From 5b28867e30a9cada823ad86ae91d39b94648940a Mon Sep 17 00:00:00 2001
From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Mon, 4 Jun 2012 14:16:05 +0000
Subject: [PATCH] semantics of the if-machine. Moved some lemmas to their
 proper places.

---
 matita/matita/lib/basics/relations.ma  |  14 +-
 matita/matita/lib/basics/star.ma       |   6 -
 matita/matita/lib/turing/if_machine.ma | 225 ++++++++++++++++++-------
 matita/matita/lib/turing/mono.ma       |  85 ++++++----
 4 files changed, 223 insertions(+), 107 deletions(-)

diff --git a/matita/matita/lib/basics/relations.ma b/matita/matita/lib/basics/relations.ma
index ff190e5e3..55b26b8ae 100644
--- a/matita/matita/lib/basics/relations.ma
+++ b/matita/matita/lib/basics/relations.ma
@@ -45,7 +45,19 @@ definition tight_apart: ∀A.∀eq,ap:relation A.Prop
 definition antisymmetric: ∀A.∀R:relation A.Prop
 ≝ λA.λR.∀x,y:A. R x y → ¬(R y x).
 
-(********** functions **********)
+(********** operations **********)
+definition Runion ≝ λA.λR1,R2:relation A.λa,b. R1 a b ∨ R2 a b.
+interpretation "union of relations" 'union R1 R2 = (Runion ? R1 R2).
+    
+definition Rintersection ≝ λA.λR1,R2:relation A.λa,b.R1 a b ∧ R2 a b.
+interpretation "interesecion of relations" 'intersects R1 R2 = (Rintersection ? R1 R2).
+
+definition inv ≝ λA.λR:relation A.λa,b.R b a.
+
+definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
+interpretation "relation inclusion" 'subseteq R S = (subR ? R S).
+
+(**********P functions **********)
 
 definition compose ≝
   λA,B,C:Type[0].λf:B→C.λg:A→B.λx:A.f (g x).
diff --git a/matita/matita/lib/basics/star.ma b/matita/matita/lib/basics/star.ma
index a80ccbea0..c8d2a9473 100644
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -11,12 +11,6 @@
 
 include "basics/relations.ma".
 
-(********** relations **********)
-
-definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
-
-definition inv ≝ λA.λR:relation A.λa,b.R b a.
-
 (* transitive closcure (plus) *)
 
 inductive TC (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
diff --git a/matita/matita/lib/turing/if_machine.ma b/matita/matita/lib/turing/if_machine.ma
index 3d6675d02..5d52dea55 100644
--- a/matita/matita/lib/turing/if_machine.ma
+++ b/matita/matita/lib/turing/if_machine.ma
@@ -14,8 +14,8 @@ include "turing/mono.ma".
 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
@@ -55,15 +55,164 @@ definition ifTM ≝ λsig. λcondM,thenM,elseM : TM sig.
 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.
+  accRealize sig 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 ?????????
+     (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 → Realize sig M2 R2 → Realize sig M3 R3 → 
-    (∀t1,t2,t3. (Rtrue t1 t3 ∧ R2 t3 t2) ∨ (Rfalse t1 t3 ∧ R3 t3 t2) → R4 t1 t2) →  
-    Realize sig 
-     (ifTM sig M1 M2 M3 acc) R4.
+  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)
@@ -72,9 +221,10 @@ lemma sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,acc.
   [* #t3 * #Hleft #Hright @(Hsub … t3) %1 /2/
   |* #t3 * #Hleft #Hright @(Hsub … t3) %2 /2/ ]
 qed.
-
+ 
+(* e ancora usato ??? *)
 axiom acc_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 → 
+  accRealize sig M1 acc Rtrue Rfalse → M2 ⊨ R2 → M3 ⊨ R3 → 
     accRealize sig 
      (ifTM sig M1 (single_finalTM … M2) M3 acc) 
      (inr … (inl … (inr … start_nop)))
@@ -97,54 +247,3 @@ lemma acc_sem_if_app: ∀sig,M1,M2,M3,Rtrue,Rfalse,R2,R3,R4,R5,acc.
      |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
   |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]
 qed.
-
-(*    
-#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 ]
-      ]
-     |
-     |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
diff --git a/matita/matita/lib/turing/mono.ma b/matita/matita/lib/turing/mono.ma
index 88ec04e1c..a192f608e 100644
--- a/matita/matita/lib/turing/mono.ma
+++ b/matita/matita/lib/turing/mono.ma
@@ -84,7 +84,18 @@ record config (sig,states:FinSet): Type[0] ≝
 { cstate : states;
   ctape: tape sig
 }.
+
+lemma config_expand: ∀sig,Q,c. 
+  c = mk_config sig Q (cstate ?? c) (ctape ?? c).
+#sig #Q * // 
+qed.
   
