X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fif_machine.ma;h=f778ff3085c6f8af5b1e6bbc2b0c18481184ee47;hb=5535cd4e08fd8d1e7e6e067eac1bb6c1bf8fcbbf;hp=28983b11eb86b4a54bc5e7efbf8f4e2cd77ea4c1;hpb=28c261aad4d710c9f218fbbbaf1ed3b45c1caf72;p=helm.git

diff --git a/matita/matita/lib/turing/if_machine.ma b/matita/matita/lib/turing/if_machine.ma
index 28983b11e..f778ff308 100644
--- a/matita/matita/lib/turing/if_machine.ma
+++ b/matita/matita/lib/turing/if_machine.ma
@@ -11,17 +11,33 @@
 
 include "turing/mono.ma".
 
+(**************************** single final machine ****************************)
+
 definition single_finalTM ≝ 
   λsig.λM:TM sig.seq ? M (nop ?).
 
-lemma sem_single_final :
-  ∀sig,M,R.Realize ? M R → Realize ? (single_finalTM sig M) R.
+lemma sem_single_final: ∀sig.∀M: TM sig.∀R.
+  M ⊨ R → single_finalTM sig M ⊨ R.
 #sig #M #R #HR #intape 
 cases (sem_seq ????? HR (sem_nop …) intape)
 #k * #outc * #Hloop * #ta * #Hta whd in ⊢ (%→?); #Houtc
 @(ex_intro ?? k) @(ex_intro ?? outc) %  [ @Hloop | >Houtc // ]
 qed.
 
+lemma single_final: ∀sig.∀M: TM sig.∀q1,q2.
+  halt ? (single_finalTM sig M) q1 = true 
+    →  halt ? (single_finalTM sig M) q2 = true → q1=q2.
+#sig #M * 
+  [#q1M #q2 whd in match (halt ???); #H destruct
+  |#q1nop *
+    [#q2M #_ whd in match (halt ???); #H destruct
+    |#q2nop #_ #_ @eq_f normalize @nop_single_state
+    ]
+  ]
+qed.
+  
+(******************************** if machine **********************************)
+
 definition if_trans ≝ λsig. λM1,M2,M3 : TM sig. λq:states sig M1.
 λp. let 〈s,a〉 ≝ p in
   match s with 
@@ -51,60 +67,384 @@ definition ifTM ≝ λsig. λcondM,thenM,elseM : TM sig.
       | inr s' ⇒ match s' with 
         [ inl s2 ⇒ halt sig thenM s2
         | inr s3 ⇒ halt sig elseM s3 ]]).
-        
-axiom daemon : ∀X:Prop.X.
-axiom tdaemon : ∀X:Type[0].X.
-
-axiom sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
-  accRealize sig M1 acc Rtrue Rfalse → Realize sig M2 R2 → Realize sig M3 R3 → 
-    Realize sig (ifTM sig M1 M2 M3 acc) (λt1,t2. (Rtrue ∘ R2) t1 t2 ∨ (Rfalse ∘ R3) t1 t2).
-(*    
+
+(****************************** lifting lemmas ********************************)
+lemma trans_if_liftM1 : ∀sig,M1,M2,M3,acc,s,a,news,move.
+  halt ? M1 s = false → 
+  trans sig M1 〈s,a〉 = 〈news,move〉 → 
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inl … news,move〉.
+#sig * #Q1 #T1 #init1 #halt1 #M2 #M3 #acc #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
+
+lemma trans_if_liftM2 : ∀sig,M1,M2,M3,acc,s,a,news,move.
+  halt ? M2 s = false → 
+  trans sig M2 〈s,a〉 = 〈news,move〉 → 
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inl … s),a〉 = 〈inr… (inl … news),move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
+
+lemma trans_if_liftM3 : ∀sig,M1,M2,M3,acc,s,a,news,move.
+  halt ? M3 s = false → 
+  trans sig M3 〈s,a〉 = 〈news,move〉 → 
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inr … (inr … s),a〉 = 〈inr… (inr … news),move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #M3 #acc #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
+
+lemma step_if_liftM1 : ∀sig,M1,M2,M3,acc,c0.
