X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fturing.ma;h=dc0a43e755968a27ea32aaea3deb2fa84548a03d;hb=d9a1ff8259a7882caa0ffd27282838c00a34cab5;hp=946999784471446c441607ab1c64ce0af9e24fb5;hpb=65a3b93b01f2d00960c56df3563b879f36f3cbfd;p=helm.git

diff --git a/matita/matita/lib/turing/turing.ma b/matita/matita/lib/turing/turing.ma
index 946999784..dc0a43e75 100644
--- a/matita/matita/lib/turing/turing.ma
+++ b/matita/matita/lib/turing/turing.ma
@@ -1,103 +1,481 @@
-(*
-    ||M||  This file is part of HELM, an Hypertextual, Electronic   
-    ||A||  Library of Mathematics, developed at the Computer Science 
-    ||T||  Department of the University of Bologna, Italy.           
-    ||I||                                                            
-    ||T||  
-    ||A||  
-    \   /  This file is distributed under the terms of the       
-     \ /   GNU General Public License Version 2   
-      V_____________________________________________________________*)
-
+include "basics/core_notation/fintersects_2.ma".
+include "turing/mono.ma".
 include "basics/vectors.ma".
 
-record tape (sig:FinSet): Type[0] ≝ 
-{ left : list sig;
-  right: list sig
-}.
-
-inductive move : Type[0] ≝
-| L : move 
-| R : move
-| N : move. 
-
 (* We do not distinuish an input tape *)
 
-record TM (sig:FinSet): Type[1] ≝ 
+(* tapes_no = number of ADDITIONAL working tapes *)
+
+record mTM (sig:FinSet) (tapes_no:nat) : Type[1] ≝ 
 { states : FinSet;
-  tapes_no: nat; (* additional working tapes *)
   trans : states × (Vector (option sig) (S tapes_no)) → 
-    states  × (Vector (sig × move) (S tapes_no)) × (option sig) ;
-  output: list sig;
+    states  × (Vector ((option sig) × move) (S tapes_no));
   start: states;
   halt : states → bool
 }.
 
-record config (sig:FinSet) (M:TM sig): Type[0] ≝
-{ state : states sig M;
-  tapes : Vector (tape sig) (S (tapes_no sig M));
-  out : list sig
+record mconfig (sig,states:FinSet) (n:nat): Type[0] ≝
+{ cstate : states;
+  ctapes : Vector (tape sig) (S n)
 }.
 
-definition option_hd ≝ λA.λl:list A.
-  match l with
-  [nil ⇒ None ?
-  |cons a _ ⇒ Some ? a
-  ].
+lemma mconfig_expand: ∀sig,n,Q,c. 
+  c = mk_mconfig sig Q n (cstate ??? c) (ctapes ??? c).
+#sig #n #Q * // 
+qed.
+  
+lemma mconfig_eq : ∀sig,n,M,c1,c2.
+  cstate sig n M c1 = cstate sig n M c2 → 
+    ctapes sig n M c1 = ctapes sig n M c2 →  c1 = c2.
+#sig #n #M1 * #s1 #t1 * #s2 #t2 //
+qed.
 
-definition tape_move ≝ λsig.λt: tape sig.λm:sig × move.
-  match \snd m with
-  [ R ⇒ mk_tape sig ((\fst m)::(left ? t)) (tail ? (right ? t))
-  | L ⇒ mk_tape sig (tail ? (left ? t)) ((\fst m)::(right ? t))
-  | N ⇒ mk_tape sig (left ? t) ((\fst m)::(tail ? (right ? t)))
-  ].
+definition current_chars ≝ λsig.λn.λtapes.
+  vec_map ?? (current sig) (S n) tapes.
 
-definition current_chars ≝ λsig.λM:TM sig.λc:config sig M.
-  vec_map ?? (λt.option_hd ? (right ? t)) (S (tapes_no sig M)) (tapes ?? c).
+lemma nth_current_chars : ∀sig,n,tapes,i.
+  nth i ? (current_chars sig n tapes) (None ?) 
+   = current sig (nth i ? tapes (niltape sig)).
+#sig #n #tapes #i >(nth_vec_map … (current sig) i (S n)) %
+qed.
 
