43 lines
1 KiB
Text
43 lines
1 KiB
Text
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import data.unit
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open nat unit
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constant f {A : Type} (a : A) {B : Type} (b : B) : nat
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constant g : unit → nat
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example (a b : unit) : g a = g b :=
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by simp
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example (a c : unit) (b d : nat) : b = d → f a b = f c d :=
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by simp
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constant h {A B : Type} : A → B → nat
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example (a b c d : unit) : h a b = h c d :=
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by simp
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definition C [reducible] : nat → Type₁
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| nat.zero := unit
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| (nat.succ a) := nat
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constant g₂ : Π {n : nat}, C n → nat → nat
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example (a b : C zero) (c d : nat) : c = d → g₂ a c = g₂ b d :=
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by simp
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example (n : nat) (h : zero = n) (a b : C n) (c d : nat) : c = d → g₂ a c = g₂ b d :=
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by simp
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-- The following one cannot be solved as is
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-- example (a c : nat) (b d : unit) : a = c → b = d → f a b = f c d :=
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-- by simp
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-- But, we can use the following trick
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definition f_aux {A B : Type} (a : A) (b : B) := f a b
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lemma to_f_aux [simp] {A B : Type} (a : A) (b : B) : f a b = f_aux a b :=
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rfl
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example (a c : nat) (b d : unit) : a = c → b = d → f a b = f c d :=
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by simp
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