42 lines
1.4 KiB
Text
42 lines
1.4 KiB
Text
import data.nat
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open algebra
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constant f {A : Type} : A → A → A
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theorem test1 {A : Type} [s : comm_ring A] (a b c : A) : f (a + 0) (f (a + 0) (a + 0)) = f a (f (0 + a) a) :=
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begin
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rewrite [add_zero at {1, 3}, -- rewrite 1st and 3rd occurrences
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{0 + _}add.comm] -- apply commutativity to (0 + _)
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end
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check @mul_zero
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axiom Ax {A : Type} [s₁ : has_mul A] [s₂ : has_zero A] (a : A) : f (a * 0) (a * 0) = 0
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theorem test2 {A : Type} [s : comm_ring A] (a b c : A) : f 0 0 = (0:A) :=
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begin
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rewrite [
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-(mul_zero a) at {1, 2}, -- - means apply symmetry, rewrite 0 ==> a * 0 at 1st and 2nd occurrences
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Ax] -- use Ax as rewrite rule
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end
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theorem test3 {A : Type} [s : comm_ring A] (a b c : A) : a * 0 + 0 * b + c * 0 + 0 * a = 0 :=
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begin
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rewrite [+mul_zero, +zero_mul, +add_zero] -- in rewrite rules, + is notation for one or more
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end
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reveal test3
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print definition test3
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theorem test4 {A : Type} [s : comm_ring A] (a b c : A) : a * 0 + 0 * b + c * 0 + 0 * a = 0 :=
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begin
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rewrite [*mul_zero, *zero_mul, *add_zero, *zero_add] -- in rewrite rules, * is notation for zero or more
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end
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theorem test5 {A : Type} [s : comm_ring A] (a b c : A) : a * 0 + 0 * b + c * 0 + 0 * a = 0 :=
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begin
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rewrite [
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2 mul_zero, -- apply mul_zero exactly twice
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2 zero_mul, -- apply zero_mul exactly twice
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5>add_zero] -- apply add_zero at most 5 times
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end
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