-definition opt_cons ≝ λA.λa:option A.λl:list A.
-  match a with
-  [ None ⇒ l
-  | Some a ⇒ a::l
-  ].
+definition tape_move_multi ≝ 
+  λsig,n,ts,mvs.
+  pmap_vec ??? (tape_move_mono sig) n ts mvs.
+  
+lemma tape_move_multi_def : ∀sig,n,ts,mvs.
+  tape_move_multi sig n ts mvs = pmap_vec ??? (tape_move_mono sig) n ts mvs.
+// qed.
 
-definition step ≝ λsig.λM:TM sig.λc:config sig M.
-  let 〈news,mvs,outchar〉 ≝ trans sig M 〈state ?? c,current_chars ?? c〉 in
-  mk_config ?? 
-    news 
-    (pmap_vec ??? (tape_move sig) ? (tapes ?? c) mvs)
-    (opt_cons ? outchar (out ?? c)).
+definition step ≝ λsig.λn.λM:mTM sig n.λc:mconfig sig (states ?? M) n.
+  let 〈news,mvs〉 ≝ trans sig n M 〈cstate ??? c,current_chars ?? (ctapes ??? c)〉 in
+  mk_mconfig ??? news (tape_move_multi sig ? (ctapes ??? c) mvs).
 
 definition empty_tapes ≝ λsig.λn.
-mk_Vector ? n (make_list (tape sig) (mk_tape sig [] []) n) ?.
+mk_Vector ? n (make_list (tape sig) (niltape sig) n) ?.
 elim n // normalize //
 qed.
 
