; };
 typedef real_bin_op<real_add_name, real_add_eval> real_add_value;
-MK_BUILTIN(real_add_fn, real_add_value);
-static value::register_deserializer_fn real_add_ds("real_add", [](deserializer & ) { return mk_real_add_fn(); });
-static register_builtin_fn real_add_blt(name({"Real", "add"}), []() { return mk_real_add_fn(); });
+MK_BUILTIN(Real_add_fn, real_add_value);
+static value::register_deserializer_fn real_add_ds("real_add", [](deserializer & ) { return mk_Real_add_fn(); });
+static register_builtin_fn real_add_blt(name({"Real", "add"}), []() { return mk_Real_add_fn(); });
 
 
 constexpr char real_mul_name[] = "mul";
 /** \brief Evaluator for * : Real -> Real -> Real */
 struct real_mul_eval { mpq operator()(mpq const & v1, mpq const & v2) { return v1 * v2; }; };
 typedef real_bin_op<real_mul_name, real_mul_eval> real_mul_value;
-MK_BUILTIN(real_mul_fn, real_mul_value);
-static value::register_deserializer_fn real_mul_ds("real_mul", [](deserializer & ) { return mk_real_mul_fn(); });
-static register_builtin_fn real_mul_blt(name({"Real", "mul"}), []() { return mk_real_mul_fn(); });
+MK_BUILTIN(Real_mul_fn, real_mul_value);
+static value::register_deserializer_fn real_mul_ds("real_mul", [](deserializer & ) { return mk_Real_mul_fn(); });
+static register_builtin_fn real_mul_blt(name({"Real", "mul"}), []() { return mk_Real_mul_fn(); });
 
 constexpr char real_div_name[] = "div";
 /** \brief Evaluator for / : Real -> Real -> Real */
@@ -109,9 +109,9 @@ struct real_div_eval {
     };
 };
 typedef real_bin_op<real_div_name, real_div_eval> real_div_value;
-MK_BUILTIN(real_div_fn, real_div_value);
-static value::register_deserializer_fn real_div_ds("real_div", [](deserializer & ) { return mk_real_div_fn(); });
-static register_builtin_fn real_div_blt(name({"Real", "div"}), []() { return mk_real_div_fn(); });
+MK_BUILTIN(Real_div_fn, real_div_value);
+static value::register_deserializer_fn real_div_ds("real_div", [](deserializer & ) { return mk_Real_div_fn(); });
+static register_builtin_fn real_div_blt(name({"Real", "div"}), []() { return mk_Real_div_fn(); });
 
 /**
    \brief Semantic attachment for less than or equal to operator with type
@@ -129,9 +129,9 @@ public:
     }
     virtual void write(serializer & s) const { s << "real_le"; }
 };
-MK_BUILTIN(real_le_fn, real_le_value);
-static value::register_deserializer_fn real_le_ds("real_le", [](deserializer & ) { return mk_real_le_fn(); });
-static register_builtin_fn real_le_btl(name({"Real", "le"}), []() { return mk_real_le_fn(); });
+MK_BUILTIN(Real_le_fn, real_le_value);
+static value::register_deserializer_fn real_le_ds("real_le", [](deserializer & ) { return mk_Real_le_fn(); });
+static register_builtin_fn real_le_btl(name({"Real", "le"}), []() { return mk_Real_le_fn(); });
 
 class int_to_real_value : public const_value {
 public:
@@ -149,14 +149,6 @@ MK_BUILTIN(int_to_real_fn,  int_to_real_value);
 static value::register_deserializer_fn int_to_real_ds("int_to_real", [](deserializer & ) { return mk_int_to_real_fn(); });
 static register_builtin_fn int_to_real_blt("int_to_real", []() { return mk_int_to_real_fn(); });
 
-MK_CONSTANT(real_sub_fn, name({"Real", "sub"}));
-MK_CONSTANT(real_neg_fn, name({"Real", "neg"}));
-MK_CONSTANT(real_abs_fn, name({"Real", "abs"}));
-MK_CONSTANT(real_ge_fn, name({"Real", "ge"}));
-MK_CONSTANT(real_lt_fn, name({"Real", "lt"}));
-MK_CONSTANT(real_gt_fn, name({"Real", "gt"}));
-MK_CONSTANT(nat_to_real_fn, name("nat_to_real"));
-
 void import_real(environment const & env, io_state const & ios) {
     env->import("Real", ios);
 }

src/library/arith/real.h

@@ -9,6 +9,7 @@ Author: Leonardo de Moura
 #include "util/numerics/mpq.h"
 #include "kernel/expr.h"
 #include "kernel/builtin.h"
+#include "library/arith/Real_decls.h"
 
 namespace lean {
 /** \brief Real numbers type */
@@ -22,48 +23,14 @@ inline expr rVal(int v) { return mk_real_value(v); }
 bool is_real_value(expr const & e);
 mpq const & real_value_numeral(expr const & e);
 
-/** \brief Addition, Real::add : Real -> Real -> Real */
-expr mk_real_add_fn();
-inline expr rAdd(expr const & e1, expr const & e2) { return mk_app(mk_real_add_fn(), e1, e2); }
-
-/** \brief Subtraction, Real::sub : Real -> Real -> Real */
-expr mk_real_sub_fn();
-inline expr rSub(expr const & e1, expr const & e2) { return mk_app(mk_real_sub_fn(), e1, e2); }
-
-/** \brief Unary minus, Real::neg : Real -> Real */
-expr mk_real_neg_fn();
-inline expr rNeg(expr const & e) { return mk_app(mk_real_neg_fn(), e); }
-
-/** \brief Multiplication, Real::mul : Real -> Real -> Real */
-expr mk_real_mul_fn();
-inline expr rMul(expr const & e1, expr const & e2) { return mk_app(mk_real_mul_fn(), e1, e2); }
-
-/** \brief Division, Real::mul : Real -> Real -> Real */
-expr mk_real_div_fn();
-inline expr rDiv(expr const & e1, expr const & e2) { return mk_app(mk_real_div_fn(), e1, e2); }
-
-/** \brief Absolute value function, Real::abs : Real -> Real */
-expr mk_real_abs_fn();
-inline expr rAbs(expr const & e) { return mk_app(mk_real_abs_fn(), e); }
-
-/** \brief Less than or equal to, Real::le : Real -> Real -> Bool */
-expr mk_real_le_fn();
-inline expr rLe(expr const & e1, expr const & e2) { return mk_app(mk_real_le_fn(), e1, e2); }
-
-/** \brief Greater than or equal to, Real::ge : Real -> Real -> Bool */
-expr mk_real_ge_fn();
-inline expr rGe(expr const & e1, expr const & e2) { return mk_app(mk_real_ge_fn(), e1, e2); }
-
-/** \brief Less than, Real::lt : Real -> Real -> Bool */
-expr mk_real_lt_fn();
-inline expr rLt(expr const & e1, expr const & e2) { return mk_app(mk_real_lt_fn(), e1, e2); }
-
-/** \brief Greater than, Real::gt : Real -> Real -> Bool */
-expr mk_real_gt_fn();
-inline expr rGt(expr const & e1, expr const & e2) { return mk_app(mk_real_gt_fn(), e1, e2); }
-
-/** \brief If-then-else for reals */
-inline expr rIf(expr const & c, expr const & t, expr const & e) { return mk_if(Real, c, t, e); }
+expr mk_Real_add_fn();
+inline expr mk_Real_add(expr const & e1, expr const & e2) { return mk_app(mk_Real_add_fn(), e1, e2); }
+expr mk_Real_mul_fn();
+inline expr mk_Real_mul(expr const & e1, expr const & e2) { return mk_app(mk_Real_mul_fn(), e1, e2); }
+expr mk_Real_div_fn();
+inline expr mk_Real_div(expr const & e1, expr const & e2) { return mk_app(mk_Real_div_fn(), e1, e2); }
+expr mk_Real_le_fn();
+inline expr mk_Real_le(expr const & e1, expr const & e2) { return mk_app(mk_Real_le_fn(), e1, e2); }
 
 class environment;
 /** \brief Import (basic) Real number library in the given environment (if it has not been imported already). */
@@ -71,10 +38,7 @@ void import_real(environment const & env, io_state const & ios);
 
 /** \brief Coercion from int to real */
 expr mk_int_to_real_fn();
-inline expr i2r(expr const & e) { return mk_app(mk_int_to_real_fn(), e); }
-/** \brief Coercion from nat to real */
-expr mk_nat_to_real_fn();
-inline expr n2r(expr const & e) { return mk_app(mk_nat_to_real_fn(), e); }
+inline expr mk_int_to_real(expr const & e) { return mk_app(mk_int_to_real_fn(), e); }
 
 /** \brief Import the coercions \c i2r and \c n2r. The Integer and (basic) Real libraries are also imported. */
 void import_int_to_real_coercions(environment const & env);

src/library/bin_op.h

@@ -30,7 +30,4 @@ inline expr And(std::initializer_list<expr> const & l) { return mk_and(l.size(),
 
 inline expr mk_or(unsigned num_args, expr const * args) { return mk_bin_rop(mk_or_fn(), False, num_args, args); }
 inline expr Or(std::initializer_list<expr> const & l) { return mk_or(l.size(), l.begin()); }
-
-inline expr mk_iff(unsigned num_args, expr const * args) { return mk_bin_rop(mk_iff_fn(), True, num_args, args); }
-inline expr Iff(std::initializer_list<expr> const & l) { return mk_iff(l.size(), l.begin()); }
 }

src/library/deep_copy.cpp

@@ -43,7 +43,7 @@ class deep_copy_fn {
                 new_args.push_back(apply(old_arg));
             return save_result(a, mk_app(new_args), sh);
         }
-        case expr_kind::Eq:       return save_result(a, mk_eq(apply(eq_lhs(a)), apply(eq_rhs(a))), sh);
+        case expr_kind::HEq:      return save_result(a, mk_heq(apply(heq_lhs(a)), apply(heq_rhs(a))), sh);
         case expr_kind::Lambda:   return save_result(a, mk_lambda(abst_name(a), apply(abst_domain(a)), apply(abst_body(a))), sh);
         case expr_kind::Pi:       return save_result(a, mk_pi(abst_name(a), apply(abst_domain(a)), apply(abst_body(a))), sh);
         case expr_kind::Let:      return save_result(a, mk_let(let_name(a), apply(let_type(a)), apply(let_value(a)), apply(let_body(a))), sh);

src/library/elaborator/elaborator.cpp

@@ -621,7 +621,7 @@ class elaborator::imp {
     }
 
     void process_eq(context const & ctx, expr & a) {
-        if (is_eq(a) && m_use_normalizer) {
+        if (is_heq(a) && m_use_normalizer) {
             a = normalize(ctx, a);
         }
     }
@@ -840,16 +840,17 @@ class elaborator::imp {
                 push_new_eq_constraint(new_state.m_active_cnstrs, ctx, update_app(a, 0, h_i), arg(b, i), new_assumption);
             }
             imitation = mk_lambda(arg_types, mk_app(imitation_args));
-        } else if (is_eq(b)) {
+        } else if (is_heq(b)) {
             // Imitation for equality
             // Assign f_a <- fun (x_1 : T_1) ... (x_{num_a} : T_{num_a}), (h_1 x_1 ... x_{num_a}) = (h_2 x_1 ... x_{num_a})
             // New constraints (h_1 a_1 ... a_{num_a}) == eq_lhs(b)
             //                 (h_2 a_1 ... a_{num_a}) == eq_rhs(b)
             expr h_1 = new_state.m_menv->mk_metavar(ctx);
             expr h_2 = new_state.m_menv->mk_metavar(ctx);
-            push_new_eq_constraint(new_state.m_active_cnstrs, ctx, update_app(a, 0, h_1), eq_lhs(b), new_assumption);
-            push_new_eq_constraint(new_state.m_active_cnstrs, ctx, update_app(a, 0, h_2), eq_rhs(b), new_assumption);
-            imitation = mk_lambda(arg_types, mk_eq(mk_app_vars(add_lift(h_1, 0, num_a - 1), num_a - 1), mk_app_vars(add_lift(h_2, 0, num_a - 1), num_a - 1)));
+            push_new_eq_constraint(new_state.m_active_cnstrs, ctx, update_app(a, 0, h_1), heq_lhs(b), new_assumption);
+            push_new_eq_constraint(new_state.m_active_cnstrs, ctx, update_app(a, 0, h_2), heq_rhs(b), new_assumption);
+            imitation = mk_lambda(arg_types, mk_heq(mk_app_vars(add_lift(h_1, 0, num_a - 1), num_a - 1),
+                                                    mk_app_vars(add_lift(h_2, 0, num_a - 1), num_a - 1)));
         } else if (is_abstraction(b)) {
             // Imitation for lambdas and Pis
             // Assign f_a <- fun (x_1 : T_1) ... (x_{num_a} : T_{num_a}),
@@ -1063,7 +1064,7 @@ class elaborator::imp {
     void imitate_equality(expr const & a, expr const & b, unification_constraint const & c) {
         lean_assert(is_metavar(a));
         static_cast<void>(b); // this line is just to avoid a warning, b is only used in an assertion
-        lean_assert(is_eq(b));
+        lean_assert(is_heq(b));
         lean_assert(!is_assigned(a));
         lean_assert(has_local_context(a));
         // imitate
@@ -1074,7 +1075,7 @@ class elaborator::imp {
         context ctx_m  = m_state.m_menv->get_context(m);
         expr h1        = m_state.m_menv->mk_metavar(ctx_m);
         expr h2        = m_state.m_menv->mk_metavar(ctx_m);
-        expr imitation = mk_eq(h1, h2);
+        expr imitation = mk_heq(h1, h2);
         justification new_jst(new imitation_justification(c));
         push_new_constraint(true, ctx_m, m, imitation, new_jst);
     }
@@ -1137,7 +1138,7 @@ class elaborator::imp {
             } else if (is_app(b) && !has_metavar(arg(b, 0))) {
                 imitate_application(a, b, c);
                 return true;
-            } else if (is_eq(b)) {
+            } else if (is_heq(b)) {
                 imitate_equality(a, b, c);
                 return true;
             }

src/library/eq_heq.cpp

@@ -9,13 +9,13 @@ Author: Leonardo de Moura
 
 namespace lean {
 bool is_eq_heq(expr const & e) {
-    return is_eq(e) || is_homo_eq(e);
+    return is_heq(e) || is_eq(e);
 }
 
 expr_pair eq_heq_args(expr const & e) {
-    lean_assert(is_eq(e) || is_homo_eq(e));
-    if (is_eq(e))
-        return expr_pair(eq_lhs(e), eq_rhs(e));
+    lean_assert(is_heq(e) || is_eq(e));
+    if (is_heq(e))
+        return expr_pair(heq_lhs(e), heq_rhs(e));
     else
         return expr_pair(arg(e, 2), arg(e, 3));
 }

src/library/expr_lt.cpp

@@ -37,11 +37,11 @@ bool is_lt(expr const & a, expr const & b, bool use_hash) {
                 return is_lt(arg(a, i), arg(b, i), use_hash);
         }
         lean_unreachable(); // LCOV_EXCL_LINE
-    case expr_kind::Eq:
-        if (eq_lhs(a) != eq_lhs(b))
-            return is_lt(eq_lhs(a), eq_lhs(b), use_hash);
+    case expr_kind::HEq:
+        if (heq_lhs(a) != heq_lhs(b))
+            return is_lt(heq_lhs(a), heq_lhs(b), use_hash);
         else
-            return is_lt(eq_rhs(a), eq_rhs(b), use_hash);
+            return is_lt(heq_rhs(a), heq_rhs(b), use_hash);
     case expr_kind::Lambda:   // Remark: we ignore get_abs_name because we want alpha-equivalence
     case expr_kind::Pi:
         if (abst_domain(a) != abst_domain(b))

src/library/fo_unify.cpp

@@ -62,7 +62,7 @@ optional<substitution> fo_unify(expr e1, expr e2) {
                         }
                     }
                     break;
-                case expr_kind::Eq:
+                case expr_kind::HEq:
                     lean_unreachable(); break; // LCOV_EXCL_LINE
                 case expr_kind::Lambda: case expr_kind::Pi:
                     todo.emplace_back(abst_body(e1), abst_body(e2));

src/library/kernel_bindings.cpp

@@ -322,8 +322,8 @@ static int expr_mk_app(lua_State * L) {
     return push_expr(L, mk_app(args));
 }
 
-static int expr_mk_eq(lua_State * L) {
-    return push_expr(L, mk_eq(to_expr(L, 1), to_expr(L, 2)));
+static int expr_mk_heq(lua_State * L) {
+    return push_expr(L, mk_heq(to_expr(L, 1), to_expr(L, 2)));
 }
 
 static int expr_mk_lambda(lua_State * L) {
@@ -437,7 +437,7 @@ static int expr_ ## P(lua_State * L) {                  \
 EXPR_PRED(is_constant)
 EXPR_PRED(is_var)
 EXPR_PRED(is_app)
-EXPR_PRED(is_eq)
+EXPR_PRED(is_heq)
 EXPR_PRED(is_lambda)
 EXPR_PRED(is_pi)
 EXPR_PRED(is_abstraction)
@@ -450,12 +450,8 @@ EXPR_PRED(has_metavar)
 EXPR_PRED(is_not)
 EXPR_PRED(is_and)
 EXPR_PRED(is_or)
-EXPR_PRED(is_if)
-EXPR_PRED(is_iff)
 EXPR_PRED(is_implies)
-EXPR_PRED(is_forall)
 EXPR_PRED(is_exists)
-EXPR_PRED(is_homo_eq)
 
