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) :=
    @DisjCases
        (@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 (@Conjunct1 (@eq N a b) (@eq N b c) H1) (@Conjunct2 (@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 (@Conjunct1 (@eq N a d) (@eq N d c) H1) (@Conjunct2 (@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) :=
    @DisjCases
        (@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 (@Conjunct1 (a == b) (@eq N b c) H1) (@Conjunct2 (@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 (@Conjunct1 (a == d) (@eq N d c) H1) (@Conjunct2 (@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 :=
    DisjCases
        H
        (λ H1 : a = b ∧ b = e ∧ b = c, CongrH (Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1))) (Refl b))
        (λ H1 : a = d ∧ d = c, CongrH (Trans (Conjunct1 H1) (Conjunct2 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 :=
    DisjCases
        H
        (λ H1 : a = b ∧ b = e ∧ b = c,
           let AeqC := Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1)) in CongrH AeqC (Symm AeqC))
        (λ H1 : a = d ∧ d = c, let AeqC := Trans (Conjunct1 H1) (Conjunct2 H1) in CongrH AeqC (Symm AeqC))