test(tests/lean): rewriting proof example

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

tests/lean/simp2.lean

@@ -0,0 +1,44 @@
+-- We can also use Lean simplifier to simplify proofs.
+
+-- First, we prove some theorems about trans. All theorems are just using the fact that
+-- we have proof irrelevance in Lean. That is, all proofs of a fact A are considered equal.
+theorem trans_refl_right {A : TypeU} {a b : A} (Hab : a = b) (Hbb : b = b) : trans Hab Hbb = Hab
+:= proof_irrel (trans Hab Hbb) Hab
+
+theorem trans_refl_left {A : TypeU} {a b : A} (Haa : a = a) (Hab : a = b) : trans Haa Hab = Hab
+:= proof_irrel (trans Haa Hab) Hab
+
+theorem trans_assoc {A : TypeU} {a b c d : A} (Hab : a = b) (Hbc : b = c) (Hcd : c = d) :
+         trans (trans Hab Hbc) Hcd = trans Hab (trans Hbc Hcd)
+:= proof_irrel (trans (trans Hab Hbc) Hcd) (trans Hab (trans Hbc Hcd))
+
+theorem symm_trans {A : TypeU} {a b c : A} (Hab : a = b) (Hbc : b = c) :
+         symm (trans Hab Hbc) = trans (symm Hbc) (symm Hab)
+:= proof_irrel (symm (trans Hab Hbc)) (trans (symm Hbc) (symm Hab))
+
+theorem trans_valley1 {A : TypeU} {a b c d : A} (Hab : a = b) (Hbc : b = c) (Hcb : c = b) (Hbd : b = d) :
+         trans Hab (trans Hbc (trans Hcb Hbd)) = trans Hab Hbd
+:= proof_irrel (trans Hab (trans Hbc (trans Hcb Hbd))) (trans Hab Hbd)
+
+theorem trans_valley2 {A : TypeU} {a b c d e : A} (Hab : a = b) (Hbc : b = c) (Hcb : c = b) (Hbd : b = d) (Hde : d = e) :
+         trans Hab (trans Hbc (trans Hcb (trans Hbd Hde))) = trans Hab (trans Hbd Hde)
+:= proof_irrel (trans Hab (trans Hbc (trans Hcb (trans Hbd Hde)))) (trans Hab (trans Hbd Hde))
+
+
+-- Now, we define a new rewrite rule set for these theorems
+rewrite_set proof_simp
+add_rewrite trans_refl_right trans_refl_left trans_assoc symm_trans : proof_simp
+
+-- Now, we create a simple example where a "messy" proof is simplified
+variables a1 a2 a3 a4 : Nat
+axiom Ax1 : a1 = a2
+axiom Ax2 : a2 = a3
+axiom Ax3 : a3 = a2
+axiom Ax4 : a2 = a4
+
+(*
+local messy_proof = parse_lean("symm (trans (trans (trans (refl a1) (trans (trans Ax1 (refl a2)) Ax2)) (refl a3)) (trans Ax3 Ax4))")
+print(messy_proof)
+clean_proof, _ = simplify(messy_proof, 'proof_simp')
+print(clean_proof)
+*)

tests/lean/simp2.lean.expected.out

@@ -0,0 +1,18 @@
+  Set: pp::colors
+  Set: pp::unicode
+  Proved: trans_refl_right
+  Proved: trans_refl_left
+  Proved: trans_assoc
+  Proved: symm_trans
+  Proved: trans_valley1
+  Proved: trans_valley2
+  Assumed: a1
+  Assumed: a2
+  Assumed: a3
+  Assumed: a4
+  Assumed: Ax1
+  Assumed: Ax2
+  Assumed: Ax3
+  Assumed: Ax4
+symm (trans (trans (trans (refl a1) (trans (trans Ax1 (refl a2)) Ax2)) (refl a3)) (trans Ax3 Ax4))
+trans (symm Ax4) (symm Ax1)