 /**
    \brief Iterator (closure base function) for application args. See \c expr_args
@@ -502,7 +498,7 @@ static int expr_fields(lua_State * L) {
     case expr_kind::Type:     return push_level(L, ty_level(e));
     case expr_kind::Value:    return to_value(e).push_lua(L);
     case expr_kind::App:      lua_pushinteger(L, num_args(e)); expr_args(L); return 2;
-    case expr_kind::Eq:       push_expr(L, eq_lhs(e)); push_expr(L, eq_rhs(e)); return 2;
+    case expr_kind::HEq:      push_expr(L, heq_lhs(e)); push_expr(L, heq_rhs(e)); return 2;
     case expr_kind::Lambda:
     case expr_kind::Pi:
         push_name(L, abst_name(e)); push_expr(L, abst_domain(e)); push_expr(L, abst_body(e)); return 3;
@@ -617,7 +613,7 @@ static const struct luaL_Reg expr_m[] = {
     {"is_var",          safe_function<expr_is_var>},
     {"is_constant",     safe_function<expr_is_constant>},
     {"is_app",          safe_function<expr_is_app>},
-    {"is_eq",           safe_function<expr_is_eq>},
+    {"is_heq",          safe_function<expr_is_heq>},
     {"is_lambda",       safe_function<expr_is_lambda>},
     {"is_pi",           safe_function<expr_is_pi>},
     {"is_abstraction",  safe_function<expr_is_abstraction>},
@@ -644,12 +640,8 @@ static const struct luaL_Reg expr_m[] = {
     {"is_not",          safe_function<expr_is_not>},
     {"is_and",          safe_function<expr_is_and>},
     {"is_or",           safe_function<expr_is_or>},
-    {"is_if",           safe_function<expr_is_if>},
-    {"is_iff",          safe_function<expr_is_iff>},
     {"is_implies",      safe_function<expr_is_implies>},
-    {"is_forall",       safe_function<expr_is_forall>},
     {"is_exists",       safe_function<expr_is_exists>},
-    {"is_home_eq",      safe_function<expr_is_homo_eq>},
     {0, 0}
 };
 
@@ -669,8 +661,8 @@ static void open_expr(lua_State * L) {
     SET_GLOBAL_FUN(expr_mk_var,      "mk_var");
     SET_GLOBAL_FUN(expr_mk_var,      "Var");
     SET_GLOBAL_FUN(expr_mk_app,      "mk_app");
-    SET_GLOBAL_FUN(expr_mk_eq,       "mk_eq");
-    SET_GLOBAL_FUN(expr_mk_eq,       "Eq");
+    SET_GLOBAL_FUN(expr_mk_heq,      "mk_heq");
+    SET_GLOBAL_FUN(expr_mk_heq,      "HEq");
     SET_GLOBAL_FUN(expr_mk_lambda,   "mk_lambda");
     SET_GLOBAL_FUN(expr_mk_pi,       "mk_pi");
     SET_GLOBAL_FUN(expr_mk_arrow,    "mk_arrow");
@@ -690,7 +682,7 @@ static void open_expr(lua_State * L) {
     SET_ENUM("Type",     expr_kind::Type);
     SET_ENUM("Value",    expr_kind::Value);
     SET_ENUM("App",      expr_kind::App);
-    SET_ENUM("Eq",       expr_kind::Eq);
+    SET_ENUM("HEq",      expr_kind::HEq);
     SET_ENUM("Lambda",   expr_kind::Lambda);
     SET_ENUM("Pi",       expr_kind::Pi);
     SET_ENUM("Let",      expr_kind::Let);

src/library/printer.cpp

@@ -18,7 +18,7 @@ bool is_atomic(expr const & e) {
     case expr_kind::Type: case expr_kind::MetaVar:
         return true;
     case expr_kind::App: case expr_kind::Lambda: case expr_kind::Pi:
-    case expr_kind::Eq: case expr_kind::Let:
+    case expr_kind::HEq: case expr_kind::Let:
         return false;
     }
     return false;
@@ -55,10 +55,10 @@ struct print_expr_fn {
         }
     }
 
-    void print_eq(expr const & a, context const & c) {
-        print_child(eq_lhs(a), c);
+    void print_heq(expr const & a, context const & c) {
+        print_child(heq_lhs(a), c);
         out() << " == ";
-        print_child(eq_rhs(a), c);
+        print_child(heq_rhs(a), c);
     }
 
     void print_app(expr const & e, context const & c) {
@@ -114,8 +114,8 @@ struct print_expr_fn {
         case expr_kind::App:
             print_app(a, c);
             break;
-        case expr_kind::Eq:
-            print_eq(a, c);
+        case expr_kind::HEq:
+            print_heq(a, c);
             break;
         case expr_kind::Lambda:
             out() << "fun " << abst_name(a) << " : ";

src/library/rewriter/fo_match.cpp

@@ -117,11 +117,11 @@ bool fo_match::match_type(expr const & p, expr const & t, unsigned, subst_map &)
     return p == t;
 }
 
-bool fo_match::match_eq(expr const & p, expr const & t, unsigned o, subst_map & s) {
+bool fo_match::match_heq(expr const & p, expr const & t, unsigned o, subst_map & s) {
     lean_trace("fo_match", tout << "match_eq : (" << p << ", " << t << ", " << o << ", " << s << ")" << endl;); // LCOV_EXCL_LINE
-    if (!is_eq(t))
+    if (!is_heq(t))
         return false;
-    return match_main(eq_lhs(p), eq_lhs(t), o, s) && match_main(eq_rhs(p), eq_rhs(t), o, s);
+    return match_main(heq_lhs(p), heq_lhs(t), o, s) && match_main(heq_rhs(p), heq_rhs(t), o, s);
 }
 
 bool fo_match::match_let(expr const & p, expr const & t, unsigned o, subst_map & s) {
@@ -176,8 +176,8 @@ bool fo_match::match_main(expr const & p, expr const & t, unsigned o, subst_map
         return match_pi(p, t, o, s);
     case expr_kind::Type:
         return match_type(p, t, o, s);
-    case expr_kind::Eq:
-        return match_eq(p, t, o, s);
+    case expr_kind::HEq:
+        return match_heq(p, t, o, s);
     case expr_kind::Let:
         return match_let(p, t, o, s);
     case expr_kind::MetaVar:

src/library/rewriter/fo_match.h

@@ -22,7 +22,7 @@ private:
     bool match_lambda(expr const & p, expr const & t, unsigned o, subst_map & s);
     bool match_pi(expr const & p, expr const & t, unsigned o, subst_map & s);
     bool match_type(expr const & p, expr const & t, unsigned o, subst_map & s);
-    bool match_eq(expr const & p, expr const & t, unsigned o, subst_map & s);
+    bool match_heq(expr const & p, expr const & t, unsigned o, subst_map & s);
     bool match_let(expr const & p, expr const & t, unsigned o, subst_map & s);
     bool match_metavar(expr const & p, expr const & t, unsigned o, subst_map & s);
     bool match_main(expr const & p, expr const & t, unsigned o, subst_map & s);

src/library/rewriter/rewriter.cpp

@@ -47,7 +47,7 @@ pair<expr, expr> rewrite_lambda_type(environment const & env, context & ctx, exp
     expr const & new_ty = result_ty.first;
     expr const & ty_v   = ti(v, ctx);
     if (ty == new_ty) {
-        return make_pair(v, Refl(ty_v, v));
+        return make_pair(v, mk_refl_th(ty_v, v));
     } else {
         name const & n         = abst_name(v);
         expr const & body      = abst_body(v);
@@ -55,10 +55,10 @@ pair<expr, expr> rewrite_lambda_type(environment const & env, context & ctx, exp
         expr const & new_v     = mk_lambda(n, new_ty, body);
         expr const & ty_ty     = ti(ty, ctx);
         lean_assert_eq(ty_ty, ti(new_ty, ctx)); // TODO(soonhok): generalize for hetreogeneous types
-        expr const & proof     = Subst(ty_ty, ty, new_ty,
+        expr const & proof     = mk_subst_th(ty_ty, ty, new_ty,
                                        Fun({Const("T"), ty_ty},
-                                           mk_eq(v, mk_lambda(n, Const("T"), body))),
-                                       Refl(ty_v, v), pf_ty);
+                                           mk_heq(v, mk_lambda(n, Const("T"), body))),
+                                       mk_refl_th(ty_v, v), pf_ty);
         return make_pair(new_v, proof);
     }
 }
@@ -83,7 +83,7 @@ pair<expr, expr> rewrite_lambda_body(environment const & env, context & ctx, exp
     expr const & new_body = result_body.first;
     expr const & ty_v     = ti(v, ctx);
     if (body == new_body) {
-        return make_pair(v, Refl(ty_v, v));
+        return make_pair(v, mk_refl_th(ty_v, v));
     } else {
         name const & n       = abst_name(v);
         expr const & ty      = abst_domain(v);
@@ -91,7 +91,7 @@ pair<expr, expr> rewrite_lambda_body(environment const & env, context & ctx, exp
         expr const & new_v   = mk_lambda(n, ty, new_body);
         expr const & ty_body = ti(body, extend(ctx, n, ty));
         lean_assert_eq(ty_body, ti(new_body, ctx)); // TODO(soonhok): generalize for hetreogeneous types
-        expr const & proof = Abst(ty, mk_lambda(n, ty, ty_body), v, new_v, mk_lambda(n, ty, pf_body));
+        expr const & proof = mk_funext_th(ty, mk_lambda(n, ty, ty_body), v, new_v, mk_lambda(n, ty, pf_body));
         return make_pair(new_v, proof);
     }
 }
@@ -128,18 +128,18 @@ pair<expr, expr> rewrite_lambda(environment const & env, context & ctx, expr con
     expr const & ty_new_v1 = ti(v, ctx);
     expr const & new_v2    = mk_lambda(n, new_ty, new_body);
     // proof1 : v = new_v1
-    expr const & proof1    = Subst(ty_ty, ty, new_ty,
-                                   Fun({Const("T"), ty_ty},
-                                       mk_eq(v, mk_lambda(n, Const("T"), body))),
-                                   Refl(ty_v, v),
-                                   pf_ty);
+    expr const & proof1    = mk_subst_th(ty_ty, ty, new_ty,
+                                         Fun({Const("T"), ty_ty},
+                                             mk_heq(v, mk_lambda(n, Const("T"), body))),
+                                         mk_refl_th(ty_v, v),
+                                         pf_ty);
     // proof2 : new_v1 = new_v2
-    expr const & proof2    = Subst(ty_body, body, new_body,
-                                   Fun({Const("e"), ty_body},
-                                       mk_eq(new_v1, mk_lambda(n, new_ty, Const("e")))),
-                                   Refl(ty_new_v1, new_v1),
-                                   pf_body);
-    expr const & proof     = Trans(ty_v, v, new_v1, new_v2, proof1, proof2);
+    expr const & proof2    = mk_subst_th(ty_body, body, new_body,
+                                         Fun({Const("e"), ty_body},
+                                             mk_heq(new_v1, mk_lambda(n, new_ty, Const("e")))),
+                                         mk_refl_th(ty_new_v1, new_v1),
+                                         pf_body);
+    expr const & proof     = mk_trans_th(ty_v, v, new_v1, new_v2, proof1, proof2);
     return make_pair(new_v2, proof);
 }
 
@@ -166,11 +166,11 @@ pair<expr, expr> rewrite_pi_type(environment const & env, context & ctx, expr co
     expr const & new_v  = mk_pi(n, new_ty, body);
     expr const & ty_ty  = ti(ty, ctx);
     expr const & ty_v   = ti(v, ctx);
-    expr const & proof  = Subst(ty_ty, ty, new_ty,
-                                Fun({Const("T"), ty_ty},
-                                    mk_eq(v, mk_pi(n, Const("T"), body))),
-                                Refl(ty_v, v),
-                                pf);
+    expr const & proof  = mk_subst_th(ty_ty, ty, new_ty,
+                                      Fun({Const("T"), ty_ty},
+                                          mk_heq(v, mk_pi(n, Const("T"), body))),
+                                      mk_refl_th(ty_v, v),
+                                      pf);
     return make_pair(new_v, proof);
 }
 
@@ -198,10 +198,10 @@ pair<expr, expr> rewrite_pi_body(environment const & env, context & ctx, expr co
     expr const & new_v    = mk_pi(n, ty, new_body);
     expr const & ty_body  = ti(body, extend(ctx, n, ty));
     expr const & ty_v     = ti(v, ctx);
-    expr const & proof    = Subst(ty_body, body, new_body,
+    expr const & proof    = mk_subst_th(ty_body, body, new_body,
                                   Fun({Const("e"), ty_body},
-                                      mk_eq(v, mk_pi(n, ty, Const("e")))),
-                                  Refl(ty_v, v),
+                                      mk_heq(v, mk_pi(n, ty, Const("e")))),
+                                  mk_refl_th(ty_v, v),
                                   pf);
     return make_pair(new_v, proof);
 }
@@ -237,17 +237,17 @@ pair<expr, expr> rewrite_pi(environment const & env, context & ctx, expr const &
     expr const & new_v1    = mk_pi(n, new_ty, body);
     expr const & ty_new_v1 = ti(v, ctx);
     expr const & new_v2    = mk_pi(n, new_ty, new_body);
-    expr const & proof1    = Subst(ty_ty, ty, new_ty,
+    expr const & proof1    = mk_subst_th(ty_ty, ty, new_ty,
                                    Fun({Const("T"), ty_ty},
-                                       mk_eq(v, mk_pi(n, Const("T"), body))),
-                                   Refl(ty_v, v),
+                                       mk_heq(v, mk_pi(n, Const("T"), body))),
+                                   mk_refl_th(ty_v, v),
                                    pf_ty);
-    expr const & proof2    = Subst(ty_body, body, new_body,
+    expr const & proof2    = mk_subst_th(ty_body, body, new_body,
                                    Fun({Const("e"), ty_body},
-                                       mk_eq(new_v1, mk_pi(n, new_ty, Const("e")))),
-                                   Refl(ty_new_v1, new_v1),
+                                       mk_heq(new_v1, mk_pi(n, new_ty, Const("e")))),
+                                   mk_refl_th(ty_new_v1, new_v1),
                                    pf_body);
-    expr const & proof     = Trans(ty_v, v, new_v1, new_v2, proof1, proof2);
+    expr const & proof     = mk_trans_th(ty_v, v, new_v1, new_v2, proof1, proof2);
     return make_pair(new_v2, proof);
 }
 
@@ -264,19 +264,19 @@ pair<expr, expr> rewrite_pi(environment const & env, context & ctx, expr const &
    \return pair of v' = (lhs' = rhs), and proof of v = v'
 */
 pair<expr, expr> rewrite_eq_lhs(environment const & env, context & ctx, expr const & v, pair<expr, expr> const & result_lhs) {
-    lean_assert(is_eq(v));
+    lean_assert(is_heq(v));
     type_inferer ti(env);
-    expr const & lhs     = eq_lhs(v);
-    expr const & rhs     = eq_rhs(v);
+    expr const & lhs     = heq_lhs(v);
+    expr const & rhs     = heq_rhs(v);
     expr const & new_lhs = result_lhs.first;
     expr const & pf      = result_lhs.second;
-    expr const & new_v   = mk_eq(new_lhs, rhs);
+    expr const & new_v   = mk_heq(new_lhs, rhs);
     expr const & ty_lhs  = ti(lhs, ctx);
     expr const & ty_v    = ti(v, ctx);
-    expr const & proof   = Subst(ty_lhs, lhs, new_lhs,
+    expr const & proof   = mk_subst_th(ty_lhs, lhs, new_lhs,
                                  Fun({Const("x"), ty_lhs},
-                                     mk_eq(v, mk_eq(Const("x"), rhs))),
-                                 Refl(ty_v, v),
+                                     mk_heq(v, mk_heq(Const("x"), rhs))),
+                                 mk_refl_th(ty_v, v),
                                  pf);
     return make_pair(new_v, proof);
 }
@@ -294,19 +294,19 @@ pair<expr, expr> rewrite_eq_lhs(environment const & env, context & ctx, expr con
    \return pair of v' = (lhs = rhs'), and proof of v = v'
 */
 pair<expr, expr> rewrite_eq_rhs(environment const & env, context & ctx, expr const & v, pair<expr, expr> const & result_rhs) {
-    lean_assert(is_eq(v));
+    lean_assert(is_heq(v));
     type_inferer ti(env);
-    expr const & lhs     = eq_lhs(v);
-    expr const & rhs     = eq_rhs(v);
+    expr const & lhs     = heq_lhs(v);
+    expr const & rhs     = heq_rhs(v);
     expr const & new_rhs = result_rhs.first;
     expr const & pf      = result_rhs.second;
-    expr const & new_v   = mk_eq(rhs, new_rhs);
+    expr const & new_v   = mk_heq(rhs, new_rhs);
     expr const & ty_rhs  = ti(rhs, ctx);
     expr const & ty_v    = ti(v, ctx);
-    expr const & proof   = Subst(ty_rhs, rhs, new_rhs,
+    expr const & proof   = mk_subst_th(ty_rhs, rhs, new_rhs,
                                  Fun({Const("x"), ty_rhs},
-                                     mk_eq(v, mk_eq(lhs, Const("x")))),
-                                 Refl(ty_v, v),
+                                     mk_heq(v, mk_heq(lhs, Const("x")))),
+                                 mk_refl_th(ty_v, v),
                                  pf);
     return make_pair(new_v, proof);
 }
@@ -326,31 +326,31 @@ pair<expr, expr> rewrite_eq_rhs(environment const & env, context & ctx, expr con
    \return pair of v' = (lhs' = rhs'), and proof of v = v'
 */
 pair<expr, expr> rewrite_eq(environment const & env, context & ctx, expr const & v, pair<expr, expr> const & result_lhs, pair<expr, expr> const & result_rhs) {
-    lean_assert(is_eq(v));
+    lean_assert(is_heq(v));
     type_inferer ti(env);
-    expr const & lhs       = eq_lhs(v);
-    expr const & rhs       = eq_rhs(v);
+    expr const & lhs       = heq_lhs(v);
+    expr const & rhs       = heq_rhs(v);
     expr const & new_lhs   = result_lhs.first;
     expr const & pf_lhs    = result_lhs.second;
     expr const & new_rhs   = result_rhs.first;
     expr const & pf_rhs    = result_rhs.second;
-    expr const & new_v1    = mk_eq(new_lhs, rhs);
-    expr const & new_v2    = mk_eq(new_lhs, new_rhs);
+    expr const & new_v1    = mk_heq(new_lhs, rhs);
+    expr const & new_v2    = mk_heq(new_lhs, new_rhs);
     expr const & ty_lhs    = ti(lhs, ctx);
     expr const & ty_rhs    = ti(rhs, ctx);
     expr const & ty_v      = ti(v, ctx);
     expr const & ty_new_v1 = ti(new_v1, ctx);
-    expr const & proof1 = Subst(ty_lhs, lhs, new_lhs,
+    expr const & proof1 = mk_subst_th(ty_lhs, lhs, new_lhs,
                                 Fun({Const("x"), ty_lhs},
-                                    mk_eq(v, mk_eq(Const("x"), rhs))),
-                                Refl(ty_v, v),
+                                    mk_heq(v, mk_heq(Const("x"), rhs))),
+                                mk_refl_th(ty_v, v),
                                 pf_lhs);
-    expr const & proof2 = Subst(ty_rhs, rhs, new_rhs,
+    expr const & proof2 = mk_subst_th(ty_rhs, rhs, new_rhs,
                                 Fun({Const("x"), ty_rhs},
-                                    mk_eq(v, mk_eq(lhs, Const("x")))),
-                                Refl(ty_new_v1, new_v1),
+                                    mk_heq(v, mk_heq(lhs, Const("x")))),
+                                mk_refl_th(ty_new_v1, new_v1),
                                 pf_rhs);
-    expr const & proof  = Trans(ty_v, v, new_v1, new_v2, proof1, proof2);
+    expr const & proof  = mk_trans_th(ty_v, v, new_v1, new_v2, proof1, proof2);
     return make_pair(new_v2, proof);
 }
 
