lean2/tests/lean/tst6.lean.expected.out
Leonardo de Moura 935c2a03a3 feat(*): change name conventions for Lean builtin libraries
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
2014-01-05 19:21:44 -08:00

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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))