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32 lines
863 B
Idris
32 lines
863 B
Idris
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------------- Example 1 ----------------
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natInjective : (x, y : Nat) -> S x === S y -> x === y
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natInjective x x Refl = Refl
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succNotZero : (x : Nat) -> Not (S x === Z)
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succNotZero x Refl impossible
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peanoNat : (a : Type
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** n0 : a
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** ns : a -> a
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** inj : ((x, y : a) -> ns x === ns y -> x === y)
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** (x : a) -> Not (ns x === n0))
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peanoNat = MkDPair
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Nat
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$ MkDPair
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Z
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$ MkDPair -- {a = Nat -> Nat}
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S
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$ MkDPair
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natInjective
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succNotZero
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------------- Example 2 ----------------
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ac : forall r. ((x : a) -> (y : b ** r x y)) -> (f : a -> b ** (x : a) -> r x (f x))
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ac g = (\x => fst (g x) ** \x => snd (g x))
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------------- Example 3 ----------------
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idid1 : forall A. (f : A -> A ** (x : A) -> f x = x)
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idid1 = MkDPair id (\x => Refl)
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