@@ -379,10 +379,10 @@ pair<expr, expr> rewrite_let_type(environment const & env, context & ctx, expr c
         expr const & new_v   = mk_let(n, new_ty, val, body);
         expr const & ty_ty   = ti(ty, ctx);
         expr const & ty_v    = ti(v, ctx);
-        expr const & proof   = Subst(ty_ty, ty, new_ty,
+        expr const & proof   = mk_subst_th(ty_ty, ty, new_ty,
                                      Fun({Const("x"), ty_ty},
-                                         mk_eq(v, mk_let(n, Const("x"), val, body))),
-                                     Refl(ty_v, v),
+                                         mk_heq(v, mk_let(n, Const("x"), val, body))),
+                                     mk_refl_th(ty_v, v),
                                      pf);
         return make_pair(new_v, proof);
     } else {
@@ -414,10 +414,10 @@ pair<expr, expr> rewrite_let_value(environment const & env, context & ctx, expr
     expr const & new_v   = mk_let(n, ty, new_val, body);
     expr const & ty_val  = ti(val, ctx);
     expr const & ty_v    = ti(v, ctx);
-    expr const & proof   = Subst(ty_val, val, new_val,
+    expr const & proof   = mk_subst_th(ty_val, val, new_val,
                                  Fun({Const("x"), ty_val},
-                                     mk_eq(v, mk_let(n, ty, Const("x"), body))),
-                                 Refl(ty_v, v),
+                                     mk_heq(v, mk_let(n, ty, Const("x"), body))),
+                                 mk_refl_th(ty_v, v),
                                  pf);
     return make_pair(new_v, proof);
 }
@@ -447,10 +447,10 @@ pair<expr, expr> rewrite_let_body(environment const & env, context & ctx, expr c
     expr const & new_v    = mk_let(n, ty, val, new_body);
     expr const & ty_body  = ti(body, extend(ctx, n, ty, body));
     expr const & ty_v     = ti(v, ctx);
-    expr const & proof    = Subst(ty_body, body, new_body,
+    expr const & proof    = mk_subst_th(ty_body, body, new_body,
                                   Fun({Const("e"), ty_body},
-                                      mk_eq(v, mk_let(n, ty, val, Const("e")))),
-                                  Refl(ty_v, v),
+                                      mk_heq(v, mk_let(n, ty, val, Const("e")))),
+                                  mk_refl_th(ty_v, v),
                                   pf);
     return make_pair(new_v, proof);
 }
@@ -483,22 +483,22 @@ pair<expr, expr> rewrite_app(environment const & env, context & ctx, expr const
         bool f_changed = f != new_f;
         if (f_changed) {
             if (arg(v, i) != results[i].first) {
-                // Congr : Pi (A : Type u) (B : A -> Type u) (f g : Pi
+                // congr : Pi (A : Type u) (B : A -> Type u) (f g : Pi
                 // (x : A) B x) (a b : A) (H1 : f = g) (H2 : a = b), f
                 // a = g b
-                pf = Congr(f_ty_domain, f_ty_body, f, new_f, e_i, new_e_i, pf, pf_e_i);
+                pf = mk_congr_th(f_ty_domain, f_ty_body, f, new_f, e_i, new_e_i, pf, pf_e_i);
             } else {
-                // Congr1 : Pi (A : Type u) (B : A -> Type u) (f g: Pi
+                // congr1 : Pi (A : Type u) (B : A -> Type u) (f g: Pi
                 // (x : A) B x) (a : A) (H : f = g), f a = g a
-                pf = Congr1(f_ty_domain, f_ty_body, f, new_f, e_i, pf);
+                pf = mk_congr1_th(f_ty_domain, f_ty_body, f, new_f, e_i, pf);
             }
         } else {
             if (arg(v, i) != results[i].first) {
-                // Congr2 : Pi (A : Type u) (B : A -> Type u)  (a b : A) (f : Pi (x : A) B x) (H : a = b), f a = f b
-                pf = Congr2(f_ty_domain, f_ty_body, e_i, new_e_i, f, pf_e_i);
+                // congr2 : Pi (A : Type u) (B : A -> Type u)  (a b : A) (f : Pi (x : A) B x) (H : a = b), f a = f b
+                pf = mk_congr2_th(f_ty_domain, f_ty_body, e_i, new_e_i, f, pf_e_i);
             } else {
-                // Refl
-                pf = Refl(ti(f(e_i), ctx), f(e_i));
+                // refl
+                pf = mk_refl_th(ti(f(e_i), ctx), f(e_i));
             }
         }
         f = f (e_i);
@@ -527,12 +527,12 @@ theorem_rewriter_cell::theorem_rewriter_cell(expr const & type, expr const & bod
         m_pattern = abst_body(m_pattern);
         m_num_args++;
     }
-    if (!is_eq(m_pattern)) {
+    if (!is_heq(m_pattern)) {
         lean_trace("rewriter", tout << "Theorem " << m_type << " is not in the form of "
                    << "Pi (x_1 : t_1 ... x_n : t_n), pattern = rhs" << endl;);
     }
-    m_rhs = eq_rhs(m_pattern);
-    m_pattern = eq_lhs(m_pattern);
+    m_rhs = heq_rhs(m_pattern);
+    m_pattern = heq_lhs(m_pattern);
 
     lean_trace("rewriter", tout << "Number of Arg = " << m_num_args << endl;);
 }
@@ -631,7 +631,7 @@ pair<expr, expr> then_rewriter_cell::operator()(environment const & env, context
         expr const & ty = type_inferer(env)(v, ctx);
         if (v != new_result.first) {
             result = make_pair(new_result.first,
-                               Trans(ty, v, result.first, new_result.first, result.second, new_result.second));
+                               mk_trans_th(ty, v, result.first, new_result.first, result.second, new_result.second));
         }
     }
     return result;
@@ -661,7 +661,7 @@ pair<expr, expr> try_rewriter_cell::operator()(environment const & env, context
     }
     // If the execution reaches here, it means every rewriter failed.
     expr const & t = type_inferer(env)(v, ctx);
-    return make_pair(v, Refl(t, v));
+    return make_pair(v, mk_refl_th(t, v));
 }
 ostream & try_rewriter_cell::display(ostream & out) const {
     out << "Try_RW({";
@@ -705,7 +705,7 @@ pair<expr, expr> lambda_type_rewriter_cell::operator()(environment const & env,
         return rewrite_lambda_type(env, ctx, v, result_ty);
     } else {
         // nothing changed
-        return make_pair(v, Refl(ti(v, ctx), v));
+        return make_pair(v, mk_refl_th(ti(v, ctx), v));
     }
 }
 ostream & lambda_type_rewriter_cell::display(ostream & out) const {
@@ -733,7 +733,7 @@ pair<expr, expr> lambda_body_rewriter_cell::operator()(environment const & env,
     } else {
         // nothing changed
         type_inferer ti(env);
-        return make_pair(v, Refl(ti(v, ctx), v));
+        return make_pair(v, mk_refl_th(ti(v, ctx), v));
     }
 }
 ostream & lambda_body_rewriter_cell::display(ostream & out) const {
@@ -774,7 +774,7 @@ pair<expr, expr> pi_type_rewriter_cell::operator()(environment const & env, cont
     } else {
         // nothing changed
         type_inferer ti(env);
-        return make_pair(v, Refl(ti(v, ctx), v));
+        return make_pair(v, mk_refl_th(ti(v, ctx), v));
     }
 }
 ostream & pi_type_rewriter_cell::display(ostream & out) const {
@@ -802,7 +802,7 @@ pair<expr, expr> pi_body_rewriter_cell::operator()(environment const & env, cont
     } else {
         // nothing changed
         type_inferer ti(env);
-        return make_pair(v, Refl(ti(v, ctx), v));
+        return make_pair(v, mk_refl_th(ti(v, ctx), v));
     }
 }
 ostream & pi_body_rewriter_cell::display(ostream & out) const {
@@ -842,7 +842,7 @@ pair<expr, expr> let_type_rewriter_cell::operator()(environment const & env, con
         return rewrite_let_type(env, ctx, v, result_ty);
     } else {
         type_inferer ti(env);
-        return make_pair(v, Refl(ti(v, ctx), v));
+        return make_pair(v, mk_refl_th(ti(v, ctx), v));
     }
 }
 ostream & let_type_rewriter_cell::display(ostream & out) const {
@@ -865,7 +865,7 @@ pair<expr, expr> let_value_rewriter_cell::operator()(environment const & env, co
         return rewrite_let_value(env, ctx, v, result_val);
     } else {
         type_inferer ti(env);
-        return make_pair(v, Refl(ti(v, ctx), v));
+        return make_pair(v, mk_refl_th(ti(v, ctx), v));
     }
 }
 ostream & let_value_rewriter_cell::display(ostream & out) const {
@@ -890,7 +890,7 @@ pair<expr, expr> let_body_rewriter_cell::operator()(environment const & env, con
         return rewrite_let_body(env, ctx, v, result_body);
     } else {
         type_inferer ti(env);
-        return make_pair(v, Refl(ti(v, ctx), v));
+        return make_pair(v, mk_refl_th(ti(v, ctx), v));
     }
 }
 ostream & let_body_rewriter_cell::display(ostream & out) const {
@@ -931,7 +931,7 @@ success_rewriter_cell::success_rewriter_cell():rewriter_cell(rewriter_kind::Succ
 success_rewriter_cell::~success_rewriter_cell() { }
 pair<expr, expr> success_rewriter_cell::operator()(environment const & env, context & ctx, expr const & v) const throw(rewriter_exception) {
     expr const & t = type_inferer(env)(v, ctx);
-    return make_pair(v, Refl(t, v));
+    return make_pair(v, mk_refl_th(t, v));
 }
 ostream & success_rewriter_cell::display(ostream & out) const {
     out << "Success_RW()";
@@ -951,7 +951,7 @@ pair<expr, expr> repeat_rewriter_cell::operator()(environment const & env, conte
                 break;
             expr const & ty = ti(v, ctx);
             result = make_pair(new_result.first,
-                               Trans(ty, v, result.first, new_result.first, result.second, new_result.second));
+                               mk_trans_th(ty, v, result.first, new_result.first, result.second, new_result.second));
         }
     } catch (rewriter_exception &) {
         return result;

src/library/rewriter/rewriter.h

@@ -382,14 +382,14 @@ class apply_rewriter_fn {
             result = rewrite_app(env, ctx, v, results);
             std::pair<expr, expr> tmp = m_rw(env, ctx, result.first);
             if (result.first != tmp.first) {
-                tmp.second = Trans(ty_v, v, result.first, tmp.first, result.second, tmp.second);
+                tmp.second = mk_trans_th(ty_v, v, result.first, tmp.first, result.second, tmp.second);
                 result = tmp;
             }
         }
             break;
-        case expr_kind::Eq: {
-            expr const & lhs = eq_lhs(v);
-            expr const & rhs = eq_rhs(v);
+        case expr_kind::HEq: {
+            expr const & lhs = heq_lhs(v);
+            expr const & rhs = heq_rhs(v);
             std::pair<expr, expr> result_lhs = apply(env, ctx, lhs);
             std::pair<expr, expr> result_rhs = apply(env, ctx, rhs);
             expr const & new_lhs = result_lhs.first;
@@ -408,12 +408,12 @@ class apply_rewriter_fn {
                     result = rewrite_eq_rhs(env, ctx, v, result_rhs);
                 } else {
                     // nothing changed
-                    result = std::make_pair(v, Refl(ti(v, ctx), v));
+                    result = std::make_pair(v, mk_refl_th(ti(v, ctx), v));
                 }
             }
             std::pair<expr, expr> tmp = m_rw(env, ctx, result.first);
             if (result.first != tmp.first) {
-                tmp.second = Trans(ty_v, v, result.first, tmp.first, result.second, tmp.second);
+                tmp.second = mk_trans_th(ty_v, v, result.first, tmp.first, result.second, tmp.second);
                 result = tmp;
             }
         }
@@ -439,12 +439,12 @@ class apply_rewriter_fn {
                     result = rewrite_lambda_body(env, ctx, v, result_body);
                 } else {
                     // nothing changed
-                    result = std::make_pair(v, Refl(ti(v, ctx), v));
+                    result = std::make_pair(v, mk_refl_th(ti(v, ctx), v));
                 }
             }
             std::pair<expr, expr> tmp = m_rw(env, ctx, result.first);
             if (result.first != tmp.first) {
-                tmp.second = Trans(ty_v, v, result.first, tmp.first, result.second, tmp.second);
+                tmp.second = mk_trans_th(ty_v, v, result.first, tmp.first, result.second, tmp.second);
                 result = tmp;
             }
         }
@@ -471,12 +471,12 @@ class apply_rewriter_fn {
                     result = rewrite_pi_body(env, ctx, v, result_body);
                 } else {
                     // nothing changed
-                    result = std::make_pair(v, Refl(ti(v, ctx), v));
+                    result = std::make_pair(v, mk_refl_th(ti(v, ctx), v));
                 }
             }
             std::pair<expr, expr> tmp = m_rw(env, ctx, result.first);
             if (result.first != tmp.first) {
-                tmp.second = Trans(ty_v, v, result.first, tmp.first, result.second, tmp.second);
+                tmp.second = mk_trans_th(ty_v, v, result.first, tmp.first, result.second, tmp.second);
                 result = tmp;
             }
         }
@@ -489,7 +489,7 @@ class apply_rewriter_fn {
 
             expr new_v = v;
             expr ty_v = ti(v, ctx);
-            expr pf = Refl(ty_v, v);
+            expr pf = mk_refl_th(ty_v, v);
             bool changed = false;
 
             if (ty) {
@@ -507,7 +507,7 @@ class apply_rewriter_fn {
             if (val != result_val.first) {
                 result = rewrite_let_value(env, ctx, new_v, result_val);
                 if (changed) {
-                    pf = Trans(ty_v, v, new_v, result.first, pf, result.second);
+                    pf = mk_trans_th(ty_v, v, new_v, result.first, pf, result.second);
                 } else {
                     pf = result.second;
                 }
@@ -520,7 +520,7 @@ class apply_rewriter_fn {
             if (body != result_body.first) {
                 result = rewrite_let_body(env, ctx, new_v, result_body);
                 if (changed) {
-                    pf = Trans(ty_v, v, new_v, result.first, pf, result.second);
+                    pf = mk_trans_th(ty_v, v, new_v, result.first, pf, result.second);
                 } else {
                     pf = result.second;
                 }
@@ -531,7 +531,7 @@ class apply_rewriter_fn {
 
             std::pair<expr, expr> tmp = m_rw(env, ctx, result.first);
             if (result.first != tmp.first) {
-                tmp.second = Trans(ty_v, v, result.first, tmp.first, result.second, tmp.second);
+                tmp.second = mk_trans_th(ty_v, v, result.first, tmp.first, result.second, tmp.second);
                 result = tmp;
             }
         }

