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::explicit N a1 b1) (H2 : eq::explicit N a2 b2) : eq::explicit
        N
        (h a1 a2)
        (h b1 b2) :=
    Congr::explicit
        N
        (λ x : N, N)
        (h a1)
        (h b1)
        a2
        b2
        (Congr::explicit N (λ x : N, N → N) h h a1 b1 (Refl::explicit (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::explicit N a b ∧ eq::explicit N b c ∨ eq::explicit N a d ∧ eq::explicit N d c) : eq::explicit
        N
        (h a b)
        (h c b) :=
    DisjCases::explicit
        (eq::explicit N a b ∧ eq::explicit N b c)
        (eq::explicit N a d ∧ eq::explicit N d c)
        (h a b == h c b)
        H
        (λ H1 : eq::explicit N a b ∧ eq::explicit N b c,
           CongrH::explicit
               a
               b
               c
               b
               (Trans::explicit
                    N
                    a
                    b
                    c
                    (Conjunct1::explicit (eq::explicit N a b) (eq::explicit N b c) H1)
                    (Conjunct2::explicit (eq::explicit N a b) (eq::explicit N b c) H1))
               (Refl::explicit N b))
        (λ H1 : eq::explicit N a d ∧ eq::explicit N d c,
           CongrH::explicit
               a
               b
               c
               b
               (Trans::explicit
                    N
                    a
                    d
                    c
                    (Conjunct1::explicit (eq::explicit N a d) (eq::explicit N d c) H1)
                    (Conjunct2::explicit (eq::explicit N a d) (eq::explicit N d c) H1))
               (Refl::explicit N b))
  Proved: Example2
  Set: lean::pp::implicit
Theorem Example2 (a b c d : N)
    (H : eq::explicit N a b ∧ eq::explicit N b c ∨ eq::explicit N a d ∧ eq::explicit N d c) : eq::explicit
        N
        (h a b)
        (h c b) :=
    DisjCases::explicit
        (eq::explicit N a b ∧ eq::explicit N b c)
        (eq::explicit N a d ∧ eq::explicit N d c)
        (eq::explicit N (h a b) (h c b))
        H
        (λ H1 : eq::explicit N a b ∧ eq::explicit N b c,
           CongrH::explicit
               a
               b
               c
               b
               (Trans::explicit
                    N
                    a
                    b
                    c
                    (Conjunct1::explicit (a == b) (b == c) H1)
                    (Conjunct2::explicit (a == b) (b == c) H1))
               (Refl::explicit N b))
        (λ H1 : eq::explicit N a d ∧ eq::explicit N d c,
           CongrH::explicit
               a
               b
               c
               b
               (Trans::explicit
                    N
                    a
                    d
                    c
                    (Conjunct1::explicit (a == d) (d == c) H1)
                    (Conjunct2::explicit (a == d) (d == c) H1))
               (Refl::explicit 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))