lean2/examples/ex15.lean

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Variable N : Type
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
(* Display the theorem showing implicit arguments *)
Set lean::pp::implicit true
Show Environment 2
(* Display the theorem hiding implicit arguments *)
Set lean::pp::implicit false
Show Environment 2
Theorem Example1 (a b c d : N) (H: (a = b ∧ b = c) (a = d ∧ d = c)) : (h a b) = (h c b) :=
DisjCases H
(fun H1 : a = b ∧ b = c,
CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
(fun H1 : a = d ∧ d = c,
CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
(* Show proof of the last theorem with all implicit arguments *)
Set lean::pp::implicit true
Show Environment 1
(* Using placeholders to hide the type of H1 *)
Theorem Example2 (a b c d : N) (H: (a = b ∧ b = c) (a = d ∧ d = c)) : (h a b) = (h c b) :=
DisjCases H
(fun H1 : _,
CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
(fun H1 : _,
CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
Set lean::pp::implicit true
Show Environment 1
(* Same example but the first conjuct has unnecessary stuff *)
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
(fun H1 : _,
CongrH (Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1))) (Refl b))
(fun H1 : _,
CongrH (Trans (Conjunct1 H1) (Conjunct2 H1)) (Refl b))
Set lean::pp::implicit false
Show Environment 1
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
(fun H1 : _,
let AeqC := Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1))
in CongrH AeqC (Symm AeqC))
(fun H1 : _,
let AeqC := Trans (Conjunct1 H1) (Conjunct2 H1)
in CongrH AeqC (Symm AeqC))
Set lean::pp::implicit false
Show Environment 1