lean2/tests/lean/tst6.lean.expected.out

  Set: pp::colors
  Set: pp::unicode
  Assumed: N
  Assumed: h
  Proved: congrH
  Set: lean::pp::implicit
variable h : N → N → N
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) :=
    @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
  Set: lean::pp::implicit
variable h : N → N → N
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
  Proved: Example1
  Set: lean::pp::implicit
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) :=
    or::@elim
        (@eq N a b ∧ @eq N b c)
        (@eq N a d ∧ @eq N d c)
        (h a b == h c b)
        H
        (λ H1 : @eq N a b ∧ @eq N b c,
           @congrH a
                   b
                   c
                   b
                   (@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))
                   (@refl N b))
        (λ H1 : @eq N a d ∧ @eq N d c,
           @congrH a
                   b
                   c
                   b
                   (@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))
                   (@refl N b))
  Proved: Example2
  Set: lean::pp::implicit
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) :=
    or::@elim
        (@eq N a b ∧ @eq N b c)
        (@eq N a d ∧ @eq N d c)
        (@eq N (h a b) (h c b))
        H
        (λ H1 : @eq N a b ∧ @eq N b c,
           @congrH a
                   b
                   c
                   b
                   (@trans N a b c (and::@eliml (a == b) (@eq N b c) H1) (and::@elimr (@eq N a b) (b == c) H1))
                   (@refl N b))
        (λ H1 : @eq N a d ∧ @eq N d c,
           @congrH a
                   b
                   c
                   b
                   (@trans N a d c (and::@eliml (a == d) (@eq N d c) H1) (and::@elimr (@eq N a d) (d == c) H1))
                   (@refl N b))
  Proved: Example3
  Set: lean::pp::implicit
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 :=
    or::elim H
             (λ H1 : a = b ∧ b = e ∧ b = c, congrH (and::eliml H1 ⋈ and::elimr (and::elimr H1)) (refl b))
             (λ H1 : a = d ∧ d = c, congrH (and::eliml H1 ⋈ and::elimr H1) (refl b))
  Proved: Example4
  Set: lean::pp::implicit
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 :=
    or::elim H
             (λ H1 : a = b ∧ b = e ∧ b = c,
                let AeqC := and::eliml H1 ⋈ and::elimr (and::elimr H1) in congrH AeqC (symm AeqC))
             (λ H1 : a = d ∧ d = c, let AeqC := and::eliml H1 ⋈ and::elimr H1 in congrH AeqC (symm AeqC))