src/library/tactic/boolean_tactics.cpp

@@ -44,7 +44,7 @@ tactic conj_tactic(bool all) {
                         for (auto nc : proof_info) {
                             name const & n = nc.first;
                             expr const & c = nc.second;
-                            new_m.insert(n, Conj(arg(c, 1), arg(c, 2), find(m, name(n, 1)), find(m, name(n, 2))));
+                            new_m.insert(n, mk_and_intro_th(arg(c, 1), arg(c, 2), find(m, name(n, 1)), find(m, name(n, 2))));
                             new_m.erase(name(n, 1));
                             new_m.erase(name(n, 2));
                         }
@@ -104,9 +104,9 @@ tactic conj_hyp_tactic(bool all) {
                                 expr const & H_1      = mk_constant(name(H_name, 1));
                                 expr const & H_2      = mk_constant(name(H_name, 2));
                                 if (occurs(H_1, pr))
-                                    pr = Let_simp(H_1, Conjunct1(arg(H_prop, 1), arg(H_prop, 2), mk_constant(H_name)), pr);
+                                    pr = Let_simp(H_1, mk_and_eliml_th(arg(H_prop, 1), arg(H_prop, 2), mk_constant(H_name)), pr);
                                 if (occurs(H_2, pr))
-                                    pr = Let_simp(H_2, Conjunct2(arg(H_prop, 1), arg(H_prop, 2), mk_constant(H_name)), pr);
+                                    pr = Let_simp(H_2, mk_and_elimr_th(arg(H_prop, 1), arg(H_prop, 2), mk_constant(H_name)), pr);
                             }
                             new_m.insert(goal_name, pr);
                         }
@@ -161,7 +161,7 @@ optional<proof_state> disj_hyp_tactic_core(name const & goal_name, name const &
             expr pr2 = find(m, name(goal_name, 2));
             pr1 = Fun(hyp_name, arg(Href, 1), pr1);
             pr2 = Fun(hyp_name, arg(Href, 2), pr2);
-            new_m.insert(goal_name, DisjCases(arg(Href, 1), arg(Href, 2), conclusion, mk_constant(hyp_name), pr1, pr2));
+            new_m.insert(goal_name, mk_or_elim_th(arg(Href, 1), arg(Href, 2), conclusion, mk_constant(hyp_name), pr1, pr2));
             new_m.erase(name(goal_name, 1));
             new_m.erase(name(goal_name, 2));
             return pb(new_m, a);
@@ -241,12 +241,12 @@ optional<proof_state_pair> disj_tactic(proof_state const & s, name gname) {
         proof_builder pb     = s.get_proof_builder();
         proof_builder new_pb1 = mk_proof_builder([=](proof_map const & m, assignment const & a) -> expr {
                 proof_map new_m(m);
-                new_m.insert(gname, Disj1(arg(*conclusion, 1), find(m, gname), arg(*conclusion, 2)));
+                new_m.insert(gname, mk_or_introl_th(arg(*conclusion, 1), find(m, gname), arg(*conclusion, 2)));
                 return pb(new_m, a);
             });
         proof_builder new_pb2 = mk_proof_builder([=](proof_map const & m, assignment const & a) -> expr {
                 proof_map new_m(m);
-                new_m.insert(gname, Disj2(arg(*conclusion, 2), arg(*conclusion, 1), find(m, gname)));
+                new_m.insert(gname, mk_or_intror_th(arg(*conclusion, 2), arg(*conclusion, 1), find(m, gname)));
                 return pb(new_m, a);
             });
         proof_state s1(precision::Over, new_gs1, s.get_menv(), new_pb1, s.get_cex_builder());
@@ -296,7 +296,7 @@ tactic absurd_tactic() {
                             expr a = arg(p1.second, 1);
                             for (auto const & p2 : g.get_hypotheses()) {
                                 if (p2.second == a) {
-                                    expr pr = AbsurdElim(a, c, mk_constant(p2.first), mk_constant(p1.first));
+                                    expr pr = mk_absurd_elim_th(a, c, mk_constant(p2.first), mk_constant(p1.first));
                                     proofs.emplace_front(gname, pr);
                                     return optional<goal>(); // remove goal
                                 }

src/library/tactic/tactic.cpp

@@ -215,7 +215,7 @@ tactic trivial_tactic() {
             goals new_gs = map_goals(s, [&](name const & gname, goal const & g) -> optional<goal> {
                     expr const & c  = env->normalize(g.get_conclusion(), context(), true);
                     if (c == True) {
-                        proofs.emplace_front(gname, Trivial);
+                        proofs.emplace_front(gname, mk_trivial());
                         return optional<goal>();
                     } else {
                         return some(g);

src/tests/kernel/environment.cpp

@@ -71,12 +71,12 @@ static void tst3() {
     } catch (exception const & ex) {
         std::cout << "expected error: " << ex.what() << "\n";
     }
-    env->add_definition("a", Int, iAdd(iVal(1), iVal(2)));
+    env->add_definition("a", Int, mk_Int_add(iVal(1), iVal(2)));
     std::cout << env << "\n";
-    expr t = iAdd(Const("a"), iVal(1));
+    expr t = mk_Int_add(Const("a"), iVal(1));
     std::cout << t << " --> " << normalize(t, env) << "\n";
     lean_assert(normalize(t, env) == iVal(4));
-    env->add_definition("b", Int, iMul(iVal(2), Const("a")));
+    env->add_definition("b", Int, mk_Int_mul(iVal(2), Const("a")));
     std::cout << "b --> " << normalize(Const("b"), env) << "\n";
     lean_assert(normalize(Const("b"), env) == iVal(6));
     try {
@@ -111,21 +111,21 @@ static void tst4() {
     std::cout << "tst4\n";
     environment env; init_test_frontend(env);
     env->add_definition("a", Int, iVal(1), true); // add opaque definition
-    expr t = iAdd(Const("a"), iVal(1));
+    expr t = mk_Int_add(Const("a"), iVal(1));
     std::cout << t << " --> " << normalize(t, env) << "\n";
     lean_assert(normalize(t, env) == t);
-    env->add_definition("b", Int, iAdd(Const("a"), iVal(1)));
-    expr t2 = iSub(Const("b"), iVal(9));
+    env->add_definition("b", Int, mk_Int_add(Const("a"), iVal(1)));
+    expr t2 = mk_Int_sub(Const("b"), iVal(9));
     std::cout << t2 << " --> " << normalize(t2, env) << "\n";
     lean_assert_eq(normalize(t2, env, context()),
-                   iSub(iAdd(Const("a"), iVal(1)), iVal(9)));
+                   mk_Int_sub(mk_Int_add(Const("a"), iVal(1)), iVal(9)));
 }
 
 static void tst5() {
     environment env; init_test_frontend(env);
     env->add_definition("a", Int, iVal(1), true); // add opaque definition
     try {
-        std::cout << type_check(iAdd(Const("a"), Int), env) << "\n";
+        std::cout << type_check(mk_Int_add(Const("a"), Int), env) << "\n";
         lean_unreachable();
     } catch (exception const & ex) {
         std::cout << "expected error: " << ex.what() << "\n";
@@ -163,10 +163,6 @@ static void tst7() {
     environment env; init_test_frontend(env);
     env->add_var("a", Int);
     env->add_var("b", Int);
-    expr t = If(Int, True, Const("a"), Const("b"));
-    std::cout << t << " --> " << normalize(t, env) << "\n";
-    std::cout << type_check(t, env) << "\n";
-    std::cout << "Environment\n" << env;
 }
 
 static void tst8() {
@@ -208,11 +204,11 @@ static void tst10() {
     expr a = Const("a");
     expr b = Const("b");
     expr c = Const("c");
-    env->add_definition("b", Int, iAdd(a, a));
+    env->add_definition("b", Int, mk_Int_add(a, a));
     lean_assert(env->get_object("b").get_weight() == 2);
-    env->add_definition("c", Int, iAdd(a, b));
+    env->add_definition("c", Int, mk_Int_add(a, b));
     lean_assert(env->get_object("c").get_weight() == 3);
-    env->add_definition("d", Int, iAdd(b, b));
+    env->add_definition("d", Int, mk_Int_add(b, b));
     lean_assert(env->get_object("d").get_weight() == 3);
 }
 

src/tests/kernel/expr.cpp

@@ -76,8 +76,8 @@ static unsigned depth2(expr const & e) {
             std::accumulate(begin_args(e), end_args(e), 0,
                             [](unsigned m, expr const & arg){ return std::max(depth2(arg), m); })
             + 1;
-    case expr_kind::Eq:
-        return std::max(depth2(eq_lhs(e)), depth2(eq_rhs(e))) + 1;
+    case expr_kind::HEq:
+        return std::max(depth2(heq_lhs(e)), depth2(heq_rhs(e))) + 1;
     case expr_kind::Lambda: case expr_kind::Pi:
         return std::max(depth2(abst_domain(e)), depth2(abst_body(e))) + 1;
     case expr_kind::Let:
@@ -135,8 +135,8 @@ static unsigned count_core(expr const & a, expr_set & s) {
     case expr_kind::App:
         return std::accumulate(begin_args(a), end_args(a), 1,
                           [&](unsigned sum, expr const & arg){ return sum + count_core(arg, s); });
-    case expr_kind::Eq:
-        return count_core(eq_lhs(a), s) + count_core(eq_rhs(a), s) + 1;
+    case expr_kind::HEq:
+        return count_core(heq_lhs(a), s) + count_core(heq_rhs(a), s) + 1;
     case expr_kind::Lambda: case expr_kind::Pi:
         return count_core(abst_domain(a), s) + count_core(abst_body(a), s) + 1;
     case expr_kind::Let:
@@ -295,7 +295,7 @@ static void tst13() {
 }
 
 static void tst14() {
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     check_serializer(t);
     std::cout << t << "\n";
     expr l = mk_let("a", none_expr(), Const("b"), Var(0));
@@ -329,9 +329,9 @@ static void tst15() {
     lean_assert(has_metavar(f(a, a, m)));
     lean_assert(has_metavar(f(a, m, a, a)));
     lean_assert(!has_metavar(f(a, a, a, a)));
-    lean_assert(!has_metavar(Eq(a, f(a))));
-    lean_assert(has_metavar(Eq(m, f(a))));
-    lean_assert(has_metavar(Eq(a, f(m))));
+    lean_assert(!has_metavar(HEq(a, f(a))));
+    lean_assert(has_metavar(HEq(m, f(a))));
+    lean_assert(has_metavar(HEq(a, f(m))));
 }
 
 static void check_copy(expr const & e) {
@@ -346,7 +346,7 @@ static void tst16() {
     expr a = Const("a");
     check_copy(iVal(10));
     check_copy(f(a));
-    check_copy(Eq(f(a), a));
+    check_copy(HEq(f(a), a));
     check_copy(mk_metavar("M"));
     check_copy(mk_lambda("x", a, Var(0)));
     check_copy(mk_pi("x", a, Var(0)));

src/tests/kernel/free_vars.cpp

@@ -75,11 +75,11 @@ static void tst4() {
     lean_assert(fn(Fun({x, Type()}, Var(0))) == 0);
     lean_assert(fn(Fun({x, Var(0)}, Var(0))) == 1);
     lean_assert(fn(Fun({x, Var(0)}, Var(2))) == 2);
-    lean_assert(fn(Fun({x, Var(0)}, Eq(Var(2), Var(1)))) == 2);
+    lean_assert(fn(Fun({x, Var(0)}, HEq(Var(2), Var(1)))) == 2);
     lean_assert(fn(Pi({x, Type()}, Var(0))) == 0);
     lean_assert(fn(Pi({x, Var(0)}, Var(0))) == 1);
     lean_assert(fn(Pi({x, Var(0)}, Var(2))) == 2);
-    lean_assert(fn(Pi({x, Var(0)}, Eq(Var(2), Var(1)))) == 2);
+    lean_assert(fn(Pi({x, Var(0)}, HEq(Var(2), Var(1)))) == 2);
     context ctx;
     ctx = extend(ctx, name("x"), Bool);
     ctx = extend(ctx, name("y"), Bool);

src/tests/kernel/metavar.cpp

@@ -275,7 +275,7 @@ static void tst14() {
     expr y = Const("y");
     env->add_var("h", Pi({N, Type()}, N >> (N >> N)));
     expr F1 = Fun({{N, Type()}, {a, N}, {f, N >> N}},
-                  (Fun({{x, N}, {y, N}}, Eq(f(m1), y)))(a));
+                  (Fun({{x, N}, {y, N}}, HEq(f(m1), y)))(a));
     metavar_env menv2 = menv.copy();
     menv2->assign(m1, h(Var(4), Var(1), Var(3)));
     normalizer norm(env);
@@ -288,10 +288,10 @@ static void tst14() {
     lean_assert(menv2->instantiate_metavars(norm(F1)) ==
                 norm(menv2->instantiate_metavars(F1)));
     expr F2 = (Fun({{N, Type()}, {f, N >> N}, {a, N}, {b, N}},
-                   (Fun({{x, N}, {y, N}}, Eq(f(m1), y)))(a, m2)))(M);
+                   (Fun({{x, N}, {y, N}}, HEq(f(m1), y)))(a, m2)))(M);
     std::cout << norm(F2) << "\n";
     expr F3 = (Fun({{N, Type()}, {f, N >> N}, {a, N}, {b, N}},
-                   (Fun({{x, N}, {y, N}}, Eq(f(m1), y)))(b, m2)))(M);
+                   (Fun({{x, N}, {y, N}}, HEq(f(m1), y)))(b, m2)))(M);
     std::cout << norm(F3) << "\n";
 }
 

src/tests/kernel/normalizer.cpp

@@ -76,8 +76,8 @@ unsigned count_core(expr const & a, expr_set & s) {
     case expr_kind::App:
         return std::accumulate(begin_args(a), end_args(a), 1,
                                [&](unsigned sum, expr const & arg){ return sum + count_core(arg, s); });
-    case expr_kind::Eq:
-        return count_core(eq_lhs(a), s) + count_core(eq_rhs(a), s) + 1;
+    case expr_kind::HEq:
+        return count_core(heq_lhs(a), s) + count_core(heq_rhs(a), s) + 1;
     case expr_kind::Lambda: case expr_kind::Pi:
         return count_core(abst_domain(a), s) + count_core(abst_body(a), s) + 1;
     case expr_kind::Let:
@@ -197,9 +197,9 @@ static void tst3() {
     env->add_var("a", Bool);
     expr t1 = Const("a");
     expr t2 = Const("a");
-    expr e = Eq(t1, t2);
+    expr e = HEq(t1, t2);
     std::cout << e << " --> " << normalize(e, env) << "\n";
-    lean_assert(normalize(e, env) == Eq(t1, t2));
+    lean_assert(normalize(e, env) == HEq(t1, t2));
 }
 
 static void tst4() {
@@ -286,7 +286,7 @@ static void tst8() {
     expr P = Const("P");
     expr v0 = Var(0);
     expr v1 = Var(1);
-    expr F = mk_forall(Int, mk_lambda("x", Int, Not(mk_lambda("x", Int, mk_exists(Int, mk_lambda("y", Int, P(v1, v0))))(v0))));
+    expr F = mk_pi("x", Int, Not(mk_lambda("x", Int, mk_exists(Int, mk_lambda("y", Int, P(v1, v0))))(v0)));
     normalizer proc(env);
     expr N1 = proc(F);
     expr N2 = proc(deep_copy(F));

src/tests/kernel/type_checker.cpp

@@ -80,7 +80,7 @@ static void tst2() {
 static void tst3() {
     environment env;
     init_test_frontend(env);
-    expr f = Fun("a", Bool, Eq(Const("a"), True));
+    expr f = Fun("a", Bool, HEq(Const("a"), True));
     std::cout << type_check(f, env) << "\n";
     lean_assert(type_check(f, env) == mk_arrow(Bool, Bool));
     expr t = mk_let("a", none_expr(), True, Var(0));
@@ -90,7 +90,7 @@ static void tst3() {
 static void tst4() {
     environment env;
     init_test_frontend(env);
-    expr a = Eq(iVal(1), iVal(2));
+    expr a = HEq(iVal(1), iVal(2));
     expr pr   = mk_lambda("x", a, Var(0));
     std::cout << type_check(pr, env) << "\n";
 }
@@ -105,7 +105,7 @@ static void tst5() {
     expr prop = P;
     expr pr   = H;
     for (unsigned i = 1; i < n; i++) {
-        pr   = Conj(P, prop, H, pr);
+        pr   = mk_and_intro_th(P, prop, H, pr);
         prop = And(P, prop);
     }
 }
@@ -191,8 +191,8 @@ static void tst10() {
     expr t1 = Let({{a, f(b)}, {a, f(a)}}, f(a));
     expr t2 = f(f(f(b)));
     std::cout << t1 << " --> " << normalize(t1, env) << "\n";
-    expr prop  = Eq(t1, t2);
-    expr proof = Refl(Int, t1);
+    expr prop  = HEq(t1, t2);
+    expr proof = mk_refl_th(Int, t1);
     env->add_theorem("simp_eq", prop, proof);
     std::cout << env->get_object("simp_eq").get_name() << "\n";
 }
@@ -215,8 +215,8 @@ static void tst11() {
         t3 = f(t3, t3);
     }
     lean_assert(t1 != t2);
-    env->add_theorem("eqs1", Eq(t1, t2), Refl(Int, t1));
-    env->add_theorem("eqs2", Eq(t1, t3), Refl(Int, t1));
+    env->add_theorem("eqs1", HEq(t1, t2), mk_refl_th(Int, t1));
+    env->add_theorem("eqs2", HEq(t1, t3), mk_refl_th(Int, t1));
 }
 
 static expr mk_big(unsigned depth) {
@@ -257,7 +257,7 @@ static void tst13() {
     env->add_var("f", Type() >> Type());
     expr f = Const("f");
     std::cout << type_check(f(Bool), env) << "\n";
-    std::cout << type_check(f(Eq(True, False)), env) << "\n";
+    std::cout << type_check(f(HEq(True, False)), env) << "\n";
 }
 