+ halt ? M1 (cstate ?? c0) = false → 
+ step sig (ifTM sig M1 M2 M3 acc) (lift_confL sig (states ? M1) ? c0) =
+ lift_confL sig (states ? M1) ? (step sig M1 c0).
+#sig #M1 #M2 #M3 #acc * #s #t
+  lapply (refl ? (trans ?? 〈s,current sig t〉))
+  cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+  #s0 #m0 cases t
+  [ #Heq #Hhalt
+  | 2,3: #s1 #l1 #Heq #Hhalt 
+  |#ls #s1 #rs #Heq #Hhalt ]
+  whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_if_liftM1 … Hhalt Heq) //
+qed.
+
+lemma step_if_liftM2 : ∀sig,M1,M2,M3,acc,c0.
+ halt ? M2 (cstate ?? c0) = false → 
+ step sig (ifTM sig M1 M2 M3 acc) (lift_confR sig ?? (lift_confL sig ?? c0)) =
+ lift_confR sig ?? (lift_confL sig ?? (step sig M2 c0)).
+#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
+  lapply (refl ? (trans ?? 〈s,current sig t〉))
+  cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+  #s0 #m0 cases t
+  [ #Heq #Hhalt
+  | 2,3: #s1 #l1 #Heq #Hhalt 
+  |#ls #s1 #rs #Heq #Hhalt ] 
+  whd in match (step ? M2 ?); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_if_liftM2 … Hhalt Heq) //
+qed.
+
+lemma step_if_liftM3 : ∀sig,M1,M2,M3,acc,c0.
+ halt ? M3 (cstate ?? c0) = false → 
+ step sig (ifTM sig M1 M2 M3 acc) (lift_confR sig ?? (lift_confR sig ?? c0)) =
+ lift_confR sig ?? (lift_confR sig ?? (step sig M3 c0)).
+#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 #M3 #acc * #s #t
+  lapply (refl ? (trans ?? 〈s,current sig t〉))
+  cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
+  #s0 #m0 cases t
+  [ #Heq #Hhalt
+  | 2,3: #s1 #l1 #Heq #Hhalt 
+  |#ls #s1 #rs #Heq #Hhalt ] 
+  whd in match (step ? M3 ?); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_if_liftM3 … Hhalt Heq) //
+qed.
+
+lemma trans_if_M1true_acc : ∀sig,M1,M2,M3,acc,s,a.
+  halt ? M1 s = true → s==acc = true →
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inl … (start ? M2)),None ?〉.
+#sig #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
+qed.
+
+lemma trans_if_M1true_notacc : ∀sig,M1,M2,M3,acc,s,a.
+  halt ? M1 s = true → s==acc = false →
+  trans sig (ifTM sig M1 M2 M3 acc) 〈inl … s,a〉 = 〈inr … (inr … (start ? M3)),None ?〉.
+#sig #M1 #M2 #M3 #acc #s #a #Hhalt #Hacc whd in ⊢ (??%?); >Hhalt >Hacc %
+qed.
+
+(******************************** semantics ***********************************)
+lemma sem_if: ∀sig.∀M1,M2,M3:TM sig.∀Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ [acc: Rtrue,Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 → 
+    ifTM sig M1 M2 M3 acc ⊨ (Rtrue ∘ R2) ∪ (Rfalse ∘ R3).
 #sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t 
 cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse 
 cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
   [cases (HR2 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM2
-   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confL … outc2)))
-   %
-   [@(loop_merge ??????????? 
-    (loop_lift ??? 
-      (lift_confL sig (states ? M1) (FinSum (states ? M2) (states ? M3)))
-      (step sig M1) (step sig (ifTM sig M1 M2 M3 acc)) 
-      (λc.halt sig M1 (cstate … c)) 
-      (λc.halt_liftL ?? (halt sig M1) (cstate … c)) 
-      … Hloop1))
-     [* *
-       [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
-       | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
-     || #c0 #Hhalt @daemon  (* <step_lift_confL // *)
-     | #x <p_halt_liftL %
-     |6:cases outc1 #s1 #t1 %
-     |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2) 
-       [ * #s2 #t2 %
-       | #c0 #Hhalt @daemon (* <step_lift_confR // *) ]
-     |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
-      generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
-      >(trans_liftL_true sig M1 M2 ??) 