-definition init ≝ λsig.λM:TM sig.λi:(list sig).
-  mk_config ??
-    (start sig M)
-    (vec_cons ? (mk_tape sig [] i) ? (empty_tapes sig (tapes_no sig M)))
-    [ ].
+(************************** Realizability *************************************)
+definition loopM ≝ λsig,n.λM:mTM sig n.λi,cin.
+  loop ? i (step sig n M) (λc.halt sig n M (cstate ??? c)) cin.
+
+lemma loopM_unfold : ∀sig,n,M,i,cin.
+  loopM sig n M i cin = loop ? i (step sig n M) (λc.halt sig n M (cstate ??? c)) cin.
+// qed.
+
+definition initc ≝ λsig,n.λM:mTM sig n.λtapes.
+  mk_mconfig sig (states sig n M) n (start sig n M) tapes.
+
+definition Realize ≝ λsig,n.λM:mTM sig n.λR:relation (Vector (tape sig) ?).
+∀t.∃i.∃outc.
+  loopM sig n M i (initc sig n M t) = Some ? outc ∧ R t (ctapes ??? outc).
+
+definition WRealize ≝ λsig,n.λM:mTM sig n.λR:relation (Vector (tape sig) ?).
+∀t,i,outc.
+  loopM sig n M i (initc sig n M t) = Some ? outc → R t (ctapes ??? outc).
+
+definition Terminate ≝ λsig,n.λM:mTM sig n.λt. ∃i,outc.
+  loopM sig n M i (initc sig n M t) = Some ? outc.
+  
+(* notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}. *)
+interpretation "multi realizability" 'models M R = (Realize ?? M R).
+
+(* notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}. *)
+interpretation "weak multi realizability" 'wmodels M R = (WRealize ?? M R).
+
+interpretation "multi termination" 'fintersects M t = (Terminate ?? M t).
+
+lemma WRealize_to_Realize : ∀sig,n .∀M: mTM sig n.∀R.
+  (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
+#sig #n #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop 
+@(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
+qed.
+
+theorem Realize_to_WRealize : ∀sig,n.∀M:mTM sig n.∀R.
+  M ⊨ R → M ⊫ R.
+#sig #n #M #R #H1 #inc #i #outc #Hloop 
+cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
+qed.
+
+definition accRealize ≝ λsig,n.λM:mTM sig n.λacc:states sig n M.λRtrue,Rfalse.
+∀t.∃i.∃outc.
+  loopM sig n M i (initc sig n M t) = Some ? outc ∧
+    (cstate ??? outc = acc → Rtrue t (ctapes ??? outc)) ∧ 
+    (cstate ??? outc ≠ acc → Rfalse t (ctapes ??? outc)).
+    
+(* notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}. *)
+interpretation "conditional multi realizability" 'cmodels M q R1 R2 = (accRealize ?? M q R1 R2).
+
+(*************************** guarded realizablity *****************************)
+definition GRealize ≝ λsig,n.λM:mTM sig n.
+ λPre:Vector (tape sig) ? →Prop.λR:relation (Vector (tape sig) ?).
+  ∀t.Pre t → ∃i.∃outc.
+   loopM sig n M i (initc sig n M t) = Some ? outc ∧ R t (ctapes ??? outc).
+  
+definition accGRealize ≝ λsig,n.λM:mTM sig n.λacc:states sig n M.
+λPre: Vector (tape sig) ? → Prop.λRtrue,Rfalse.
+∀t.Pre t → ∃i.∃outc.
+  loopM sig n M i (initc sig n M t) = Some ? outc ∧
+    (cstate ??? outc = acc → Rtrue t (ctapes ??? outc)) ∧ 
+    (cstate ??? outc ≠ acc → Rfalse t (ctapes ??? outc)).
+    
+lemma WRealize_to_GRealize : ∀sig,n.∀M: mTM sig n.∀Pre,R.
+  (∀t.Pre t → M ↓ t) → M ⊫ R → GRealize sig n M Pre R.
+#sig #n #M #Pre #R #HT #HW #t #HPre cases (HT … t HPre) #i * #outc #Hloop 
+@(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
+qed.
+
+lemma Realize_to_GRealize : ∀sig,n.∀M: mTM sig n.∀P,R. 
+  M ⊨ R → GRealize sig n M P R.
+#alpha #n #M #Pre #R #HR #t #HPre
+cases (HR t) -HR #k * #outc * #Hloop #HR 
+@(ex_intro ?? k) @(ex_intro ?? outc) % 
+  [ @Hloop | @HR ]
+qed.
+
+lemma acc_Realize_to_acc_GRealize: ∀sig,n.∀M:mTM sig n.∀q:states sig n M.∀P,R1,R2. 
+  M ⊨ [q:R1,R2] → accGRealize sig n M q P R1 R2.
+#alpha #n #M #q #Pre #R1 #R2 #HR #t #HPre
+cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse 
+@(ex_intro ?? k) @(ex_intro ?? outc) % 
+  [ % [@Hloop] @HRtrue | @HRfalse]
+qed.
+
+(******************************** monotonicity ********************************)
+lemma Realize_to_Realize : ∀sig,n.∀M:mTM sig n.∀R1,R2.
+  R1 ⊆ R2 → M ⊨ R1 → M ⊨ R2.
+#alpha #n #M #R1 #R2 #Himpl #HR1 #intape
+cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma WRealize_to_WRealize: ∀sig,n.∀M:mTM sig n.∀R1,R2.
+  R1 ⊆ R2 → WRealize sig n M R1 → WRealize sig n M R2.
+#alpha #n #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
+@Hsub @(HR1 … i) @Hloop
+qed.
+
+lemma GRealize_to_GRealize : ∀sig,n.∀M:mTM sig n.∀P,R1,R2.
+  R1 ⊆ R2 → GRealize sig n M P R1 → GRealize sig n M P R2.
+#alpha #n #M #P #R1 #R2 #Himpl #HR1 #intape #HP
+cases (HR1 intape HP) -HR1 #k * #outc * #Hloop #HR1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
 
-definition stop ≝ λsig.λM:TM sig.λc:config sig M.
-  halt sig M (state sig M c).
+lemma GRealize_to_GRealize_2 : ∀sig,n.∀M:mTM sig n.∀P1,P2,R1,R2.
+  P2 ⊆ P1 → R1 ⊆ R2 → GRealize sig n M P1 R1 → GRealize sig n M P2 R2.
+#alpha #n #M #P1 #P2 #R1 #R2 #Himpl1 #Himpl2 #H1 #intape #HP
+cases (H1 intape (Himpl1 … HP)) -H1 #k * #outc * #Hloop #H1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma acc_Realize_to_acc_Realize: ∀sig,n.∀M:mTM sig n.∀q:states sig n M.
+ ∀R1,R2,R3,R4. 
+  R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
+#alpha #n #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
+cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse 
+@(ex_intro ?? k) @(ex_intro ?? outc) % 
+  [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
+qed.
 