+lemma config_eq : ∀sig,M,c1,c2.
+  cstate sig M c1 = cstate sig M c2 → 
+    ctape sig M c1 = ctape sig M c2 →  c1 = c2.
+#sig #M1 * #s1 #t1 * #s2 #t2 //
+qed.
+
 definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
   let current_char ≝ current ? (ctape ?? c) in
   let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
@@ -172,6 +183,28 @@ lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
 ]
 qed.
 
+lemma loop_Some : 
+  ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
+#A #k #f #p elim k 
+  [#a #b normalize #Hfalse destruct
+  |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
+    [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
+  ]
+qed. 
+
+lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
+  (∀x.hlift (lift x) = h x) → 
+  (∀x.h x = false → lift (f x) = g (lift x)) → 
+  loop A k f h c1 = Some ? c2 → 
+  loop B k g hlift (lift c1) = Some ? (lift … c2).
+#A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
+generalize in match c1; elim k
+[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
+|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
+ cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
+ [ #Heq destruct (Heq) % | <Hhlift // @IH ]
+qed.
+
 (************************** Realizability *************************************)
 definition loopM ≝ λsig,M,i,cin.
   loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
@@ -290,7 +323,7 @@ lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
 #sig #S1 #S2 #halt #c cases c #s #t %
 qed.
 
-lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,move.
   halt ? M1 s = false → 
   trans sig M1 〈s,a〉 = 〈news,move〉 → 
   trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
@@ -298,7 +331,7 @@ lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
 
-lemma trans_liftR : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,move.
   halt ? M2 s = false → 
   trans sig M2 〈s,a〉 = 〈news,move〉 → 
   trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
@@ -306,14 +339,7 @@ lemma trans_liftR : ∀sig,M1,M2,s,a,news,move.
 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
 
-lemma config_eq : 
-  ∀sig,M,c1,c2.
-  cstate sig M c1 = cstate sig M c2 → 
-  ctape sig M c1 = ctape sig M c2 →  c1 = c2.
-#sig #M1 * #s1 #t1 * #s2 #t2 //
-qed.
-
-lemma step_lift_confR : ∀sig,M1,M2,c0.
+lemma step_seq_liftR : ∀sig,M1,M2,c0.
  halt ? M2 (cstate ?? c0) = false → 
  step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
  lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
@@ -325,10 +351,10 @@ lemma step_lift_confR : ∀sig,M1,M2,c0.
   | 2,3: #s1 #l1 #Heq #Hhalt 
   |#ls #s1 #rs #Heq #Hhalt ]
   whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
-  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_liftR … Heq) //
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
 qed.
 
-lemma step_lift_confL : ∀sig,M1,M2,c0.
+lemma step_seq_liftL : ∀sig,M1,M2,c0.
  halt ? M1 (cstate ?? c0) = false → 
  step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
  lift_confL sig ?? (step sig M1 c0).
@@ -340,31 +366,9 @@ lemma step_lift_confL : ∀sig,M1,M2,c0.
   | 2,3: #s1 #l1 #Heq #Hhalt 
   |#ls #s1 #rs #Heq #Hhalt ]
   whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
-  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_liftL … Heq) //
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
 qed.
 
-lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
-  (∀x.hlift (lift x) = h x) → 
-  (∀x.h x = false → lift (f x) = g (lift x)) → 
-  loop A k f h c1 = Some ? c2 → 
-  loop B k g hlift (lift c1) = Some ? (lift … c2).
-#A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
-generalize in match c1; elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
- [ #Heq destruct (Heq) % | <Hhlift // @IH ]
-qed.
-
-lemma loop_Some : 
-  ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
-#A #k #f #p elim k 
-  [#a #b normalize #Hfalse destruct
-  |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
-    [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
-  ]
-qed. 
-
 lemma trans_liftL_true : ∀sig,M1,M2,s,a.
   halt ? M1 s = true → 
   trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?〉.
@@ -396,12 +400,12 @@ cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
   [ * *
    [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
    | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
-  || #c0 #Hhalt <step_lift_confL //
+  || #c0 #Hhalt <step_seq_liftL //
   | #x <p_halt_liftL %
   |6:cases outc1 #s1 #t1 %
   |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2) 
     [ * #s2 #t2 %
-    | #c0 #Hhalt <step_lift_confR // ]
+    | #c0 #Hhalt <step_seq_liftR // ]
   |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
    generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
    >(trans_liftL_true sig M1 M2 ??) 
@@ -414,3 +418,10 @@ cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
 ]
 qed.
 
+theorem sem_seq_app: ∀sig.∀M1,M2:TM sig.∀R1,R2,R3.
+  M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
+#sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
+#t cases (sem_seq … HR1 HR2 t)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop |@Hsub @Houtc]
+qed.
-- 
2.39.2