-       [ whd in ⊢ (??%?); whd in ⊢ (???%);
-         @daemon
-(*         @config_eq whd in ⊢ (???%); // *)
-       | @(loop_Some ?????? Hloop10)
-       | @tdaemon 
-       | skip ]
+   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+    [@(loop_merge ?????????
+     (mk_config ? (FinSum (states sig M1) (FinSum (states sig M2) (states sig M3)))
+      (inr (states sig M1) ? (inl (states sig M2) (states sig M3) (start sig M2))) (ctape ?? outc1) )
+     ? 
+     (loop_lift ??? 
+       (lift_confL sig (states ? M1) (FinSum (states ? M2) (states ? M3)))
+       (step sig M1) (step sig (ifTM sig M1 M2 M3 acc)) 
+       (λc.halt sig M1 (cstate … c)) 
+       (λc.halt_liftL ?? (halt sig M1) (cstate … c)) 
+       … Hloop1))
+      [* *
+        [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+        | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+      |#c0 #Hhalt >(step_if_liftM1 … Hhalt) // 
+      |#x <p_halt_liftL %
+      |whd in ⊢ (??%?); >(config_expand ?? outc1);
+       whd in match (lift_confL ????);
+       >(trans_if_M1true_acc … Hacc) 
+        [% | @(loop_Some ?????? Hloop1)]
+      |cases outc1 #s1 #t1 %
+      |@(loop_lift ??? 
+         (λc.(lift_confR … (lift_confL sig (states ? M2) (states ? M3) c)))
+         … Hloop2) 
+        [ * #s2 #t2 %
+        | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+      ]
+    |%1 @(ex_intro … (ctape ?? outc1)) % 
+      [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
+    ]
+  |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+    [@(loop_merge ?????????
+     (mk_config ? (FinSum (states sig M1) (FinSum (states sig M2) (states sig M3)))
+      (inr (states sig M1) ? (inr (states sig M2) (states sig M3) (start sig M3))) (ctape ?? outc1) )
+     ? 
+     (loop_lift ??? 
+       (lift_confL sig (states ? M1) (FinSum (states ? M2) (states ? M3)))
+       (step sig M1) (step sig (ifTM sig M1 M2 M3 acc)) 
+       (λc.halt sig M1 (cstate … c)) 
+       (λc.halt_liftL ?? (halt sig M1) (cstate … c)) 
+       … Hloop1))
+      [* *
+        [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+        | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+      |#c0 #Hhalt >(step_if_liftM1 … Hhalt) // 
+      |#x <p_halt_liftL %
+      |whd in ⊢ (??%?); >(config_expand ?? outc1);
+       whd in match (lift_confL ????);
+       >(trans_if_M1true_notacc … Hacc) 
+        [% | @(loop_Some ?????? Hloop1)]
+      |cases outc1 #s1 #t1 %
+      |@(loop_lift ??? 
+         (λc.(lift_confR … (lift_confR sig (states ? M2) (states ? M3) c)))
+         … Hloop2) 
+        [ * #s2 #t2 %
+        | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+      ]
+    |%2 @(ex_intro … (ctape ?? outc1)) % 
+      [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
+    ]
+  ]
+qed.
+
+lemma sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
+  accRealize sig M1 acc Rtrue Rfalse  → M2 ⊨ R2  → M3 ⊨ R3 →  
+    (∀t1,t2,t3. (Rtrue t1 t3 → R2 t3 t2) ∨ (Rfalse t1 t3 → R3 t3 t2) → R4 t1 t2) → 
+    ifTM sig M1 M2 M3 acc ⊨ R4.
