lean2/tests/lean/hott/rewriter1.hlean

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import algebra.group
open path_algebra
constant f {A : Type} : A → A → A
theorem test1 {A : Type} [s : add_comm_group A] (a b c : A) : f (a + 0) (f (a + 0) (a + 0)) = f a (f (0 + a) a) :=
begin
rewrite [
add_right_id at {1, 3}, -- rewrite 1st and 3rd occurrences
{0 + _}add_comm] -- apply commutativity to (0 + _)
end
axiom Ax {A : Type} [s₁ : has_mul A] [s₂ : has_one A] (a : A) : f (a * 1) (a * 1) = 1
theorem test2 {A : Type} [s : comm_group A] (a b c : A) : f a a = 1 :=
begin
rewrite [-(mul_right_id a), -- - means apply symmetry, rewrite 0 ==> a * 0 at 1st and 2nd occurrences
Ax] -- use Ax as rewrite rule
end