From 4fa6ecf53dd170e4079edb582c68a55448047c6a Mon Sep 17 00:00:00 2001
From: Helmut Grohne <helmut@subdivi.de>
Date: Mon, 23 Jan 2012 11:50:28 +0100
Subject: rewrite lemma-1 using propositional equality

---
 Bidir.agda | 25 ++++++++++++++++++-------
 1 file changed, 18 insertions(+), 7 deletions(-)

diff --git a/Bidir.agda b/Bidir.agda
index 32397af..07193c7 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -79,15 +79,26 @@ lemma-insert-same [] () a?
 lemma-insert-same (.(just x) ∷ xs) zero (is-just x) = refl
 lemma-insert-same (x ∷ xs) (suc f′) a? = cong (_∷_ x) (lemma-insert-same xs f′ a?)
 
+lemma-checkInsert-generate : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (i : Fin n) → (is : List (Fin n)) → checkInsert eq i (f i) (generate f is) ≡ just (generate f (i ∷ is))
+lemma-checkInsert-generate eq f i is with lookupM i (generate f is)
+lemma-checkInsert-generate eq f i is | nothing = refl
+lemma-checkInsert-generate eq f i is | just x with eq (f i) x
+lemma-checkInsert-generate eq f i is | just .(f i) | yes refl = cong just (lemma-insert-same (generate f is) i {!!})
+lemma-checkInsert-generate eq f i is | just x | no ¬p = {!!}
+
 lemma-1 : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (is : List (Fin n)) → assoc eq is (map f is) ≡ just (generate f is)
 lemma-1 eq f []        = refl
-lemma-1 eq f (i ∷ is′) with assoc eq is′ (map f is′) | generate f is′ | lemma-1 eq f is′
-lemma-1 eq f (i ∷ is′) | nothing | _ | ()
-lemma-1 eq f (i ∷ is′) | just m | .m | refl with lookup i m
-lemma-1 eq f (i ∷ is′) | just m | .m | refl | nothing = refl
-lemma-1 eq f (i ∷ is′) | just m | .m | refl | just x with eq (f i) x
-lemma-1 eq f (i ∷ is′) | just m | .m | refl | just .(f i) | yes refl = cong just (lemma-insert-same m i {!!})
-lemma-1 eq f (i ∷ is′) | just m | .m | refl | just x | no ¬p = {!!}
+lemma-1 eq f (i ∷ is′) = begin
+  (assoc eq (i ∷ is′) (map f (i ∷ is′)))
+    ≡⟨ refl ⟩
+  (assoc eq is′ (map f is′) >>= checkInsert eq i (f i))
+    ≡⟨ cong (λ m → m >>= checkInsert eq i (f i)) (lemma-1 eq f is′) ⟩
+  (just (generate f is′) >>= (checkInsert eq i (f i)))
+    ≡⟨ refl ⟩
+  (checkInsert eq i (f i) (generate f is′))
+    ≡⟨ lemma-checkInsert-generate eq f i is′ ⟩
+  just (generate f (i ∷ is′)) ∎
+     where open Relation.Binary.PropositionalEquality.≡-Reasoning
 
 lemma-2 : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (is : List (Fin n)) → (v : List τ) → (h : FinMapMaybe n τ) → just h ≡ assoc eq is v → map (flip lookup h) is ≡ map just v
 lemma-2 eq is v h p = {!!}
-- 
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