 static void tst14() {
@@ -383,8 +383,8 @@ static void tst18() {
     lean_assert(type_of(f(a, a)) == Int);
     lean_assert(type_of(f(a)) == Int >> Int);
     lean_assert(is_bool(type_of(And(a, f(a)))));
-    lean_assert(type_of(Fun({a, Int}, iAdd(a, iVal(1)))) == Int >> Int);
-    lean_assert(type_of(Let({a, iVal(10)}, iAdd(a, b))) == Int);
+    lean_assert(type_of(Fun({a, Int}, mk_Int_add(a, iVal(1)))) == Int >> Int);
+    lean_assert(type_of(Let({a, iVal(10)}, mk_Int_add(a, b))) == Int);
     lean_assert(type_of(Type()) == Type(level() + 1));
     lean_assert(type_of(Bool) == Type());
     lean_assert(type_of(Pi({a, Type()}, Type(level() + 2))) == Type(level() + 3));
@@ -399,7 +399,7 @@ static expr mk_big(unsigned val, unsigned depth) {
     if (depth == 0)
         return iVal(val);
     else
-        return iAdd(mk_big(val*2, depth-1), mk_big(val*2 + 1, depth-1));
+        return mk_Int_add(mk_big(val*2, depth-1), mk_big(val*2 + 1, depth-1));
 }
 
 static void tst19() {

src/tests/library/arith.cpp

@@ -22,28 +22,28 @@ static void tst0() {
     env->add_var("n", Nat);
     expr n = Const("n");
     lean_assert_eq(mk_nat_type(), Nat);
-    lean_assert_eq(norm(nMul(nVal(2), nVal(3))), nVal(6));
-    lean_assert_eq(norm(nLe(nVal(2), nVal(3))), True);
-    lean_assert_eq(norm(nLe(nVal(5), nVal(3))), False);
-    lean_assert_eq(norm(nLe(nVal(2), nVal(3))), True);
-    lean_assert_eq(norm(nLe(n, nVal(3))), nLe(n, nVal(3)));
+    lean_assert_eq(norm(mk_Nat_mul(nVal(2), nVal(3))), nVal(6));
+    lean_assert_eq(norm(mk_Nat_le(nVal(2), nVal(3))), True);
+    lean_assert_eq(norm(mk_Nat_le(nVal(5), nVal(3))), False);
+    lean_assert_eq(norm(mk_Nat_le(nVal(2), nVal(3))), True);
+    lean_assert_eq(norm(mk_Nat_le(n, nVal(3))), mk_Nat_le(n, nVal(3)));
     env->add_var("x", Real);
     expr x = Const("x");
     lean_assert_eq(mk_real_type(), Real);
-    lean_assert_eq(norm(rMul(rVal(2), rVal(3))), rVal(6));
-    lean_assert_eq(norm(rDiv(rVal(2), rVal(0))), rVal(0));
-    lean_assert_eq(norm(rLe(rVal(2),  rVal(3))), True);
-    lean_assert_eq(norm(rLe(rVal(5),  rVal(3))), False);
-    lean_assert_eq(norm(rLe(rVal(2),  rVal(3))), True);
-    lean_assert_eq(norm(rLe(x, rVal(3))), rLe(x, rVal(3)));
+    lean_assert_eq(norm(mk_Real_mul(rVal(2), rVal(3))), rVal(6));
+    lean_assert_eq(norm(mk_Real_div(rVal(2), rVal(0))), rVal(0));
+    lean_assert_eq(norm(mk_Real_le(rVal(2),  rVal(3))), True);
+    lean_assert_eq(norm(mk_Real_le(rVal(5),  rVal(3))), False);
+    lean_assert_eq(norm(mk_Real_le(rVal(2),  rVal(3))), True);
+    lean_assert_eq(norm(mk_Real_le(x, rVal(3))), mk_Real_le(x, rVal(3)));
     env->add_var("i", Int);
     expr i = Const("i");
-    lean_assert_eq(norm(i2r(i)), i2r(i));
-    lean_assert_eq(norm(i2r(iVal(2))), rVal(2));
+    lean_assert_eq(norm(mk_int_to_real(i)), mk_int_to_real(i));
+    lean_assert_eq(norm(mk_int_to_real(iVal(2))), rVal(2));
     lean_assert_eq(mk_int_type(), Int);
-    lean_assert_eq(norm(n2i(n)), n2i(n));
-    lean_assert_eq(norm(n2i(nVal(2))), iVal(2));
-    lean_assert_eq(norm(iDiv(iVal(2), iVal(0))), iVal(0));
+    lean_assert_eq(norm(mk_nat_to_int(n)), mk_nat_to_int(n));
+    lean_assert_eq(norm(mk_nat_to_int(nVal(2))), iVal(2));
+    lean_assert_eq(norm(mk_Int_div(iVal(2), iVal(0))), iVal(0));
 }
 
 static void tst1() {
@@ -58,76 +58,76 @@ static void tst1() {
 static void tst2() {
     environment env;
     init_test_frontend(env);
-    expr e = iAdd(iVal(10), iVal(30));
+    expr e = mk_Int_add(iVal(10), iVal(30));
     std::cout << e << "\n";
     std::cout << normalize(e, env) << "\n";
     lean_assert(normalize(e, env) == iVal(40));
-    std::cout << type_check(mk_int_add_fn(), env) << "\n";
+    std::cout << type_check(mk_Int_add_fn(), env) << "\n";
     lean_assert(type_check(e, env) == Int);
-    lean_assert(type_check(mk_app(mk_int_add_fn(), iVal(10)), env) == (Int >> Int));
+    lean_assert(type_check(mk_app(mk_Int_add_fn(), iVal(10)), env) == (Int >> Int));
     lean_assert(is_int_value(normalize(e, env)));
-    expr e2 = Fun("a", Int, iAdd(Const("a"), iAdd(iVal(10), iVal(30))));
+    expr e2 = Fun("a", Int, mk_Int_add(Const("a"), mk_Int_add(iVal(10), iVal(30))));
     std::cout << e2 << " --> " << normalize(e2, env) << "\n";
     lean_assert(type_check(e2, env) == mk_arrow(Int, Int));
-    lean_assert(normalize(e2, env) == Fun("a", Int, iAdd(Const("a"), iVal(40))));
+    lean_assert(normalize(e2, env) == Fun("a", Int, mk_Int_add(Const("a"), iVal(40))));
 }
 
 static void tst3() {
     environment env;
     init_test_frontend(env);
-    expr e = iMul(iVal(10), iVal(30));
+    expr e = mk_Int_mul(iVal(10), iVal(30));
     std::cout << e << "\n";
     std::cout << normalize(e, env) << "\n";
     lean_assert(normalize(e, env) == iVal(300));
-    std::cout << type_check(mk_int_mul_fn(), env) << "\n";
+    std::cout << type_check(mk_Int_mul_fn(), env) << "\n";
     lean_assert(type_check(e, env) == Int);
-    lean_assert(type_check(mk_app(mk_int_mul_fn(), iVal(10)), env) == mk_arrow(Int, Int));
+    lean_assert(type_check(mk_app(mk_Int_mul_fn(), iVal(10)), env) == mk_arrow(Int, Int));
     lean_assert(is_int_value(normalize(e, env)));
-    expr e2 = Fun("a", Int, iMul(Const("a"), iMul(iVal(10), iVal(30))));
+    expr e2 = Fun("a", Int, mk_Int_mul(Const("a"), mk_Int_mul(iVal(10), iVal(30))));
     std::cout << e2 << " --> " << normalize(e2, env) << "\n";
     lean_assert(type_check(e2, env) == (Int >> Int));
-    lean_assert(normalize(e2, env) == Fun("a", Int, iMul(Const("a"), iVal(300))));
+    lean_assert(normalize(e2, env) == Fun("a", Int, mk_Int_mul(Const("a"), iVal(300))));
 }
 
 static void tst4() {
     environment env;
     init_test_frontend(env);
-    expr e = iSub(iVal(10), iVal(30));
+    expr e = mk_Int_sub(iVal(10), iVal(30));
     std::cout << e << "\n";
     std::cout << normalize(e, env) << "\n";
     lean_assert(normalize(e, env, context(), true) == iVal(-20));
-    std::cout << type_check(mk_int_sub_fn(), env) << "\n";
+    std::cout << type_check(mk_Int_sub_fn(), env) << "\n";
     lean_assert(type_check(e, env) == Int);
-    lean_assert(type_check(mk_app(mk_int_sub_fn(), iVal(10)), env) == mk_arrow(Int, Int));
+    lean_assert(type_check(mk_app(mk_Int_sub_fn(), iVal(10)), env) == mk_arrow(Int, Int));
     lean_assert(is_int_value(normalize(e, env, context(), true)));
-    expr e2 = Fun("a", Int, iSub(Const("a"), iSub(iVal(10), iVal(30))));
+    expr e2 = Fun("a", Int, mk_Int_sub(Const("a"), mk_Int_sub(iVal(10), iVal(30))));
     std::cout << e2 << " --> " << normalize(e2, env) << "\n";
     lean_assert(type_check(e2, env) == (Int >> Int));
-    lean_assert_eq(normalize(e2, env, context(), true), Fun("a", Int, iAdd(Const("a"), iVal(20))));
+    lean_assert_eq(normalize(e2, env, context(), true), Fun("a", Int, mk_Int_add(Const("a"), iVal(20))));
 }
 
 static void tst5() {
     environment env;
     init_test_frontend(env);
     env->add_var(name("a"), Int);
-    expr e = Eq(iVal(3), iVal(4));
+    expr e = HEq(iVal(3), iVal(4));
     std::cout << e << " --> " << normalize(e, env) << "\n";
     lean_assert(normalize(e, env) == False);
-    lean_assert(normalize(Eq(Const("a"), iVal(3)), env) == Eq(Const("a"), iVal(3)));
+    lean_assert(normalize(HEq(Const("a"), iVal(3)), env) == HEq(Const("a"), iVal(3)));
 }
 
 static void tst6() {
     std::cout << "tst6\n";
-    std::cout << mk_int_add_fn().raw() << "\n";
-    std::cout << mk_int_add_fn().raw() << "\n";
+    std::cout << mk_Int_add_fn().raw() << "\n";
+    std::cout << mk_Int_add_fn().raw() << "\n";
 
     #ifndef __APPLE__
-    thread t1([](){ save_stack_info(); std::cout << "t1: " << mk_int_add_fn().raw() << "\n"; });
+    thread t1([](){ save_stack_info(); std::cout << "t1: " << mk_Int_add_fn().raw() << "\n"; });
     t1.join();
-    thread t2([](){ save_stack_info(); std::cout << "t2: " << mk_int_add_fn().raw() << "\n"; });
+    thread t2([](){ save_stack_info(); std::cout << "t2: " << mk_Int_add_fn().raw() << "\n"; });
     t2.join();
     #endif
-    std::cout << mk_int_add_fn().raw() << "\n";
+    std::cout << mk_Int_add_fn().raw() << "\n";
 }
 
 int main() {

src/tests/library/deep_copy.cpp

@@ -18,7 +18,7 @@ static void tst1() {
     expr z = Const("z");
     local_context lctx{mk_lift(0, 1), mk_inst(0, a)};
     expr m = mk_metavar("a", lctx);
-    expr F = mk_let("z", Type(), Type(level()+1), mk_pi("y", t, mk_lambda("x", t, f(f(f(x, a), Const("10")), Eq(x, m)))));
+    expr F = mk_let("z", Type(), Type(level()+1), mk_pi("y", t, mk_lambda("x", t, f(f(f(x, a), Const("10")), HEq(x, m)))));
     expr G = deep_copy(F);
     lean_assert(F == G);
     lean_assert(!is_eqp(F, G));

src/tests/library/elaborator/elaborator.cpp

@@ -95,7 +95,7 @@ static void tst2() {
     expr F  = m1(g(m2, m3(a)), m4(nVal(0)));
     std::cout << F << "\n";
     std::cout << checker.check(F, context(), menv, ucs) << "\n";
-    ucs.push_back(mk_choice_constraint(context(), m1, { mk_nat_le_fn(), mk_int_le_fn(), mk_real_le_fn() }, justification()));
+    ucs.push_back(mk_choice_constraint(context(), m1, { mk_Nat_le_fn(), mk_Int_le_fn(), mk_Real_le_fn() }, justification()));
     ucs.push_back(mk_choice_constraint(context(), m3, { int_id, mk_int_to_real_fn() }, justification()));
     ucs.push_back(mk_choice_constraint(context(), m4, { nat_id, mk_nat_to_int_fn(), mk_nat_to_real_fn() }, justification()));
     elaborator elb(env, menv, ucs.size(), ucs.data());
@@ -138,7 +138,7 @@ static void tst3() {
     expr F = Fun({x, m1}, m2(f(m3, x), m4(nVal(10))))(m5(a));
     std::cout << F << "\n";
     std::cout << checker.check(F, context(), menv, ucs) << "\n";
-    ucs.push_back(mk_choice_constraint(context(), m2, { mk_nat_le_fn(), mk_int_le_fn(), mk_real_le_fn() }, justification()));
+    ucs.push_back(mk_choice_constraint(context(), m2, { mk_Nat_le_fn(), mk_Int_le_fn(), mk_Real_le_fn() }, justification()));
     ucs.push_back(mk_choice_constraint(context(), m4, { nat_id, mk_nat_to_int_fn(), mk_nat_to_real_fn() }, justification()));
     ucs.push_back(mk_choice_constraint(context(), m5, { int_id, mk_int_to_real_fn() }, justification()));
     elaborator elb(env, menv, ucs.size(), ucs.data());
@@ -183,7 +183,7 @@ static void tst4() {
     expr F = Fun({{x, m1}, {y, m2}}, m3(f(m4, x), f(m5, y)))(m6(a), b);
     std::cout << F << "\n";
     std::cout << checker.check(F, context(), menv, ucs) << "\n";
-    ucs.push_back(mk_choice_constraint(context(), m3, { mk_nat_le_fn(), mk_int_le_fn(), mk_real_le_fn() }, justification()));
+    ucs.push_back(mk_choice_constraint(context(), m3, { mk_Nat_le_fn(), mk_Int_le_fn(), mk_Real_le_fn() }, justification()));
     ucs.push_back(mk_choice_constraint(context(), m6, { int_id, mk_int_to_real_fn() }, justification()));
     elaborator elb(env, menv, ucs.size(), ucs.data());
     elb.next();
@@ -257,10 +257,10 @@ static void tst6() {
     env->add_var("f", Int >> (Int >> Int));
     env->add_var("a", Int);
     env->add_var("b", Int);
-    env->add_axiom("H1", Eq(f(a, f(a, b)), a));
-    env->add_axiom("H2", Eq(a, b));
-    expr V = Subst(m1, m2, m3, m4, H1, H2);
-    expr expected = Eq(f(a, f(b, b)), a);
+    env->add_axiom("H1", HEq(f(a, f(a, b)), a));
+    env->add_axiom("H2", HEq(a, b));
+    expr V = mk_subst_th(m1, m2, m3, m4, H1, H2);
+    expr expected = HEq(f(a, f(b, b)), a);
     expr given    = checker.check(V, context(), menv, ucs);
     ucs.push_back(mk_eq_constraint(context(), expected, given, justification()));
     elaborator elb(env, menv, ucs.size(), ucs.data());
@@ -339,17 +339,17 @@ static void tst8() {
     env->add_var("a", Bool);
     env->add_var("b", Bool);
     env->add_var("c", Bool);
-    env->add_axiom("H1", Eq(a, b));
-    env->add_axiom("H2", Eq(b, c));
-    success(Trans(_, _, _, _, H1, H2), Trans(Bool, a, b, c, H1, H2), env);
-    success(Trans(_, _, _, _, Symm(_, _, _, H2), Symm(_, _, _, H1)),
-            Trans(Bool, c, b, a, Symm(Bool, b, c, H2), Symm(Bool, a, b, H1)), env);
-    success(Symm(_, _, _, Trans(_, _ , _ , _ , Symm(_, _, _, H2), Symm(_, _, _, H1))),
-            Symm(Bool, c, a, Trans(Bool, c, b, a, Symm(Bool, b, c, H2), Symm(Bool, a, b, H1))), env);
+    env->add_axiom("H1", HEq(a, b));
+    env->add_axiom("H2", HEq(b, c));
+    success(mk_trans_th(_, _, _, _, H1, H2), mk_trans_th(Bool, a, b, c, H1, H2), env);
+    success(mk_trans_th(_, _, _, _, mk_symm_th(_, _, _, H2), mk_symm_th(_, _, _, H1)),
+            mk_trans_th(Bool, c, b, a, mk_symm_th(Bool, b, c, H2), mk_symm_th(Bool, a, b, H1)), env);
+    success(mk_symm_th(_, _, _, mk_trans_th(_, _ , _ , _ , mk_symm_th(_, _, _, H2), mk_symm_th(_, _, _, H1))),
+            mk_symm_th(Bool, c, a, mk_trans_th(Bool, c, b, a, mk_symm_th(Bool, b, c, H2), mk_symm_th(Bool, a, b, H1))), env);
     env->add_axiom("H3", a);
     expr H3 = Const("H3");
-    success(EqTIntro(_, EqMP(_, _, Symm(_, _, _, Trans(_, _, _, _, Symm(_, _, _, H2), Symm(_, _, _, H1))), H3)),
-            EqTIntro(c, EqMP(a, c, Symm(Bool, c, a, Trans(Bool, c, b, a, Symm(Bool, b, c, H2), Symm(Bool, a, b, H1))), H3)),
+    success(mk_eqt_intro_th(_, mk_eqmp_th(_, _, mk_symm_th(_, _, _, mk_trans_th(_, _, _, _, mk_symm_th(_, _, _, H2), mk_symm_th(_, _, _, H1))), H3)),
+            mk_eqt_intro_th(c, mk_eqmp_th(a, c, mk_symm_th(Bool, c, a, mk_trans_th(Bool, c, b, a, mk_symm_th(Bool, b, c, H2), mk_symm_th(Bool, a, b, H1))), H3)),
             env);
 }
 
