107 lines
3.7 KiB
Text
107 lines
3.7 KiB
Text
Set option: pp::colors
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Assumed: N
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Assumed: h
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Proved: CongrH
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Set option: lean::pp::implicit
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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::explicit
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N
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(λ x : N, N)
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(h a1)
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(h b1)
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a2
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b2
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(Congr::explicit N (λ x : N, N → N) h h a1 b1 (Refl::explicit (N → N → N) h) H1)
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H2
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Theorem CongrH::explicit (a1 a2 b1 b2 : N) (H1 : a1 = b1) (H2 : a2 = b2) : (h a1 a2) = (h b1 b2) :=
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CongrH::explicit a1 a2 b1 b2 H1 H2
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Set option: lean::pp::implicit
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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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Theorem CongrH::explicit (a1 a2 b1 b2 : N) (H1 : a1 = b1) (H2 : a2 = b2) : (h a1 a2) = (h b1 b2) := CongrH H1 H2
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Proved: Example1
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Set option: lean::pp::implicit
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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::explicit
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(a = b ∧ b = c)
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(a = d ∧ d = c)
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((h a b) = (h c b))
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H
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(λ H1 : a = b ∧ b = c,
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CongrH::explicit
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a
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b
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c
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b
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(Trans::explicit
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N
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a
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b
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c
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(Conjunct1::explicit (a = b) (b = c) H1)
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(Conjunct2::explicit (a = b) (b = c) H1))
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(Refl::explicit N b))
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(λ H1 : a = d ∧ d = c,
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CongrH::explicit
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a
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b
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c
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b
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(Trans::explicit
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N
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a
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d
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c
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(Conjunct1::explicit (a = d) (d = c) H1)
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(Conjunct2::explicit (a = d) (d = c) H1))
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(Refl::explicit N b))
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Proved: Example2
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Set option: lean::pp::implicit
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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::explicit
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(a = b ∧ b = c)
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(a = d ∧ d = c)
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((h a b) = (h c b))
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H
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(λ H1 : a = b ∧ b = c,
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CongrH::explicit
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a
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b
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c
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b
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(Trans::explicit
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N
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a
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b
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c
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(Conjunct1::explicit (a = b) (b = c) H1)
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(Conjunct2::explicit (a = b) (b = c) H1))
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(Refl::explicit N b))
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(λ H1 : a = d ∧ d = c,
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CongrH::explicit
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a
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b
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c
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b
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(Trans::explicit
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N
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a
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d
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c
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(Conjunct1::explicit (a = d) (d = c) H1)
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(Conjunct2::explicit (a = d) (d = c) H1))
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(Refl::explicit N b))
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Proved: Example3
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Set option: 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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DisjCases
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H
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(λ H1 : a = b ∧ b = e ∧ b = c, CongrH (Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1))) (Refl b))
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(λ H1 : a = d ∧ d = c, CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
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Proved: Example4
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Set option: 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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DisjCases
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H
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(λ H1 : a = b ∧ b = e ∧ b = c,
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let AeqC := Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1)) in CongrH AeqC (Symm AeqC))
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(λ H1 : a = d ∧ d = c, let AeqC := Trans (Conjunct1 H1) (Conjunct2 H1) in CongrH AeqC (Symm AeqC))
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