feat(builtin/kernel): prove false_elim without using case

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

src/builtin/kernel.lean

@@ -37,7 +37,6 @@ set_opaque true true
 definition false : Bool
 := ∀ x : Bool, x
 
-set_opaque false true
 alias ⊤ : true
 alias ⊥ : false
 
@@ -89,10 +88,10 @@ definition Exists (A : (Type U)) (P : A → Bool)
 definition exists_unique {A : (Type U)} (p : A → Bool)
 := ∃ x, p x ∧ ∀ y, y ≠ x → ¬ p y
 
-axiom case (P : Bool → Bool) (H1 : P true) (H2 : P false) (a : Bool) : P a
-
 theorem false_elim (a : Bool) (H : false) : a
-:= case (λ x, x) trivial H a
+:= H a
+
+set_opaque false true
 
 theorem mt {a b : Bool} (H1 : a → b) (H2 : ¬ b) : ¬ a
 := assume Ha : a, absurd (H1 Ha) H2
@@ -110,18 +109,6 @@ theorem or_introl {a : Bool} (H : a) (b : Bool) : a ∨ b
 theorem or_intror {b : Bool} (a : Bool) (H : b) : a ∨ b
 := assume H1 : ¬ a, H
 
-theorem boolcomplete (a : Bool) : a = true ∨ a = false
-:= case (λ x, x = true ∨ x = false)
-        (or_introl (refl true) (true = false))
-        (or_intror (false = true) (refl false))
-        a
-
-theorem boolcomplete_swapped (a : Bool) : a = false ∨ a = true
-:= case (λ x, x = false ∨ x = true)
-        (or_intror (true = false) (refl true))
-        (or_introl (refl false) (false = true))
-        a
-
 theorem resolve1 {a b : Bool} (H1 : a ∨ b) (H2 : ¬ a) : b
 := H1 H2
 
@@ -208,6 +195,20 @@ theorem eqf_elim {a : Bool} (H : a = false) : ¬ a
 theorem heqt_elim {a : Bool} (H : a == true) : a
 := eqt_elim (to_eq H)
 
+axiom case (P : Bool → Bool) (H1 : P true) (H2 : P false) (a : Bool) : P a
+
+theorem boolcomplete (a : Bool) : a = true ∨ a = false
+:= case (λ x, x = true ∨ x = false)
+        (or_introl (refl true) (true = false))
+        (or_intror (false = true) (refl false))
+        a
+
+theorem boolcomplete_swapped (a : Bool) : a = false ∨ a = true
+:= case (λ x, x = false ∨ x = true)
+        (or_intror (true = false) (refl true))
+        (or_introl (refl false) (false = true))
+        a
+
 theorem not_true : (¬ true) = false
 := let aux : ¬ (¬ true) = true
            := assume H : (¬ true) = true,

src/kernel/kernel_decls.cpp

@@ -27,15 +27,12 @@ MK_CONSTANT(not_intro_fn, name("not_intro"));
 MK_CONSTANT(absurd_fn, name("absurd"));
 MK_CONSTANT(exists_fn, name("exists"));
 MK_CONSTANT(exists_unique_fn, name("exists_unique"));
-MK_CONSTANT(case_fn, name("case"));
 MK_CONSTANT(false_elim_fn, name("false_elim"));
 MK_CONSTANT(mt_fn, name("mt"));
 MK_CONSTANT(contrapos_fn, name("contrapos"));
 MK_CONSTANT(absurd_elim_fn, name("absurd_elim"));
 MK_CONSTANT(or_introl_fn, name("or_introl"));
 MK_CONSTANT(or_intror_fn, name("or_intror"));
-MK_CONSTANT(boolcomplete_fn, name("boolcomplete"));
-MK_CONSTANT(boolcomplete_swapped_fn, name("boolcomplete_swapped"));
 MK_CONSTANT(resolve1_fn, name("resolve1"));
 MK_CONSTANT(subst_fn, name("subst"));
 MK_CONSTANT(substp_fn, name("substp"));
@@ -63,6 +60,9 @@ MK_CONSTANT(ne_eq_trans_fn, name("ne_eq_trans"));
 MK_CONSTANT(eqt_elim_fn, name("eqt_elim"));
 MK_CONSTANT(eqf_elim_fn, name("eqf_elim"));
 MK_CONSTANT(heqt_elim_fn, name("heqt_elim"));
+MK_CONSTANT(case_fn, name("case"));
+MK_CONSTANT(boolcomplete_fn, name("boolcomplete"));
+MK_CONSTANT(boolcomplete_swapped_fn, name("boolcomplete_swapped"));
 MK_CONSTANT(not_true, name("not_true"));
 MK_CONSTANT(not_false, name("not_false"));
 MK_CONSTANT(not_not_eq_fn, name("not_not_eq"));

src/kernel/kernel_decls.h

@@ -72,9 +72,6 @@ expr mk_exists_unique_fn();
 bool is_exists_unique_fn(expr const & e);
 inline bool is_exists_unique(expr const & e) { return is_app(e) && is_exists_unique_fn(arg(e, 0)) && num_args(e) == 3; }
 inline expr mk_exists_unique(expr const & e1, expr const & e2) { return mk_app({mk_exists_unique_fn(), e1, e2}); }
-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_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}); }
@@ -93,12 +90,6 @@ inline expr mk_or_introl_th(expr const & e1, expr const & e2, expr const & 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_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_boolcomplete_swapped_fn();
-bool is_boolcomplete_swapped_fn(expr const & e);
-inline expr mk_boolcomplete_swapped_th(expr const & e1) { return mk_app({mk_boolcomplete_swapped_fn(), e1}); }
 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}); }
@@ -178,6 +169,15 @@ inline expr mk_eqf_elim_th(expr const & e1, expr const & e2) { return mk_app({mk
 expr mk_heqt_elim_fn();
 bool is_heqt_elim_fn(expr const & e);
 inline expr mk_heqt_elim_th(expr const & e1, expr const & e2) { return mk_app({mk_heqt_elim_fn(), e1, e2}); }
+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_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_boolcomplete_swapped_fn();
+bool is_boolcomplete_swapped_fn(expr const & e);
+inline expr mk_boolcomplete_swapped_th(expr const & e1) { return mk_app({mk_boolcomplete_swapped_fn(), e1}); }
 expr mk_not_true();
 bool is_not_true(expr const & e);
 expr mk_not_false();