+#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #R4 #acc
+#HRacc #HRtrue #HRfalse #Hsub
+#t cases (sem_if … HRacc HRtrue HRfalse t)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop] cases Houtc
+  [* #t3 * #Hleft #Hright @(Hsub … t3) %1 /2/
+  |* #t3 * #Hleft #Hright @(Hsub … t3) %2 /2/ ]
+qed.
+ 
+(* we can probably use acc_sem_if to prove sem_if *)
+(* for sure we can use acc_sem_if_guarded to prove acc_sem_if *)
+lemma acc_sem_if: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ [acc: Rtrue, Rfalse]  → M2 ⊨ R2 → M3 ⊨ R3 → 
+  ifTM sig M1 (single_finalTM … M2) M3 acc ⊨
+     [inr … (inl … (inr … start_nop)): Rtrue ∘ R2, Rfalse ∘ R3].
+#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HR3 #t 
+cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse 
+cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
+  [lapply (sem_single_final … HR2) -HR2 #HR2
+   cases (HR2 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM2
+   @(ex_intro … (k1+k2)) 
+   @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+    [%
+      [@(loop_merge ?????????
+         (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+          (inr (states sig M1) ? (inl ? (states sig M3) (start sig (single_finalTM sig M2)))) (ctape ?? outc1) )
+         ? 
+         (loop_lift ??? 
+          (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+          (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? M2) M3 acc)) 
+          (λc.halt sig M1 (cstate … c)) 
+          (λc.halt_liftL ?? (halt sig M1) (cstate … c)) 
+          … Hloop1))
+        [* *
+          [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+          | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+        |#c0 #Hhalt >(step_if_liftM1 … Hhalt) // 
+        |#x <p_halt_liftL %
+        |whd in ⊢ (??%?); >(config_expand ?? outc1);
+         whd in match (lift_confL ????);
+         >(trans_if_M1true_acc … Hacc) 
+          [% | @(loop_Some ?????? Hloop1)]
+        |cases outc1 #s1 #t1 %
+        |@(loop_lift ??? 
+           (λc.(lift_confR … (lift_confL sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+           … Hloop2) 
+          [ * #s2 #t2 %
+          | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+        ]
+      |#_ @(ex_intro … (ctape ?? outc1)) % 
+        [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
+      ]
+    |>(config_expand ?? outc2) whd in match (lift_confR ????);
+     * #H @False_ind @H @eq_f @eq_f >(config_expand ?? outc2)
+     @single_final // @(loop_Some ?????? Hloop2)
+    ]
+  |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+    [%
+      [@(loop_merge ?????????
+         (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+          (inr (states sig M1) ? (inr (states sig (single_finalTM ? M2)) ? (start sig M3))) (ctape ?? outc1) )
+         ? 
+         (loop_lift ??? 
+          (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+          (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? M2) M3 acc)) 
+          (λc.halt sig M1 (cstate … c)) 
+          (λc.halt_liftL ?? (halt sig M1) (cstate … c)) 
+          … Hloop1))
+        [* *
+          [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+          | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+        |#c0 #Hhalt >(step_if_liftM1 … Hhalt) // 
+        |#x <p_halt_liftL %
+        |whd in ⊢ (??%?); >(config_expand ?? outc1);
+         whd in match (lift_confL ????);
+         >(trans_if_M1true_notacc … Hacc) 
+          [% | @(loop_Some ?????? Hloop1)]
+        |cases outc1 #s1 #t1 %
+        |@(loop_lift ??? 
+           (λc.(lift_confR … (lift_confR sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+           … Hloop2) 
+          [ * #s2 #t2 %
+          | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+        ]
+      |>(config_expand ?? outc2) whd in match (lift_confR ????);
+       #H destruct (H) 
+      ]
+    |#_ @(ex_intro … (ctape ?? outc1)) % 
+      [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
+    ]
+  ]
+qed.
+    
+lemma acc_sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,R5,acc.