@@ -370,19 +370,19 @@ static void tst9() {
     expr fact = Const("fact");
     env->add_var("a", Nat);
     env->add_var("b", Nat);
-    env->add_definition("fact", Bool, Eq(a, b));
+    env->add_definition("fact", Bool, HEq(a, b));
     env->add_axiom("H", fact);
-    success(Congr2(_, _, _, _, f, H),
-            Congr2(Nat, Fun({n, Nat}, vec(n) >> Nat), a, b, f, H), env);
+    success(mk_congr2_th(_, _, _, _, f, H),
+            mk_congr2_th(Nat, Fun({n, Nat}, vec(n) >> Nat), a, b, f, H), env);
     env->add_var("g", Pi({n, Nat}, vec(n) >> Nat));
     expr g = Const("g");
-    env->add_axiom("H2", Eq(f, g));
+    env->add_axiom("H2", HEq(f, g));
     expr H2 = Const("H2");
-    success(Congr(_, _, _, _, _, _, H2, H),
-            Congr(Nat, Fun({n, Nat}, vec(n) >> Nat), f, g, a, b, H2, H), env);
-    success(Congr(_, _, _, _, _, _, Refl(_, f), H),
-            Congr(Nat, Fun({n, Nat}, vec(n) >> Nat), f, f, a, b, Refl(Pi({n, Nat}, vec(n) >> Nat), f), H), env);
-    success(Refl(_, a), Refl(Nat, a), env);
+    success(mk_congr_th(_, _, _, _, _, _, H2, H),
+            mk_congr_th(Nat, Fun({n, Nat}, vec(n) >> Nat), f, g, a, b, H2, H), env);
+    success(mk_congr_th(_, _, _, _, _, _, mk_refl_th(_, f), H),
+            mk_congr_th(Nat, Fun({n, Nat}, vec(n) >> Nat), f, f, a, b, mk_refl_th(Pi({n, Nat}, vec(n) >> Nat), f), H), env);
+    success(mk_refl_th(_, a), mk_refl_th(Nat, a), env);
 }
 
 static void tst10() {
@@ -402,11 +402,11 @@ static void tst10() {
     expr z   = Const("z");
     success(Fun({{x, _}, {y, _}}, f(x, y)),
             Fun({{x, Nat}, {y, R >> Nat}}, f(x, y)), env);
-    success(Fun({{x, _}, {y, _}, {z, _}}, Eq(f(x, y), f(x, z))),
-            Fun({{x, Nat}, {y, R >> Nat}, {z, R >> Nat}}, Eq(f(x, y), f(x, z))), env);
+    success(Fun({{x, _}, {y, _}, {z, _}}, HEq(f(x, y), f(x, z))),
+            Fun({{x, Nat}, {y, R >> Nat}, {z, R >> Nat}}, HEq(f(x, y), f(x, z))), env);
     expr A   = Const("A");
-    success(Fun({{A, Type()}, {x, _}, {y, _}, {z, _}}, Eq(f(x, y), f(x, z))),
-            Fun({{A, Type()}, {x, Nat}, {y, R >> Nat}, {z, R >> Nat}}, Eq(f(x, y), f(x, z))), env);
+    success(Fun({{A, Type()}, {x, _}, {y, _}, {z, _}}, HEq(f(x, y), f(x, z))),
+            Fun({{A, Type()}, {x, Nat}, {y, R >> Nat}, {z, R >> Nat}}, HEq(f(x, y), f(x, z))), env);
 }
 
 static void tst11() {
@@ -464,11 +464,11 @@ static void tst13() {
             Fun({{A, Type()}, {x, A}}, f(A, x)), env);
     success(Fun({{A, Type()}, {B, Type()}, {x, _}}, f(A, x)),
             Fun({{A, Type()}, {B, Type()}, {x, A}}, f(A, x)), env);
-    success(Fun({{A, Type()}, {B, Type()}, {x, _}}, Eq(f(B, x), f(_, x))),
-            Fun({{A, Type()}, {B, Type()}, {x, B}}, Eq(f(B, x), f(B, x))), env);
-    success(Fun({{A, Type()}, {B, Type()}, {x, _}}, Eq(f(B, x), f(_, x))),
-            Fun({{A, Type()}, {B, Type()}, {x, B}}, Eq(f(B, x), f(B, x))), env);
-    unsolved(Fun({{A, _}, {B, _}, {x, _}}, Eq(f(B, x), f(_, x))), env);
+    success(Fun({{A, Type()}, {B, Type()}, {x, _}}, HEq(f(B, x), f(_, x))),
+            Fun({{A, Type()}, {B, Type()}, {x, B}}, HEq(f(B, x), f(B, x))), env);
+    success(Fun({{A, Type()}, {B, Type()}, {x, _}}, HEq(f(B, x), f(_, x))),
+            Fun({{A, Type()}, {B, Type()}, {x, B}}, HEq(f(B, x), f(B, x))), env);
+    unsolved(Fun({{A, _}, {B, _}, {x, _}}, HEq(f(B, x), f(_, x))), env);
 }
 
 static void tst14() {
@@ -495,20 +495,20 @@ static void tst14() {
             env);
     success(Fun({g, Pi({A, Type()}, A >> (A >> Bool))},
                 g(_,
-                  Fun({{x, _}, {y, _}}, Eq(f(_, x), f(_, y))),
+                  Fun({{x, _}, {y, _}}, HEq(f(_, x), f(_, y))),
                   Fun({{x, N}, {y, Bool}}, True))),
             Fun({g, Pi({A, Type()}, A >> (A >> Bool))},
                 g((N >> (Bool >> Bool)),
-                  Fun({{x, N}, {y, Bool}}, Eq(f(N, x), f(Bool, y))),
+                  Fun({{x, N}, {y, Bool}}, HEq(f(N, x), f(Bool, y))),
                   Fun({{x, N}, {y, Bool}}, True))), env);
 
     success(Fun({g, Pi({A, Type()}, A >> (A >> Bool))},
                 g(_,
-                  Fun({{x, N}, {y, _}}, Eq(f(_, x), f(_, y))),
+                  Fun({{x, N}, {y, _}}, HEq(f(_, x), f(_, y))),
                   Fun({{x, _}, {y, Bool}}, True))),
             Fun({g, Pi({A, Type()}, A >> (A >> Bool))},
                 g((N >> (Bool >> Bool)),
-                  Fun({{x, N}, {y, Bool}}, Eq(f(N, x), f(Bool, y))),
+                  Fun({{x, N}, {y, Bool}}, HEq(f(N, x), f(Bool, y))),
                   Fun({{x, N}, {y, Bool}}, True))), env);
 }
 
@@ -550,29 +550,29 @@ static void tst16() {
     env->add_var("a", Bool);
     env->add_var("b", Bool);
     env->add_var("c", Bool);
-    success(Fun({{H1, Eq(a, b)}, {H2, Eq(b, c)}},
-                Trans(_, _, _, _, H1, H2)),
-            Fun({{H1, Eq(a, b)}, {H2, Eq(b, c)}},
-                Trans(Bool, a, b, c, H1, H2)),
+    success(Fun({{H1, HEq(a, b)}, {H2, HEq(b, c)}},
+                mk_trans_th(_, _, _, _, H1, H2)),
+            Fun({{H1, HEq(a, b)}, {H2, HEq(b, c)}},
+                mk_trans_th(Bool, a, b, c, H1, H2)),
             env);
     expr H3 = Const("H3");
-    success(Fun({{H1, Eq(a, b)}, {H2, Eq(b, c)}, {H3, a}},
-                EqTIntro(_, EqMP(_, _, Symm(_, _, _, Trans(_, _, _, _, Symm(_, _, _, H2), Symm(_, _, _, H1))), H3))),
-            Fun({{H1, Eq(a, b)}, {H2, Eq(b, c)}, {H3, a}},
-                EqTIntro(c, EqMP(a, c, Symm(Bool, c, a, Trans(Bool, c, b, a, Symm(Bool, b, c, H2), Symm(Bool, a, b, H1))), H3))),
+    success(Fun({{H1, HEq(a, b)}, {H2, HEq(b, c)}, {H3, a}},
+                mk_eqt_intro_th(_, mk_eqmp_th(_, _, mk_symm_th(_, _, _, mk_trans_th(_, _, _, _, mk_symm_th(_, _, _, H2), mk_symm_th(_, _, _, H1))), H3))),
+            Fun({{H1, HEq(a, b)}, {H2, HEq(b, c)}, {H3, a}},
+                mk_eqt_intro_th(c, mk_eqmp_th(a, c, mk_symm_th(Bool, c, a, mk_trans_th(Bool, c, b, a, mk_symm_th(Bool, b, c, H2), mk_symm_th(Bool, a, b, H1))), H3))),
             env);
     environment env2;
     init_test_frontend(env2);
-    success(Fun({{a, Bool}, {b, Bool}, {c, Bool}, {H1, Eq(a, b)}, {H2, Eq(b, c)}, {H3, a}},
-                EqTIntro(_, EqMP(_, _, Symm(_, _, _, Trans(_, _, _, _, Symm(_, _, _, H2), Symm(_, _, _, H1))), H3))),
-            Fun({{a, Bool}, {b, Bool}, {c, Bool}, {H1, Eq(a, b)}, {H2, Eq(b, c)}, {H3, a}},
-                EqTIntro(c, EqMP(a, c, Symm(Bool, c, a, Trans(Bool, c, b, a, Symm(Bool, b, c, H2), Symm(Bool, a, b, H1))), H3))),
+    success(Fun({{a, Bool}, {b, Bool}, {c, Bool}, {H1, HEq(a, b)}, {H2, HEq(b, c)}, {H3, a}},
+                mk_eqt_intro_th(_, mk_eqmp_th(_, _, mk_symm_th(_, _, _, mk_trans_th(_, _, _, _, mk_symm_th(_, _, _, H2), mk_symm_th(_, _, _, H1))), H3))),
+            Fun({{a, Bool}, {b, Bool}, {c, Bool}, {H1, HEq(a, b)}, {H2, HEq(b, c)}, {H3, a}},
+                mk_eqt_intro_th(c, mk_eqmp_th(a, c, mk_symm_th(Bool, c, a, mk_trans_th(Bool, c, b, a, mk_symm_th(Bool, b, c, H2), mk_symm_th(Bool, a, b, H1))), H3))),
             env2);
     expr A = Const("A");
-    success(Fun({{A, Type()}, {a, A}, {b, A}, {c, A}, {H1, Eq(a, b)}, {H2, Eq(b, c)}},
-                Symm(_, _, _, Trans(_, _, _, _, Symm(_, _, _, H2), Symm(_, _, _, H1)))),
-            Fun({{A, Type()}, {a, A}, {b, A}, {c, A}, {H1, Eq(a, b)}, {H2, Eq(b, c)}},
-                Symm(A, c, a, Trans(A, c, b, a, Symm(A, b, c, H2), Symm(A, a, b, H1)))),
+    success(Fun({{A, Type()}, {a, A}, {b, A}, {c, A}, {H1, HEq(a, b)}, {H2, HEq(b, c)}},
+                mk_symm_th(_, _, _, mk_trans_th(_, _, _, _, mk_symm_th(_, _, _, H2), mk_symm_th(_, _, _, H1)))),
+            Fun({{A, Type()}, {a, A}, {b, A}, {c, A}, {H1, HEq(a, b)}, {H2, HEq(b, c)}},
+                mk_symm_th(A, c, a, mk_trans_th(A, c, b, a, mk_symm_th(A, b, c, H2), mk_symm_th(A, a, b, H1)))),
             env2);
 }
 
@@ -682,7 +682,7 @@ void tst21() {
     expr b  = Const("b");
     expr m1 = menv->mk_metavar();
     expr l = m1(b, a);
-    expr r = Fun({x, N}, f(x, Eq(a, b)));
+    expr r = Fun({x, N}, f(x, HEq(a, b)));
     elaborator elb(env, menv, context(), l, r);
     while (true) {
         try {
@@ -715,8 +715,8 @@ void tst22() {
     expr f  = Const("f");
     expr a  = Const("a");
     expr b  = Const("b");
-    expr l = f(m1(a), iAdd(m3, iAdd(iVal(1), iVal(1))));
-    expr r = f(m2(b), iAdd(iVal(1), iVal(2)));
+    expr l = f(m1(a), mk_Int_add(m3, mk_Int_add(iVal(1), iVal(1))));
+    expr r = f(m2(b), mk_Int_add(iVal(1), iVal(2)));
     elaborator elb(env, menv, context(), l, r);
     while (true) {
         try {
@@ -745,8 +745,8 @@ void tst23() {
     expr f  = Const("f");
     expr m1 = menv->mk_metavar();
     expr m2 = menv->mk_metavar();
-    expr l  = Fun({{x, N}, {y, N}}, Eq(y, f(x, m1)));
-    expr r  = Fun({{x, N}, {y, N}}, Eq(m2, f(m1, x)));
+    expr l  = Fun({{x, N}, {y, N}}, HEq(y, f(x, m1)));
+    expr r  = Fun({{x, N}, {y, N}}, HEq(m2, f(m1, x)));
     elaborator elb(env, menv, context(), l, r);
     while (true) {
         try {
@@ -832,13 +832,13 @@ void tst26() {
     expr a = Const("a");
     env->add_var("a", Type(level()+1));
     expr m1 = menv->mk_metavar();
-    expr F  = Eq(g(m1, a), a);
+    expr F  = HEq(g(m1, a), a);
     std::cout << F << "\n";
     std::cout << checker.check(F, context(), menv, ucs) << "\n";
     elaborator elb(env, menv, ucs.size(), ucs.data());
     metavar_env s = elb.next();
     std::cout << s->instantiate_metavars(F) << "\n";
-    lean_assert_eq(s->instantiate_metavars(F), Eq(g(Type(level()+1), a), a));
+    lean_assert_eq(s->instantiate_metavars(F), HEq(g(Type(level()+1), a), a));
 }
 
 void tst27() {

src/tests/library/expr_lt.cpp

@@ -34,10 +34,10 @@ static void tst1() {
     lt(Const("a"), Const("b"), true);
     lt(Const("a"), Const("a"), false);
     lt(Var(1), Const("a"), true);
-    lt(Eq(Var(0), Var(1)), Eq(Var(1), Var(1)), true);
-    lt(Eq(Var(1), Var(0)), Eq(Var(1), Var(1)), true);
-    lt(Eq(Var(1), Var(1)), Eq(Var(1), Var(1)), false);
-    lt(Eq(Var(2), Var(1)), Eq(Var(1), Var(1)), false);
+    lt(HEq(Var(0), Var(1)), HEq(Var(1), Var(1)), true);
+    lt(HEq(Var(1), Var(0)), HEq(Var(1), Var(1)), true);
+    lt(HEq(Var(1), Var(1)), HEq(Var(1), Var(1)), false);
+    lt(HEq(Var(2), Var(1)), HEq(Var(1), Var(1)), false);
     lt(Const("f")(Var(0)), Const("f")(Var(0), Const("a")), true);
     lt(Const("f")(Var(0), Const("a"), Const("b")), Const("f")(Var(0), Const("a")), false);
     lt(Const("f")(Var(0), Const("a")), Const("g")(Var(0), Const("a")), true);

src/tests/library/formatter.cpp

@@ -30,13 +30,13 @@ static void tst1() {
     expr x  = Const("x");
     expr y  = Const("y");
     expr N  = Const("N");
-    expr F  = Fun({x, Pi({x, N}, x >> x)}, Let({y, f(a)}, f(Eq(x, f(y, a)))));
+    expr F  = Fun({x, Pi({x, N}, x >> x)}, Let({y, f(a)}, f(HEq(x, f(y, a)))));
     check(fmt(F), "fun x : (Pi x : N, (x#0 -> x#1)), (let y := f a in (f (x#1 == (f y#0 a))))");
     check(fmt(env->get_object("N")), "variable N : Type");
     context ctx;
     ctx = extend(ctx, "x", f(a));
     ctx = extend(ctx, "y", f(Var(0), N >> N));
-    ctx = extend(ctx, "z", N, Eq(Var(0), Var(1)));
+    ctx = extend(ctx, "z", N, HEq(Var(0), Var(1)));
     check(fmt(ctx), "[x : f a; y : f x#0 (N -> N); z : N := y#0 == x#1]");
     check(fmt(ctx, f(Var(0), Var(2))), "f z#0 x#2");
     check(fmt(ctx, f(Var(0), Var(2)), true), "[x : f a; y : f x#0 (N -> N); z : N := y#0 == x#1] |- f z#0 x#2");

src/tests/library/max_sharing.cpp

@@ -15,10 +15,10 @@ static void tst1() {
     max_sharing_fn max_fn;
     expr a1 = Const("a");
     expr a2 = Const("a");
-    expr F1 = Eq(a1, a2);
-    lean_assert(!is_eqp(eq_lhs(F1), eq_rhs(F1)));
+    expr F1 = HEq(a1, a2);
+    lean_assert(!is_eqp(heq_lhs(F1), heq_rhs(F1)));
     expr F2 = max_fn(F1);
-    lean_assert(is_eqp(eq_lhs(F2), eq_rhs(F2)));
+    lean_assert(is_eqp(heq_lhs(F2), heq_rhs(F2)));
     expr x = Const("x");
     expr y = Const("y");
     expr f = Const("f");

src/tests/library/rewriter/fo_match.cpp

@@ -273,7 +273,7 @@ static void match_let_tst1() {
 
 static void match_let_tst2() {
     cout << "--- match_let_tst2() ---" << endl;
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     expr l = mk_let("a", Type(), Const("b"), Var(0));
     subst_map s;
     bool result = test_match(l, t, 0, s);
@@ -283,7 +283,7 @@ static void match_let_tst2() {
 
