69 lines
2.9 KiB
Text
69 lines
2.9 KiB
Text
Set: pp::colors
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Set: pp::unicode
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Assumed: N
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Assumed: h
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Proved: congrH
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Set: lean::pp::implicit
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variable h : N → N → N
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theorem congrH {a1 a2 b1 b2 : N} (H1 : @eq N a1 b1) (H2 : @eq N a2 b2) : @eq N (h a1 a2) (h b1 b2) :=
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@congr N (λ x : N, N) (h a1) (h b1) a2 b2 (@congr N (λ x : N, N → N) h h a1 b1 (@refl (N → N → N) h) H1) H2
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Set: lean::pp::implicit
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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 := congr (congr (refl h) H1) H2
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Proved: Example1
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Set: lean::pp::implicit
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theorem Example1 (a b c d : N) (H : @eq N a b ∧ @eq N b c ∨ @eq N a d ∧ @eq N d c) : @eq N (h a b) (h c b) :=
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or::@elim
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(@eq N a b ∧ @eq N b c)
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(@eq N a d ∧ @eq N d c)
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(h a b == h c b)
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H
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(λ H1 : @eq N a b ∧ @eq N b c,
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@congrH a
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b
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c
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b
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(@trans N a b c (and::@eliml (@eq N a b) (@eq N b c) H1) (and::@elimr (@eq N a b) (@eq N b c) H1))
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(@refl N b))
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(λ H1 : @eq N a d ∧ @eq N d c,
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@congrH a
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b
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c
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b
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(@trans N a d c (and::@eliml (@eq N a d) (@eq N d c) H1) (and::@elimr (@eq N a d) (@eq N d c) H1))
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(@refl N b))
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Proved: Example2
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Set: lean::pp::implicit
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theorem Example2 (a b c d : N) (H : @eq N a b ∧ @eq N b c ∨ @eq N a d ∧ @eq N d c) : @eq N (h a b) (h c b) :=
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or::@elim
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(@eq N a b ∧ @eq N b c)
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(@eq N a d ∧ @eq N d c)
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(@eq N (h a b) (h c b))
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H
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(λ H1 : @eq N a b ∧ @eq N b c,
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@congrH a
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b
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c
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b
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(@trans N a b c (and::@eliml (a == b) (@eq N b c) H1) (and::@elimr (@eq N a b) (b == c) H1))
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(@refl N b))
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(λ H1 : @eq N a d ∧ @eq N d c,
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@congrH a
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b
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c
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b
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(@trans N a d c (and::@eliml (a == d) (@eq N d c) H1) (and::@elimr (@eq N a d) (d == c) H1))
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(@refl N b))
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Proved: Example3
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Set: lean::pp::implicit
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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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or::elim H
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(λ H1 : a = b ∧ b = e ∧ b = c, congrH (and::eliml H1 ⋈ and::elimr (and::elimr H1)) (refl b))
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(λ H1 : a = d ∧ d = c, congrH (and::eliml H1 ⋈ and::elimr H1) (refl b))
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Proved: Example4
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Set: lean::pp::implicit
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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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or::elim H
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(λ H1 : a = b ∧ b = e ∧ b = c,
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let AeqC := and::eliml H1 ⋈ and::elimr (and::elimr H1) in congrH AeqC (symm AeqC))
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(λ H1 : a = d ∧ d = c, let AeqC := and::eliml H1 ⋈ and::elimr H1 in congrH AeqC (symm AeqC))
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