-let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
-  match n with 
-  [ O ⇒ None ?
-  | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
+(**************************** A canonical relation ****************************)
+
+definition R_mTM ≝ λsig,n.λM:mTM sig n.λq.λt1,t2.
+∃i,outc.
+  loopM ? n M i (mk_mconfig ??? q t1) = Some ? outc ∧ 
+  t2 = (ctapes ??? outc).
+  
+lemma R_mTM_to_R: ∀sig,n.∀M:mTM sig n.∀R. ∀t1,t2. 
+  M ⊫ R → R_mTM ?? M (start sig n M) t1 t2 → R t1 t2.
+#sig #n #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
+#Hloop #Ht2 >Ht2 @(HMR … Hloop)
+qed.
+
+(******************************** NOP Machine *********************************)
+
+(* NO OPERATION
+   t1 = t2 
+  
+definition nop_states ≝ initN 1.
+definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1). *)
+
+definition nop ≝ 
+  λalpha:FinSet.λn.mk_mTM alpha n nop_states
+  (λp.let 〈q,a〉 ≝ p in 〈q,mk_Vector ? (S n) (make_list ? (〈None ?,N〉) (S n)) ?〉)
+  start_nop (λ_.true).
+elim n normalize //
+qed.
+  
+definition R_nop ≝ λalpha,n.λt1,t2:Vector (tape alpha) (S n).t2 = t1.
+
+lemma sem_nop :
+  ∀alpha,n.nop alpha n⊨ R_nop alpha n.
+#alpha #n #intapes @(ex_intro ?? 1) 
+@(ex_intro … (mk_mconfig ??? start_nop intapes)) % % 
+qed.
+
+lemma nop_single_state: ∀sig,n.∀q1,q2:states ? n (nop sig n). q1 = q2.
+normalize #sig #n0 * #n #ltn1 * #m #ltm1 
+generalize in match ltn1; generalize in match ltm1;
+<(le_n_O_to_eq … (le_S_S_to_le … ltn1)) <(le_n_O_to_eq … (le_S_S_to_le … ltm1)) 
+// qed.
+
+(************************** Sequential Composition ****************************)
+definition null_action ≝ λsig.λn.
+mk_Vector ? (S n) (make_list (option sig × move) (〈None ?,N〉) (S n)) ?.
+elim (S n) // normalize //
+qed.
+
+lemma tape_move_null_action: ∀sig,n,tapes.
+  tape_move_multi sig (S n) tapes (null_action sig n) = tapes.
+#sig #n #tapes cases tapes -tapes #tapes whd in match (null_action ??);
+#Heq @Vector_eq <Heq -Heq elim tapes //
+#a #tl #Hind whd in ⊢ (??%?); @eq_f2 // @Hind
+qed.
+
+definition seq_trans ≝ λsig,n. λM1,M2 : mTM sig n. 
+λp. let 〈s,a〉 ≝ p in
+  match s with 
+  [ inl s1 ⇒ 
+      if halt sig n M1 s1 then 〈inr … (start sig n M2), null_action sig n〉
+      else let 〈news1,m〉 ≝ trans sig n M1 〈s1,a〉 in 〈inl … news1,m〉
+  | inr s2 ⇒ let 〈news2,m〉 ≝ trans sig n M2 〈s2,a〉 in 〈inr … news2,m〉
   ].
+ 
+definition seq ≝ λsig,n. λM1,M2 : mTM sig n. 
+  mk_mTM sig n
+    (FinSum (states sig n M1) (states sig n M2))
+    (seq_trans sig n M1 M2) 
+    (inl … (start sig n M1))
+    (λs.match s with 
+      [ inl _ ⇒ false | inr s2 ⇒ halt sig n M2 s2]). 
+
+(* notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}. *)
+interpretation "sequential composition" 'middot a b = (seq ?? a b).
+
+definition lift_confL ≝ 
+  λsig,n,S1,S2,c.match c with 
+  [ mk_mconfig s t ⇒ mk_mconfig sig (FinSum S1 S2) n (inl … s) t ].
+  
+definition lift_confR ≝ 
+  λsig,n,S1,S2,c.match c with
+  [ mk_mconfig s t ⇒ mk_mconfig sig (FinSum S1 S2) n (inr … s) t ].
+
+(* 
+definition halt_liftL ≝ 
+  λS1,S2,halt.λs:FinSum S1 S2.
+  match s with
+  [ inl s1 ⇒ halt s1
+  | inr _ ⇒ true ]. (* should be vacuous in all cases we use halt_liftL *)
 