+  M1 ⊨ [acc: Rtrue, Rfalse] → M2 ⊨ R2 → M3 ⊨ R3 → 
+    (∀t1,t2,t3. Rtrue t1 t3 → R2 t3 t2 → R4 t1 t2) → 
+    (∀t1,t2,t3. Rfalse t1 t3 → R3 t3 t2 → R5 t1 t2) → 
+     ifTM sig M1 (single_finalTM … M2) M3 acc ⊨ 
+       [inr … (inl … (inr … start_nop)): R4, R5].    
+#sig #M1 #M2 #M3 #Rtrue #Rfalse #R2 #R3 #R4 #R5 #acc
+#HRacc #HRtrue #HRfalse #Hsub1 #Hsub2 
+#t cases (acc_sem_if … HRacc HRtrue HRfalse t)
+#k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
+% [% [@Hloop
+     |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
+  |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]
+qed.
+
+lemma sem_single_final_guarded: ∀sig.∀M: TM sig.∀Pre,R.
+  GRealize sig M Pre R → GRealize sig (single_finalTM sig M) Pre R.
+#sig #M #Pre #R #HR #intape #HPre 
+cases (sem_seq_guarded ??????? HR (Realize_to_GRealize ?? (λt.True) ? (sem_nop …)) ?? HPre) //
+#k * #outc * #Hloop * #ta * #Hta whd in ⊢ (%→?); #Houtc
+@(ex_intro ?? k) @(ex_intro ?? outc) %  [ @Hloop | >Houtc // ]
+qed.
+    
+lemma acc_sem_if_guarded: ∀sig,M1,M2,M3,P,P2,Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ [acc: Rtrue, Rfalse]  → 
+  (GRealize ? M2 P2 R2) → (∀t,t0.P t → Rtrue t t0 → P2 t0) → 
+  M3 ⊨ R3 → 
+  accGRealize ? (ifTM sig M1 (single_finalTM … M2) M3 acc) 
+    (inr … (inl … (inr … start_nop))) P (Rtrue ∘ R2) (Rfalse ∘ R3).
+#sig #M1 #M2 #M3 #P #P2 #Rtrue #Rfalse #R2 #R3 #acc #HaccR #HR2 #HP2 #HR3 #t #HPt 
+cases (HaccR t) #k1 * #outc1 * * #Hloop1 #HMtrue #HMfalse 
+cases (true_or_false (cstate ?? outc1 == acc)) #Hacc
+  [lapply (sem_single_final_guarded … HR2) -HR2 #HR2
+   cases (HR2 (ctape sig ? outc1) ?)
+   [|@HP2 [||@HMtrue @(\P Hacc)] // ]
+   #k2 * #outc2 * #Hloop2 #HM2
+   @(ex_intro … (k1+k2)) 
+   @(ex_intro … (lift_confR … (lift_confL … outc2))) %
+    [%
+      [@(loop_merge ?????????
+         (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+          (inr (states sig M1) ? (inl ? (states sig M3) (start sig (single_finalTM sig M2)))) (ctape ?? outc1) )
+         ? 
+         (loop_lift ??? 
+          (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+          (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? M2) M3 acc)) 
+          (λc.halt sig M1 (cstate … c)) 
+          (λc.halt_liftL ?? (halt sig M1) (cstate … c)) 
+          … Hloop1))
+        [* *
+          [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+          | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+        |#c0 #Hhalt >(step_if_liftM1 … Hhalt) // 
+        |#x <p_halt_liftL %
+        |whd in ⊢ (??%?); >(config_expand ?? outc1);
+         whd in match (lift_confL ????);
+         >(trans_if_M1true_acc … Hacc) 
+          [% | @(loop_Some ?????? Hloop1)]
+        |cases outc1 #s1 #t1 %
+        |@(loop_lift ??? 
+           (λc.(lift_confR … (lift_confL sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+           … Hloop2) 
+          [ * #s2 #t2 %
+          | #c0 #Hhalt >(step_if_liftM2 … Hhalt) // ]
+        ]
+      |#_ @(ex_intro … (ctape ?? outc1)) % 
+        [@HMtrue @(\P Hacc) | >(config_expand ?? outc2) @HM2 ]
       ]
-     |
-     |4:cases outc1 #s1 #t1 %
-     |5:
-         
-         @(loop_liftR … Hloop2) 
-         |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
-          generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
-          >(trans_liftL_true sig M1 M2 ??) 
