58 lines
2.1 KiB
Text
58 lines
2.1 KiB
Text
Variable N : Type
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Variable h : N -> N -> N
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Theorem CongrH {a1 a2 b1 b2 : N} (H1 : a1 = b1) (H2 : a2 = b2) : (h a1 a2) = (h b1 b2) :=
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Congr (Congr (Refl h) H1) H2
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(* Display the theorem showing implicit arguments *)
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Set lean::pp::implicit true
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Show Environment 2
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(* Display the theorem hiding implicit arguments *)
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Set lean::pp::implicit false
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Show Environment 2
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Theorem Example1 (a b c d : N) (H: (a = b ∧ b = c) ∨ (a = d ∧ d = c)) : (h a b) = (h c b) :=
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DisjCases H
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(fun H1 : a = b ∧ b = c,
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CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
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(fun H1 : a = d ∧ d = c,
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CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
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(* Show proof of the last theorem with all implicit arguments *)
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Set lean::pp::implicit true
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Show Environment 1
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(* Using placeholders to hide the type of H1 *)
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Theorem Example2 (a b c d : N) (H: (a = b ∧ b = c) ∨ (a = d ∧ d = c)) : (h a b) = (h c b) :=
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DisjCases H
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(fun H1 : _,
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CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
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(fun H1 : _,
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CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
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Set lean::pp::implicit true
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Show Environment 1
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(* Same example but the first conjuct has unnecessary stuff *)
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Theorem Example3 (a b c d e : N) (H: (a = b ∧ b = e ∧ b = c) ∨ (a = d ∧ d = c)) : (h a b) = (h c b) :=
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DisjCases H
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(fun H1 : _,
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CongrH (Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1))) (Refl b))
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(fun H1 : _,
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CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
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Set lean::pp::implicit false
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Show Environment 1
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Theorem Example4 (a b c d e : N) (H: (a = b ∧ b = e ∧ b = c) ∨ (a = d ∧ d = c)) : (h a c) = (h c a) :=
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DisjCases H
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(fun H1 : _,
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let AeqC := Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1))
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in CongrH AeqC (Symm AeqC))
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(fun H1 : _,
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let AeqC := Trans (Conjunct1 H1) (Conjunct2 H1)
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in CongrH AeqC (Symm AeqC))
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Set lean::pp::implicit false
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Show Environment 1
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