-(* Compute ? M f states that f is computed by M *)
-definition Compute ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
-∀l.∃i.∃c.
-  loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
-  out ?? c = f l.
+definition halt_liftR ≝ 
+  λS1,S2,halt.λs:FinSum S1 S2.
+  match s with
+  [ inl _ ⇒ false 
+  | inr s2 ⇒ halt s2 ]. *)
+      
+lemma p_halt_liftL : ∀sig,n,S1,S2,halt,c.
+  halt (cstate sig S1 n c) =
+     halt_liftL S1 S2 halt (cstate … (lift_confL … c)).
+#sig #n #S1 #S2 #halt #c cases c #s #t %
+qed.
 
-(* for decision problems, we accept a string if on termination
-output is not empty *)
+lemma trans_seq_liftL : ∀sig,n,M1,M2,s,a,news,move.
+  halt ?? M1 s = false → 
+  trans sig n M1 〈s,a〉 = 〈news,move〉 → 
+  trans sig n (seq sig n M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
+#sig #n (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
 
-definition ComputeB ≝ λsig.λM:TM sig.λf:(list sig) → bool.
-∀l.∃i.∃c.
-  loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
-  (isnilb ? (out ?? c) = false).
+lemma trans_seq_liftR : ∀sig,n,M1,M2,s,a,news,move.
+  halt ?? M2 s = false → 
+  trans sig n M2 〈s,a〉 = 〈news,move〉 → 
+  trans sig n (seq sig n M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
+#sig #n #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
+#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
+qed.
+
+lemma step_seq_liftR : ∀sig,n,M1,M2,c0.
+ halt ?? M2 (cstate ??? c0) = false → 
+ step sig n (seq sig n M1 M2) (lift_confR sig n (states ?? M1) (states ?? M2) c0) =
+ lift_confR sig n (states ?? M1) (states ?? M2) (step sig n M2 c0).
+#sig #n #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
+lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
+cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
+#s0 #m0 #Heq #Hhalt whd in ⊢ (???(?????%)); >Heq  whd in ⊢ (???%);
+whd in ⊢ (??(????%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
+qed.
+
+lemma step_seq_liftL : ∀sig,n,M1,M2,c0.
+ halt ?? M1 (cstate ??? c0) = false → 
+ step sig n (seq sig n M1 M2) (lift_confL sig n (states ?? M1) (states ?? M2) c0) =
+ lift_confL sig n ?? (step sig n M1 c0).
+#sig #n #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
+  lapply (refl ? (trans ??? 〈s,current_chars sig n t〉))
+  cases (trans ??? 〈s,current_chars sig n t〉) in ⊢ (???% → %);
+  #s0 #m0 #Heq #Hhalt
+  whd in ⊢ (???(?????%)); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(????%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
+qed.
+
+lemma trans_liftL_true : ∀sig,n,M1,M2,s,a.
+  halt ?? M1 s = true → 
+  trans sig n (seq sig n M1 M2) 〈inl … s,a〉 = 〈inr … (start ?? M2),null_action sig n〉.
+#sig #n #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
+qed.
+
+lemma eq_ctape_lift_conf_L : ∀sig,n,S1,S2,outc.
+  ctapes sig (FinSum S1 S2) n (lift_confL … outc) = ctapes … outc.
+#sig #n #S1 #S2 #outc cases outc #s #t %
+qed.
+  