 static void match_let_tst3() {
     cout << "--- match_let_tst3() ---" << endl;
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     expr l1 = mk_let("a", Var(0), Const("b"), Var(0));
     expr l2 = mk_let("a", Int, Const("b"), Var(0));
     subst_map s;
@@ -295,7 +295,7 @@ static void match_let_tst3() {
 
 static void match_let_tst4() {
     cout << "--- match_let_tst4() ---" << endl;
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     expr l1 = mk_let("a", Nat, Const("b"), Var(0));
     expr l2 = mk_let("a", Int, Const("b"), Var(0));
     subst_map s;
@@ -306,7 +306,7 @@ static void match_let_tst4() {
 
 static void match_let_tst5() {
     cout << "--- match_let_tst5() ---" << endl;
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     expr l1 = mk_let("a", Int, Var(3), Var(0));
     expr l2 = mk_let("a", Int, Const("b"), Const("b"));
     subst_map s;
@@ -317,7 +317,7 @@ static void match_let_tst5() {
 
 static void match_let_tst6() {
     cout << "--- match_let_tst6() ---" << endl;
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     expr l1 = mk_let("a", Var(0), Var(1), Var(0));
     expr l2 = mk_let("a", Int, Const("b"), Const("b"));
     subst_map s;
@@ -328,7 +328,7 @@ static void match_let_tst6() {
 
 static void match_let_tst7() {
     cout << "--- match_let_tst7() ---" << endl;
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     expr l1 = mk_let("a", Var(0), Var(1), Var(0));
     expr l2 = mk_let("a", Int, Const("b"), Var(0));
     subst_map s;
@@ -341,7 +341,7 @@ static void match_let_tst7() {
 
 static void match_let_tst8() {
     cout << "--- match_let_tst8() ---" << endl;
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     expr l1 = mk_let("a", Nat, Var(1), Var(0));
     expr l2 = mk_let("a", Int, Const("b"), Var(0));
     subst_map s;
@@ -352,7 +352,7 @@ static void match_let_tst8() {
 
 static void match_let_tst9() {
     cout << "--- match_let_tst9() ---" << endl;
-    expr t = Eq(Const("a"), Const("b"));
+    expr t = HEq(Const("a"), Const("b"));
     expr l1 = mk_let("a", Var(0), Var(0), Var(0));
     expr l2 = mk_let("a", Nat, Const("b"), Const("a"));
     subst_map s;
@@ -367,7 +367,7 @@ static void match_eq_tst1() {
     expr f = Const("f");
     expr a = Const("a");
     subst_map s;
-    bool result = test_match(mk_eq(x, a), mk_eq(f, a), 0, s);
+    bool result = test_match(mk_heq(x, a), mk_heq(f, a), 0, s);
     lean_assert_eq(result, true);
     lean_assert_eq(s.find(0)->second, f);
     lean_assert_eq(s.size(), 1);
@@ -379,7 +379,7 @@ static void match_eq_tst2() {
     expr f = Const("f");
     expr a = Const("a");
     subst_map s;
-    bool result = test_match(mk_eq(nAdd(x, a), x), mk_eq(nAdd(f, a), f), 0, s);
+    bool result = test_match(mk_heq(mk_Nat_add(x, a), x), mk_heq(mk_Nat_add(f, a), f), 0, s);
     lean_assert_eq(result, true);
     lean_assert_eq(s.find(0)->second, f);
     lean_assert_eq(s.size(), 1);
@@ -391,7 +391,7 @@ static void match_eq_tst3() {
     expr f = Const("f");
     expr a = Const("a");
     subst_map s;
-    bool result = test_match(mk_eq(nAdd(x, a), x), f, 0, s);
+    bool result = test_match(mk_heq(mk_Nat_add(x, a), x), f, 0, s);
     lean_assert_eq(result, false);
     lean_assert_eq(s.empty(), true);
 }

src/tests/library/rewriter/rewriter.cpp

@@ -32,11 +32,11 @@ static void theorem_rewriter1_tst() {
     // Result :     (b + a, ADD_COMM a b)
     expr a        = Const("a");                  // a  : Nat
     expr b        = Const("b");                  // b  : Nat
-    expr a_plus_b = nAdd(a, b);
-    expr b_plus_a = nAdd(b, a);
+    expr a_plus_b = mk_Nat_add(a, b);
+    expr b_plus_a = mk_Nat_add(b, a);
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
 
     environment env; init_test_frontend(env);
@@ -48,13 +48,13 @@ static void theorem_rewriter1_tst() {
     rewriter add_comm_thm_rewriter = mk_theorem_rewriter(add_comm_thm_type, add_comm_thm_body);
     context ctx;
     pair<expr, expr> result = add_comm_thm_rewriter(env, ctx, a_plus_b);
-    expr concl = mk_eq(a_plus_b, result.first);
+    expr concl = mk_heq(a_plus_b, result.first);
     expr proof = result.second;
 
     cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(a_plus_b, b_plus_a));
+    lean_assert_eq(concl, mk_heq(a_plus_b, b_plus_a));
     lean_assert_eq(proof, Const("ADD_COMM")(a, b));
     env->add_theorem("New_theorem1", concl, proof);
 }
@@ -66,9 +66,9 @@ static void theorem_rewriter2_tst() {
     // Result :     (a, ADD_ID a)
     expr a           = Const("a");                  // a    : at
     expr zero        = nVal(0);                     // zero : Nat
-    expr a_plus_zero = nAdd(a, zero);
+    expr a_plus_zero = mk_Nat_add(a, zero);
     expr add_id_thm_type = Pi("x", Nat,
-                           Eq(nAdd(Const("x"), zero), Const("x")));
+                           HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -79,13 +79,13 @@ static void theorem_rewriter2_tst() {
     rewriter add_id_thm_rewriter = mk_theorem_rewriter(add_id_thm_type, add_id_thm_body);
     context ctx;
     pair<expr, expr> result = add_id_thm_rewriter(env, ctx, a_plus_zero);
-    expr concl = mk_eq(a_plus_zero, result.first);
+    expr concl = mk_heq(a_plus_zero, result.first);
     expr proof = result.second;
 
     cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(a_plus_zero, a));
+    lean_assert_eq(concl, mk_heq(a_plus_zero, a));
     lean_assert_eq(proof, Const("ADD_ID")(a));
     env->add_theorem("New_theorem2", concl, proof);
 }
@@ -99,14 +99,14 @@ static void then_rewriter1_tst() {
 
     expr a           = Const("a");                  // a  : Nat
     expr zero        = nVal(0);                     // zero : Nat
-    expr a_plus_zero = nAdd(a, zero);
-    expr zero_plus_a = nAdd(zero, a);
+    expr a_plus_zero = mk_Nat_add(a, zero);
+    expr zero_plus_a = mk_Nat_add(zero, a);
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
     expr add_id_thm_type = Pi("x", Nat,
-                           Eq(nAdd(Const("x"), zero), Const("x")));
+                           HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -120,15 +120,15 @@ static void then_rewriter1_tst() {
     rewriter then_rewriter1 = mk_then_rewriter(add_comm_thm_rewriter, add_id_thm_rewriter);
     context ctx;
     pair<expr, expr> result = then_rewriter1(env, ctx, zero_plus_a);
-    expr concl = mk_eq(zero_plus_a, result.first);
+    expr concl = mk_heq(zero_plus_a, result.first);
     expr proof = result.second;
 
     cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
     cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(zero_plus_a, a));
-    lean_assert(proof == Trans(Nat, zero_plus_a, a_plus_zero, a,
+    lean_assert_eq(concl, mk_heq(zero_plus_a, a));
+    lean_assert(proof == mk_trans_th(Nat, zero_plus_a, a_plus_zero, a,
                                Const("ADD_COMM")(zero, a), Const("ADD_ID")(a)));
 
     env->add_theorem("New_theorem3", concl, proof);
@@ -147,22 +147,22 @@ static void then_rewriter2_tst() {
 
     expr a           = Const("a");                  // a  : Nat
     expr zero        = nVal(0);                     // zero : Nat
-    expr zero_plus_a  = nAdd(zero, a);
-    expr a_plus_zero  = nAdd(a, zero);
-    expr zero_plus_a_plus_zero  = nAdd(zero, nAdd(a, zero));
-    expr zero_plus_a_plus_zero_ = nAdd(nAdd(zero, a), zero);
+    expr zero_plus_a  = mk_Nat_add(zero, a);
+    expr a_plus_zero  = mk_Nat_add(a, zero);
+    expr zero_plus_a_plus_zero  = mk_Nat_add(zero, mk_Nat_add(a, zero));
+    expr zero_plus_a_plus_zero_ = mk_Nat_add(mk_Nat_add(zero, a), zero);
     expr add_assoc_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
                                    Pi("z", Nat,
-                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
-                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+                                      HEq(mk_Nat_add(Const("x"), mk_Nat_add(Const("y"), Const("z"))),
+                                         mk_Nat_add(mk_Nat_add(Const("x"), Const("y")), Const("z"))))));
     expr add_assoc_thm_body = Const("ADD_ASSOC");
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
     expr add_id_thm_type = Pi("x", Nat,
-                           Eq(nAdd(Const("x"), zero), Const("x")));
+                           HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -181,17 +181,17 @@ static void then_rewriter2_tst() {
                                                 add_id_thm_rewriter});
     context ctx;
     pair<expr, expr> result = then_rewriter2(env, ctx, zero_plus_a_plus_zero);
-    expr concl = mk_eq(zero_plus_a_plus_zero, result.first);
+    expr concl = mk_heq(zero_plus_a_plus_zero, result.first);
     expr proof = result.second;
     cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
     cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
     cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(zero_plus_a_plus_zero, a));
-    lean_assert(proof == Trans(Nat, zero_plus_a_plus_zero, a_plus_zero, a,
-                               Trans(Nat, zero_plus_a_plus_zero, zero_plus_a, a_plus_zero,
-                                     Trans(Nat, zero_plus_a_plus_zero, zero_plus_a_plus_zero_, zero_plus_a,
+    lean_assert_eq(concl, mk_heq(zero_plus_a_plus_zero, a));
+    lean_assert(proof == mk_trans_th(Nat, zero_plus_a_plus_zero, a_plus_zero, a,
+                               mk_trans_th(Nat, zero_plus_a_plus_zero, zero_plus_a, a_plus_zero,
+                                     mk_trans_th(Nat, zero_plus_a_plus_zero, zero_plus_a_plus_zero_, zero_plus_a,
                                            Const("ADD_ASSOC")(zero, a, zero), Const("ADD_ID")(zero_plus_a)),
                                      Const("ADD_COMM")(zero, a)),
                                Const("ADD_ID")(a)));
@@ -208,20 +208,20 @@ static void orelse_rewriter1_tst() {
     expr a        = Const("a");                  // a  : Nat
     expr b        = Const("b");                  // b  : Nat
     expr zero     = nVal(0);                     // zero : Nat
-    expr a_plus_b = nAdd(a, b);
-    expr b_plus_a = nAdd(b, a);
+    expr a_plus_b = mk_Nat_add(a, b);
+    expr b_plus_a = mk_Nat_add(b, a);
     expr add_assoc_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
                                    Pi("z", Nat,
-                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
-                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+                                      HEq(mk_Nat_add(Const("x"), mk_Nat_add(Const("y"), Const("z"))),
+                                         mk_Nat_add(mk_Nat_add(Const("x"), Const("y")), Const("z"))))));
     expr add_assoc_thm_body = Const("ADD_ASSOC");
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
     expr add_id_thm_type = Pi("x", Nat,
-                              Eq(nAdd(Const("x"), zero), Const("x")));
+                              HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -238,7 +238,7 @@ static void orelse_rewriter1_tst() {
                                                                   add_id_thm_rewriter});
     context ctx;
     pair<expr, expr> result = add_assoc_or_comm_thm_rewriter(env, ctx, a_plus_b);
-    expr concl = mk_eq(a_plus_b, result.first);
+    expr concl = mk_heq(a_plus_b, result.first);
     expr proof = result.second;
 
     cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
@@ -246,7 +246,7 @@ static void orelse_rewriter1_tst() {
     cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(a_plus_b, b_plus_a));
+    lean_assert_eq(concl, mk_heq(a_plus_b, b_plus_a));
     lean_assert_eq(proof, Const("ADD_COMM")(a, b));
     env->add_theorem("New_theorem5", concl, proof);
 }
@@ -259,16 +259,16 @@ static void orelse_rewriter2_tst() {
     expr a        = Const("a");                  // a  : Nat
     expr b        = Const("b");                  // b  : Nat
     expr zero     = nVal(0);                     // zero : Nat
-    expr a_plus_b = nAdd(a, b);
-    expr b_plus_a = nAdd(b, a);
+    expr a_plus_b = mk_Nat_add(a, b);
+    expr b_plus_a = mk_Nat_add(b, a);
     expr add_assoc_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
                                    Pi("z", Nat,
-                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
-                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+                                      HEq(mk_Nat_add(Const("x"), mk_Nat_add(Const("y"), Const("z"))),
+                                         mk_Nat_add(mk_Nat_add(Const("x"), Const("y")), Const("z"))))));
     expr add_assoc_thm_body = Const("ADD_ASSOC");
     expr add_id_thm_type = Pi("x", Nat,
-                              Eq(nAdd(Const("x"), zero), Const("x")));
+                              HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -304,20 +304,20 @@ static void try_rewriter1_tst() {
     expr a        = Const("a");                  // a  : Nat
     expr b        = Const("b");                  // b  : Nat
     expr zero     = nVal(0);                     // zero : Nat
-    expr a_plus_b = nAdd(a, b);
-    expr b_plus_a = nAdd(b, a);
+    expr a_plus_b = mk_Nat_add(a, b);
+    expr b_plus_a = mk_Nat_add(b, a);
     expr add_assoc_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
                                    Pi("z", Nat,
-                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
-                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+                                      HEq(mk_Nat_add(Const("x"), mk_Nat_add(Const("y"), Const("z"))),
+                                         mk_Nat_add(mk_Nat_add(Const("x"), Const("y")), Const("z"))))));
     expr add_assoc_thm_body = Const("ADD_ASSOC");
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
     expr add_id_thm_type = Pi("x", Nat,
-                              Eq(nAdd(Const("x"), zero), Const("x")));
+                              HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -333,7 +333,7 @@ static void try_rewriter1_tst() {
                                                  add_id_thm_rewriter});
     context ctx;
     pair<expr, expr> result = add_try_rewriter(env, ctx, a_plus_b);
-    expr concl = mk_eq(a_plus_b, result.first);
+    expr concl = mk_heq(a_plus_b, result.first);
     expr proof = result.second;
 
     cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
@@ -341,7 +341,7 @@ static void try_rewriter1_tst() {
     cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(a_plus_b, a_plus_b));
+    lean_assert_eq(concl, mk_heq(a_plus_b, a_plus_b));
     lean_assert_eq(proof, Const("refl")(Nat, a_plus_b));
     env->add_theorem("New_theorem6", concl, proof);
 }
@@ -355,20 +355,20 @@ static void try_rewriter2_tst() {
     expr a        = Const("a");                  // a  : Nat
     expr b        = Const("b");                  // b  : Nat
     expr zero     = nVal(0);                     // zero : Nat
-    expr a_plus_b = nAdd(a, b);
-    expr b_plus_a = nAdd(b, a);
+    expr a_plus_b = mk_Nat_add(a, b);
+    expr b_plus_a = mk_Nat_add(b, a);
     expr add_assoc_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
                                    Pi("z", Nat,
-                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
-                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+                                      HEq(mk_Nat_add(Const("x"), mk_Nat_add(Const("y"), Const("z"))),
+                                         mk_Nat_add(mk_Nat_add(Const("x"), Const("y")), Const("z"))))));
     expr add_assoc_thm_body = Const("ADD_ASSOC");
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
     expr add_id_thm_type = Pi("x", Nat,
-                              Eq(nAdd(Const("x"), zero), Const("x")));
+                              HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -385,7 +385,7 @@ static void try_rewriter2_tst() {
                                                  add_id_thm_rewriter});
     context ctx;
     pair<expr, expr> result = add_try_rewriter(env, ctx, a_plus_b);
-    expr concl = mk_eq(a_plus_b, result.first);
+    expr concl = mk_heq(a_plus_b, result.first);
     expr proof = result.second;
 
     cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
@@ -393,7 +393,7 @@ static void try_rewriter2_tst() {
     cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(a_plus_b, b_plus_a));
+    lean_assert_eq(concl, mk_heq(a_plus_b, b_plus_a));
     lean_assert_eq(proof, Const("ADD_COMM")(a, b));
     env->add_theorem("try2", concl, proof);
 }
@@ -410,11 +410,11 @@ static void app_rewriter1_tst() {
     expr f3       = Const("f3");  // f  : Nat -> Nat -> Nat -> Nat
     expr f4       = Const("f4");  // f  : Nat -> Nat -> Nat -> Nat -> Nat
     expr zero     = nVal(0);      // zero : Nat
-    expr a_plus_b = nAdd(a, b);
-    expr b_plus_a = nAdd(b, a);
+    expr a_plus_b = mk_Nat_add(a, b);
+    expr b_plus_a = mk_Nat_add(b, a);
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
 
     environment env; init_test_frontend(env);
@@ -436,32 +436,32 @@ static void app_rewriter1_tst() {
 
