feat(builtin/kernel): add left_comm theorem

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

src/builtin/kernel.lean

@@ -365,6 +365,20 @@ theorem exists_unfold {A : TypeU} (P : A → Bool) (a : A) : (∃ x : A, P x) =
 := boolext (λ H : (∃ x : A, P x), exists_unfold1 a H)
            (λ H : (P a ∨ (∃ x : A, x ≠ a ∧ P x)), exists_unfold2 a H)
 
+
+-- Remark: ordered rewriting + assoc + comm + left_comm sorts a term lexicographically
+theorem left_comm {A : TypeU} {R : A -> A -> A} (comm : ∀ x y, R x y = R y x) (assoc : ∀ x y z, R (R x y) z = R x (R y z)) :
+        ∀ x y z, R x (R y z) = R y (R x z)
+:= λ x y z, calc R x (R y z) = R (R x y) z : symm (assoc x y z)
+                      ...    = R (R y x) z : { comm x y }
+                      ...    = R y (R x z) : assoc y x z
+
+theorem and_left_comm (a b c : Bool) : (a ∧ (b ∧ c)) = (b ∧ (a ∧ c))
+:= left_comm and_comm and_assoc a b c
+
+theorem or_left_comm (a b c : Bool) : (a ∨ (b ∨ c)) = (b ∨ (a ∨ c))
+:= left_comm or_comm or_assoc a b c
+
 -- Congruence theorems for contextual simplification
 
 -- Simplify a → b, by first simplifying a to c using the fact that ¬ b is true, and then

src/kernel/kernel_decls.cpp

@@ -100,6 +100,9 @@ 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"));
+MK_CONSTANT(left_comm_fn, name("left_comm"));
+MK_CONSTANT(and_left_comm_fn, name("and_left_comm"));
+MK_CONSTANT(or_left_comm_fn, name("or_left_comm"));
 MK_CONSTANT(imp_congrr_fn, name("imp_congrr"));
 MK_CONSTANT(imp_congrl_fn, name("imp_congrl"));
 MK_CONSTANT(imp_congr_fn, name("imp_congr"));

src/kernel/kernel_decls.h

@@ -291,6 +291,15 @@ inline expr mk_exists_unfold2_th(expr const & e1, expr const & e2, expr const &
 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}); }
+expr mk_left_comm_fn();
+bool is_left_comm_fn(expr const & e);
+inline expr mk_left_comm_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6, expr const & e7) { return mk_app({mk_left_comm_fn(), e1, e2, e3, e4, e5, e6, e7}); }
+expr mk_and_left_comm_fn();
+bool is_and_left_comm_fn(expr const & e);
+inline expr mk_and_left_comm_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_and_left_comm_fn(), e1, e2, e3}); }
+expr mk_or_left_comm_fn();
+bool is_or_left_comm_fn(expr const & e);
+inline expr mk_or_left_comm_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_or_left_comm_fn(), e1, e2, e3}); }
 expr mk_imp_congrr_fn();
 bool is_imp_congrr_fn(expr const & e);
 inline expr mk_imp_congrr_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4, expr const & e5, expr const & e6) { return mk_app({mk_imp_congrr_fn(), e1, e2, e3, e4, e5, e6}); }