+lemma eq_ctape_lift_conf_R : ∀sig,n,S1,S2,outc.
+  ctapes sig (FinSum S1 S2) n (lift_confR … outc) = ctapes … outc.
+#sig #n #S1 #S2 #outc cases outc #s #t %
+qed.
+
+theorem sem_seq: ∀sig,n.∀M1,M2:mTM sig n.∀R1,R2.
+  M1 ⊨ R1 → M2 ⊨ R2 → M1 · M2 ⊨ R1 ∘ R2.
+#sig #n #M1 #M2 #R1 #R2 #HR1 #HR2 #t 
+cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
+cases (HR2 (ctapes sig (states ?? M1) n outc1)) #k2 * #outc2 * #Hloop2 #HM2
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+%
+[@(loop_merge ??????????? 
+   (loop_lift ??? (lift_confL sig n (states sig n M1) (states sig n M2))
+   (step sig n M1) (step sig n (seq sig n M1 M2)) 
+   (λc.halt sig n M1 (cstate … c)) 
+   (λc.halt_liftL ?? (halt sig n M1) (cstate … c)) … Hloop1))
+  [ * *
+   [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+   | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+  || #c0 #Hhalt <step_seq_liftL //
+  | #x <p_halt_liftL %
+  |6:cases outc1 #s1 #t1 %
+  |7:@(loop_lift … (initc ??? (ctapes … outc1)) … Hloop2) 
+    [ * #s2 #t2 %
+    | #c0 #Hhalt <step_seq_liftR // ]
+  |whd in ⊢ (??(????%)?);whd in ⊢ (??%?);
+   generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
+   >(trans_liftL_true sig n M1 M2 ??) 
+    [ whd in ⊢ (??%?); whd in ⊢ (???%);
+      @mconfig_eq whd in ⊢ (???%); // 
+    | @(loop_Some ?????? Hloop10) ]
+ ]
+| @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) ? (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
+]
+qed.
+
+theorem sem_seq_app: ∀sig,n.∀M1,M2:mTM sig n.∀R1,R2,R3.
+  M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
+#sig #n #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.
+
+(* composition with guards *)
+theorem sem_seq_guarded: ∀sig,n.∀M1,M2:mTM sig n.∀Pre1,Pre2,R1,R2.
+  GRealize sig n M1 Pre1 R1 → GRealize sig n M2 Pre2 R2 → 
+  (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → 
+  GRealize sig n (M1 · M2) Pre1 (R1 ∘ R2).
+#sig #n #M1 #M2 #Pre1 #Pre2 #R1 #R2 #HGR1 #HGR2 #Hinv #t1 #HPre1
+cases (HGR1 t1 HPre1) #k1 * #outc1 * #Hloop1 #HM1
+cases (HGR2 (ctapes sig (states ?? M1) n outc1) ?) 
+  [2: @(Hinv … HPre1 HM1)]  
+#k2 * #outc2 * #Hloop2 #HM2
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+%
+[@(loop_merge ??????????? 
+   (loop_lift ??? (lift_confL sig n (states sig n M1) (states sig n M2))
+   (step sig n M1) (step sig n (seq sig n M1 M2)) 
+   (λc.halt sig n M1 (cstate … c)) 
+   (λc.halt_liftL ?? (halt sig n M1) (cstate … c)) … Hloop1))
+  [ * *
+   [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+   | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+  || #c0 #Hhalt <step_seq_liftL //
+  | #x <p_halt_liftL %
+  |6:cases outc1 #s1 #t1 %
+  |7:@(loop_lift … (initc ??? (ctapes … outc1)) … Hloop2) 
+    [ * #s2 #t2 %
+    | #c0 #Hhalt <step_seq_liftR // ]
+  |whd in ⊢ (??(????%)?);whd in ⊢ (??%?);
+   generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
+   >(trans_liftL_true sig n M1 M2 ??) 