     expr v = f1(nVal(0));
     pair<expr, expr> result = app_try_comm_rewriter(env, ctx, v);
-    expr concl = mk_eq(v, result.first);
+    expr concl = mk_heq(v, result.first);
     expr proof = result.second;
     cout << "Concl = " << concl << std::endl
          << "Proof = " << proof << std::endl;
-    lean_assert_eq(concl, mk_eq(v, f1(nVal(0))));
-    lean_assert_eq(proof, Refl(Nat, f1(nVal(0))));
+    lean_assert_eq(concl, mk_heq(v, f1(nVal(0))));
+    lean_assert_eq(proof, mk_refl_th(Nat, f1(nVal(0))));
     env->add_theorem("app_rewriter1", concl, proof);
     cout << "====================================================" << std::endl;
     v = f1(a_plus_b);
     result = app_try_comm_rewriter(env, ctx, v);
-    concl = mk_eq(v, result.first);
+    concl = mk_heq(v, result.first);
     proof = result.second;
     cout << "Concl = " << concl << std::endl
          << "Proof = " << proof << std::endl;
-    lean_assert_eq(concl, mk_eq(v, f1(b_plus_a)));
+    lean_assert_eq(concl, mk_heq(v, f1(b_plus_a)));
     lean_assert_eq(proof,
                    Const("congr2")(Nat, Fun(name("_"), Nat, Nat), a_plus_b, b_plus_a, f1, Const("ADD_COMM")(a, b)));
     env->add_theorem("app_rewriter2", concl, proof);
     cout << "====================================================" << std::endl;
     v = f4(nVal(0), a_plus_b, nVal(0), b_plus_a);
     result = app_try_comm_rewriter(env, ctx, v);
-    concl = mk_eq(v, result.first);
+    concl = mk_heq(v, result.first);
     proof = result.second;
     cout << "Concl = " << concl << std::endl
          << "Proof = " << proof << std::endl;
-    lean_assert_eq(concl, mk_eq(v, f4(nVal(0), b_plus_a, nVal(0), a_plus_b)));
+    lean_assert_eq(concl, mk_heq(v, f4(nVal(0), b_plus_a, nVal(0), a_plus_b)));
     // Congr Nat (fun _ : Nat, Nat) (f4 0 (Nat::add a b) 0) (f4 0 (Nat::add b a) 0) (Nat::add b a) (Nat::add a b) (Congr1 Nat (fun _ : Nat, (Nat -> Nat)) (f4 0 (Nat::add a b)) (f4 0 (Nat::add b a)) 0 (Congr2 Nat (fun _ : Nat, (Nat -> Nat -> Nat)) (Nat::add a b) (Nat::add b a) (f4 0) (ADD_COMM a b))) (ADD_COMM b a)
 
     lean_assert_eq(proof,
@@ -489,22 +489,22 @@ static void repeat_rewriter1_tst() {
 
     expr a           = Const("a");                  // a  : Nat
     expr zero        = nVal(0);                     // zero : Nat
-    expr zero_plus_a  = nAdd(zero, a);
-    expr a_plus_zero  = nAdd(a, zero);
-    expr zero_plus_a_plus_zero  = nAdd(zero, nAdd(a, zero));
-    expr zero_plus_a_plus_zero_ = nAdd(nAdd(zero, a), zero);
+    expr zero_plus_a  = mk_Nat_add(zero, a);
+    expr a_plus_zero  = mk_Nat_add(a, zero);
+    expr zero_plus_a_plus_zero  = mk_Nat_add(zero, mk_Nat_add(a, zero));
+    expr zero_plus_a_plus_zero_ = mk_Nat_add(mk_Nat_add(zero, a), zero);
     expr add_assoc_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
                                    Pi("z", Nat,
-                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
-                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+                                      HEq(mk_Nat_add(Const("x"), mk_Nat_add(Const("y"), Const("z"))),
+                                         mk_Nat_add(mk_Nat_add(Const("x"), Const("y")), Const("z"))))));
     expr add_assoc_thm_body = Const("ADD_ASSOC");
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
     expr add_id_thm_type = Pi("x", Nat,
-                           Eq(nAdd(Const("x"), zero), Const("x")));
+                           HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -523,14 +523,14 @@ static void repeat_rewriter1_tst() {
     rewriter repeat_rw = mk_repeat_rewriter(or_rewriter);
     context ctx;
     pair<expr, expr> result = repeat_rw(env, ctx, zero_plus_a_plus_zero);
-    expr concl = mk_eq(zero_plus_a_plus_zero, result.first);
+    expr concl = mk_heq(zero_plus_a_plus_zero, result.first);
     expr proof = result.second;
     cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
     cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
     cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(zero_plus_a_plus_zero, a));
+    lean_assert_eq(concl, mk_heq(zero_plus_a_plus_zero, a));
     env->add_theorem("repeat_thm1", concl, proof);
 }
 
@@ -547,22 +547,22 @@ static void repeat_rewriter2_tst() {
 
     expr a           = Const("a");                  // a  : Nat
     expr zero        = nVal(0);                     // zero : Nat
-    expr zero_plus_a  = nAdd(zero, a);
-    expr a_plus_zero  = nAdd(a, zero);
-    expr zero_plus_a_plus_zero  = nAdd(zero, nAdd(a, zero));
-    expr zero_plus_a_plus_zero_ = nAdd(nAdd(zero, a), zero);
+    expr zero_plus_a  = mk_Nat_add(zero, a);
+    expr a_plus_zero  = mk_Nat_add(a, zero);
+    expr zero_plus_a_plus_zero  = mk_Nat_add(zero, mk_Nat_add(a, zero));
+    expr zero_plus_a_plus_zero_ = mk_Nat_add(mk_Nat_add(zero, a), zero);
     expr add_assoc_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
                                    Pi("z", Nat,
-                                      Eq(nAdd(Const("x"), nAdd(Const("y"), Const("z"))),
-                                         nAdd(nAdd(Const("x"), Const("y")), Const("z"))))));
+                                      HEq(mk_Nat_add(Const("x"), mk_Nat_add(Const("y"), Const("z"))),
+                                         mk_Nat_add(mk_Nat_add(Const("x"), Const("y")), Const("z"))))));
     expr add_assoc_thm_body = Const("ADD_ASSOC");
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
     expr add_id_thm_type = Pi("x", Nat,
-                           Eq(nAdd(Const("x"), zero), Const("x")));
+                           HEq(mk_Nat_add(Const("x"), zero), Const("x")));
     expr add_id_thm_body = Const("ADD_ID");
 
     environment env; init_test_frontend(env);
@@ -582,14 +582,14 @@ static void repeat_rewriter2_tst() {
     rewriter repeat_rw = mk_repeat_rewriter(try_rw);
     context ctx;
     pair<expr, expr> result = repeat_rw(env, ctx, zero_plus_a_plus_zero);
-    expr concl = mk_eq(zero_plus_a_plus_zero, result.first);
+    expr concl = mk_heq(zero_plus_a_plus_zero, result.first);
     expr proof = result.second;
     cout << "Theorem: " << add_assoc_thm_type << " := " << add_assoc_thm_body << std::endl;
     cout << "Theorem: " << add_comm_thm_type << " := " << add_comm_thm_body << std::endl;
     cout << "Theorem: " << add_id_thm_type << " := " << add_id_thm_body << std::endl;
     cout << "         " << concl << " := " << proof << std::endl;
 
-    lean_assert_eq(concl, mk_eq(zero_plus_a_plus_zero, a));
+    lean_assert_eq(concl, mk_heq(zero_plus_a_plus_zero, a));
     env->add_theorem("repeat_thm2", concl, proof);
 }
 
@@ -605,11 +605,11 @@ static void depth_rewriter1_tst() {
     expr f3       = Const("f3");  // f  : Nat -> Nat -> Nat -> Nat
     expr f4       = Const("f4");  // f  : Nat -> Nat -> Nat -> Nat -> Nat
     expr zero     = nVal(0);      // zero : Nat
-    expr a_plus_b = nAdd(a, b);
-    expr b_plus_a = nAdd(b, a);
+    expr a_plus_b = mk_Nat_add(a, b);
+    expr b_plus_a = mk_Nat_add(b, a);
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
 
     environment env; init_test_frontend(env);
@@ -629,13 +629,13 @@ static void depth_rewriter1_tst() {
 
     cout << "RW = " << depth_rewriter << std::endl;
 
-    expr v = nAdd(f1(nAdd(a, b)), f3(a, b, nAdd(a, b)));
+    expr v = mk_Nat_add(f1(mk_Nat_add(a, b)), f3(a, b, mk_Nat_add(a, b)));
     pair<expr, expr> result = depth_rewriter(env, ctx, v);
-    expr concl = mk_eq(v, result.first);
+    expr concl = mk_heq(v, result.first);
     expr proof = result.second;
     cout << "Concl = " << concl << std::endl
          << "Proof = " << proof << std::endl;
-    lean_assert_eq(concl, mk_eq(v, nAdd(f3(a, b, nAdd(b, a)), f1(nAdd(b, a)))));
+    lean_assert_eq(concl, mk_heq(v, mk_Nat_add(f3(a, b, mk_Nat_add(b, a)), f1(mk_Nat_add(b, a)))));
     env->add_theorem("depth_rewriter1", concl, proof);
     cout << "====================================================" << std::endl;
 }
@@ -652,11 +652,11 @@ static void lambda_body_rewriter_tst() {
     expr f3       = Const("f3");  // f  : Nat -> Nat -> Nat -> Nat
     expr f4       = Const("f4");  // f  : Nat -> Nat -> Nat -> Nat -> Nat
     expr zero     = nVal(0);      // zero : Nat
-    expr a_plus_b = nAdd(a, b);
-    expr b_plus_a = nAdd(b, a);
+    expr a_plus_b = mk_Nat_add(a, b);
+    expr b_plus_a = mk_Nat_add(b, a);
     expr add_comm_thm_type = Pi("x", Nat,
                                 Pi("y", Nat,
-                                   Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+                                   HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
 
     environment env; init_test_frontend(env);
@@ -673,27 +673,27 @@ static void lambda_body_rewriter_tst() {
     rewriter lambda_rewriter = mk_lambda_body_rewriter(add_comm_thm_rewriter);
     context ctx;
     cout << "RW = " << lambda_rewriter << std::endl;
-    expr v = mk_lambda("x", Nat, nAdd(b, a));
+    expr v = mk_lambda("x", Nat, mk_Nat_add(b, a));
     pair<expr, expr> result = lambda_rewriter(env, ctx, v);
-    expr concl = mk_eq(v, result.first);
+    expr concl = mk_heq(v, result.first);
     expr proof = result.second;
     cout << "v     = " << v     << std::endl;
     cout << "Concl = " << concl << std::endl
          << "Proof = " << proof << std::endl;
-    lean_assert_eq(concl, mk_eq(v, mk_lambda("x", Nat, nAdd(a, b))));
+    lean_assert_eq(concl, mk_heq(v, mk_lambda("x", Nat, mk_Nat_add(a, b))));
     env->add_theorem("lambda_rewriter1", concl, proof);
 
     // Theorem:     Pi(x y : N),  x + y       = y + x       := ADD_COMM x y
     // Term   :     fun (x : Nat), (x + a)
     // Result :     fun (x : Nat), (a + x)
-    v = mk_lambda("x", Nat, nAdd(Var(0), a));
+    v = mk_lambda("x", Nat, mk_Nat_add(Var(0), a));
     result = lambda_rewriter(env, ctx, v);
-    concl = mk_eq(v, result.first);
+    concl = mk_heq(v, result.first);
     proof = result.second;
     cout << "v     = " << v     << std::endl;
     cout << "Concl = " << concl << std::endl
          << "Proof = " << proof << std::endl;
-    lean_assert_eq(concl, mk_eq(v, mk_lambda("x", Nat, nAdd(a, Var(0)))));
+    lean_assert_eq(concl, mk_heq(v, mk_lambda("x", Nat, mk_Nat_add(a, Var(0)))));
     env->add_theorem("lambda_rewriter2", concl, proof);
     cout << "====================================================" << std::endl;
 }
@@ -711,7 +711,7 @@ static void lambda_type_rewriter_tst() {
     env->add_var("b", Nat);
     expr vec = Const("vec");
     env->add_var("vec", Type() >> (Nat >> Type())); // vec : Type -> Nat -> Type
-    expr add_comm_thm_type = Pi("x", Nat, Pi("y", Nat, Eq(nAdd(Const("x"), Const("y")), nAdd(Const("y"), Const("x")))));
+    expr add_comm_thm_type = Pi("x", Nat, Pi("y", Nat, HEq(mk_Nat_add(Const("x"), Const("y")), mk_Nat_add(Const("y"), Const("x")))));
     expr add_comm_thm_body = Const("ADD_COMM");
     env->add_axiom("ADD_COMM", add_comm_thm_type); // ADD_COMM : Pi (x, y: N), x + y = y + z
     rewriter add_comm_thm_rewriter = mk_theorem_rewriter(add_comm_thm_type, add_comm_thm_body);
@@ -719,14 +719,14 @@ static void lambda_type_rewriter_tst() {
     rewriter depth_rewriter = mk_depth_rewriter(try_rewriter);
     rewriter lambda_rewriter = mk_lambda_type_rewriter(depth_rewriter);
 
-    expr v = mk_lambda("x", vec(Nat, nAdd(a, b)), Var(0));
+    expr v = mk_lambda("x", vec(Nat, mk_Nat_add(a, b)), Var(0));
     pair<expr, expr> result = lambda_rewriter(env, ctx, v);
-    expr concl = mk_eq(v, result.first);
+    expr concl = mk_heq(v, result.first);
     expr proof = result.second;
     cout << "v     = " << v     << std::endl;
     cout << "Concl = " << concl << std::endl
          << "Proof = " << proof << std::endl;
-    lean_assert_eq(concl, mk_eq(v, mk_lambda("x", vec(Nat, nAdd(b, a)), Var(0))));
+    lean_assert_eq(concl, mk_heq(v, mk_lambda("x", vec(Nat, mk_Nat_add(b, a)), Var(0))));
     env->add_theorem("lambda_type_rewriter", concl, proof);
     cout << "====================================================" << std::endl;
 }

src/tests/library/update_expr.cpp

@@ -12,11 +12,11 @@ using namespace lean;
 static void tst1() {
     expr a = Const("a");
     expr b = Const("b");
-    expr eq1 = Eq(a, b);
-    expr eq2 = update_eq(eq1, a, a);
-    expr eq3 = update_eq(eq1, a, b);
-    lean_assert(eq_lhs(eq3) == a);
-    lean_assert(eq_rhs(eq3) == b);
+    expr eq1 = HEq(a, b);
+    expr eq2 = update_heq(eq1, a, a);
+    expr eq3 = update_heq(eq1, a, b);
+    lean_assert(heq_lhs(eq3) == a);
+    lean_assert(heq_rhs(eq3) == b);
     lean_assert(is_eqp(eq1, eq3));
 }
 

tests/lua/env4.lua

@@ -26,8 +26,8 @@ local ok, msg = pcall(function() child:add_definition("val3", Const("Int"), Cons
 assert(not ok)
 print(msg)
 assert(child:normalize(Const("val2")) == Const("val2"))
-child:add_theorem("Th1", Eq(iVal(0), iVal(0)), Const("trivial"))
-child:add_axiom("H1", Eq(Const("x"), iVal(0)))
+child:add_theorem("Th1", HEq(iVal(0), iVal(0)), Const("trivial"))
+child:add_axiom("H1", HEq(Const("x"), iVal(0)))
 assert(child:has_object("H1"))
 local ctx = context(context(), "x", Const("Int"), iVal(10))
 assert(child:normalize(Var(0), ctx) == iVal(10))

tests/lua/expr5.lua

@@ -1,9 +1,9 @@
 e = context_entry("a", Const("a"))
 n = e:get_body()
-check_error(function() mk_eq(n, Const("a")) end)
+check_error(function() mk_heq(n, Const("a")) end)
 print(Const("a") < Const("b"))
 check_error(function() mk_app(Const("a")) end)
-print(mk_eq(Const("a"), Const("b")))
+print(mk_heq(Const("a"), Const("b")))
 print(mk_pi("x", Const("N"), Var(0)))
 print(Pi("x", Const("N"), Const("x")))
 assert(mk_pi("x", Const("N"), Var(0)) == Pi("x", Const("N"), Const("x")))

tests/lua/expr6.lua

@@ -22,7 +22,7 @@ function print_leaves(e, ctx)
       print("abstraction var name: " .. tostring(name))
       print_leaves(domain, ctx)
       print_leaves(body, ctx:extend(name, domain))
-   elseif k == expr_kind.Eq then
+   elseif k == expr_kind.HEq then
       local lhs, rhs = e:fields()
       print_leaves(lhs, ctx)
       print_leaves(rhs, ctx)
@@ -43,6 +43,6 @@ local x, y, z = Consts("x, y, z")
 assert(is_expr(f))
 
 local F = fun(h, mk_arrow(N, N),
-              Let(x, h(a), Eq(f(x), h(x))))
+              Let(x, h(a), HEq(f(x), h(x))))
 print(F)
 print_leaves(F, context())

tests/lua/expr8.lua

@@ -25,7 +25,7 @@ assert(mk_metavar("M"):is_metavar())
 assert(mk_real_value(mpq(10)):is_value())
 assert(not F:has_metavar())
 assert(f(mk_metavar("M")):has_metavar())
-assert(Eq(a, b):is_eq())
+assert(HEq(a, b):is_heq())
 assert(F:num_args() == 3)
 assert(F:arg(0) == f)
 assert(F:arg(1) == g(x, a))

tests/lua/unify1.lua

@@ -16,7 +16,7 @@ function must_unify(t1, t2)
    assert(s:apply(t1) == s:apply(t2))
 end
 Bool = Const("Bool")
-must_unify(Eq(a, m1), Eq(m2, m2))
+must_unify(HEq(a, m1), HEq(m2, m2))
 must_unify(Type(), m1)
 must_unify(fun(x, Bool, x), fun(x, Bool, m1))
 must_unify(Pi(x, Bool, x), Pi(x, Bool, m1))