-           [ whd in ⊢ (??%?); whd in ⊢ (???%);
-             @config_eq //
-           | @(loop_Some ?????? Hloop10) ]
-           ]
-| @(ex_intro … (ctape ? (seq sig M1 M2) (lift_confL … outc1)))
-  % //
-]
-qed.
-*)
\ No newline at end of file
+    |>(config_expand ?? outc2) whd in match (lift_confR ????);
+     * #H @False_ind @H @eq_f @eq_f >(config_expand ?? outc2)
+     @single_final // @(loop_Some ?????? Hloop2)
+    ]
+  |cases (HR3 (ctape sig ? outc1)) #k2 * #outc2 * #Hloop2 #HM3
+   @(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … (lift_confR … outc2))) %
+    [%
+      [@(loop_merge ?????????
+         (mk_config ? (states sig (ifTM sig M1 (single_finalTM … M2) M3 acc))
+          (inr (states sig M1) ? (inr (states sig (single_finalTM ? M2)) ? (start sig M3))) (ctape ?? outc1) )
+         ? 
+         (loop_lift ??? 
+          (lift_confL sig (states ? M1) (FinSum (states ? (single_finalTM … M2)) (states ? M3)))
+          (step sig M1) (step sig (ifTM sig M1 (single_finalTM ? M2) M3 acc)) 
+          (λc.halt sig M1 (cstate … c)) 
+          (λc.halt_liftL ?? (halt sig M1) (cstate … c)) 
+          … Hloop1))
+        [* *
+          [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+          | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+        |#c0 #Hhalt >(step_if_liftM1 … Hhalt) // 
+        |#x <p_halt_liftL %
+        |whd in ⊢ (??%?); >(config_expand ?? outc1);
+         whd in match (lift_confL ????);
+         >(trans_if_M1true_notacc … Hacc) 
+          [% | @(loop_Some ?????? Hloop1)]
+        |cases outc1 #s1 #t1 %
+        |@(loop_lift ??? 
+           (λc.(lift_confR … (lift_confR sig (states ? (single_finalTM ? M2)) (states ? M3) c)))
+           … Hloop2) 
+          [ * #s2 #t2 %
+          | #c0 #Hhalt >(step_if_liftM3 … Hhalt) // ]
+        ]
+      |>(config_expand ?? outc2) whd in match (lift_confR ????);
+       #H destruct (H) 
+      ]
+    |#_ @(ex_intro … (ctape ?? outc1)) % 
+      [@HMfalse @(\Pf Hacc) | >(config_expand ?? outc2) @HM3 ]
+    ]
+  ]
+qed.
+    
+lemma acc_sem_if_app_guarded: ∀sig,M1,M2,M3,P,P2,Rtrue,Rfalse,R2,R3,R4,R5,acc.
+  M1 ⊨ [acc: Rtrue, Rfalse] → 
+  (GRealize ? M2 P2 R2) → (∀t,t0.P t → Rtrue t t0 → P2 t0) → 
+  M3 ⊨ R3 → 
+  (∀t1,t2,t3. Rtrue t1 t3 → R2 t3 t2 → R4 t1 t2) → 
+  (∀t1,t2,t3. Rfalse t1 t3 → R3 t3 t2 → R5 t1 t2) → 
+  accGRealize ? (ifTM sig M1 (single_finalTM … M2) M3 acc) 
+    (inr … (inl … (inr … start_nop))) P R4 R5 .
+#sig #M1 #M2 #M3 #P #P2 #Rtrue #Rfalse #R2 #R3 #R4 #R5 #acc
+#HRacc #HRtrue #Hinv #HRfalse #Hsub1 #Hsub2 
+#t #HPt cases (acc_sem_if_guarded … HRacc HRtrue Hinv HRfalse t HPt)
+#k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
+% [% [@Hloop
+     |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
+  |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]
+qed.
+
+