+    [ whd in ⊢ (??%?); whd in ⊢ (???%);
+      @mconfig_eq whd in ⊢ (???%); //
+    | @(loop_Some ?????? Hloop10) ]
+ ]
+| @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) n (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
+]
+qed.
+
+theorem sem_seq_app_guarded: ∀sig,n.∀M1,M2:mTM sig n.∀Pre1,Pre2,R1,R2,R3.
+  GRealize sig n M1 Pre1 R1 → GRealize sig n M2 Pre2 R2 → 
+  (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → R1 ∘ R2 ⊆ R3 →
+  GRealize sig n (M1 · M2) Pre1 R3.
+#sig #n #M1 #M2 #Pre1 #Pre2 #R1 #R2 #R3 #HR1 #HR2 #Hinv #Hsub
+#t #HPre1 cases (sem_seq_guarded … HR1 HR2 Hinv t HPre1)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop |@Hsub @Houtc]
+qed.
+
+theorem acc_sem_seq : ∀sig,n.∀M1,M2:mTM sig n.∀R1,Rtrue,Rfalse,acc.
+  M1 ⊨ R1 → M2 ⊨ [ acc: Rtrue, Rfalse ] → 
+  M1 · M2 ⊨ [ inr … acc: R1 ∘ Rtrue, R1 ∘ Rfalse ].
+#sig #n #M1 #M2 #R1 #Rtrue #Rfalse #acc #HR1 #HR2 #t 
+cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
+cases (HR2 (ctapes sig (states ?? M1) n outc1)) #k2 * #outc2 * * #Hloop2 
+#HMtrue #HMfalse
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+% [ %
+[@(loop_merge ??????????? 
+   (loop_lift ??? (lift_confL sig n (states sig n M1) (states sig n M2))
+   (step sig n M1) (step sig n (seq sig n M1 M2)) 
+   (λc.halt sig n M1 (cstate … c)) 
+   (λc.halt_liftL ?? (halt sig n M1) (cstate … c)) … Hloop1))
+  [ * *
+   [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+   | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+  || #c0 #Hhalt <step_seq_liftL //
+  | #x <p_halt_liftL %
+  |6:cases outc1 #s1 #t1 %
+  |7:@(loop_lift … (initc ??? (ctapes … outc1)) … Hloop2) 
+    [ * #s2 #t2 %
+    | #c0 #Hhalt <step_seq_liftR // ]
+  |whd in ⊢ (??(????%)?);whd in ⊢ (??%?);
+   generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
+   >(trans_liftL_true sig n M1 M2 ??) 
+    [ whd in ⊢ (??%?); whd in ⊢ (???%);
+      @mconfig_eq whd in ⊢ (???%); // 
+    | @(loop_Some ?????? Hloop10) ]
+ ]
+| >(mconfig_expand … outc2) in ⊢ (%→?); whd in ⊢ (??%?→?); 
+  #Hqtrue destruct (Hqtrue)
+  @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) ? (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R /2/ ]
+| >(mconfig_expand … outc2) in ⊢ (%→?); whd in ⊢ (?(??%?)→?); #Hqfalse
+  @(ex_intro … (ctapes ? (FinSum (states ?? M1) (states ?? M2)) ? (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R @HMfalse
+  @(not_to_not … Hqfalse) //
+]
+qed.
+
+lemma acc_sem_seq_app : ∀sig,n.∀M1,M2:mTM sig n.∀R1,Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ R1 → M2 ⊨ [acc: Rtrue, Rfalse] → 
+    (∀t1,t2,t3. R1 t1 t3 → Rtrue t3 t2 → R2 t1 t2) → 
+    (∀t1,t2,t3. R1 t1 t3 → Rfalse t3 t2 → R3 t1 t2) → 
+    M1 · M2 ⊨ [inr … acc : R2, R3].    
+#sig #n #M1 #M2 #R1 #Rtrue #Rfalse #R2 #R3 #acc
+#HR1 #HRacc #Hsub1 #Hsub2 
+#t cases (acc_sem_seq … HR1 HRacc 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.