+++ /dev/null
-(*
- ||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 "turing/universal/tuples.ma".
-include "turing/universal/marks.ma".
-
-(*
-l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
- ^ ^
-
-if current (* x *) = #
- then
- else if x = 0
- then move_right; ----
- adv_to_mark_r;
- if current (* x0 *) = 0
- then advance_mark ----
- adv_to_mark_l;
- advance_mark
- else STOP
- else x = 1 (* analogo *)
-
-*)
-
-
-(*
- MARK NEXT TUPLE machine
- (partially axiomatized)
-
- marks the first character after the first bar (rightwards)
- *)
-
-definition mark_next_tuple ≝
- adv_to_mark_r ? bar_or_grid ·
- (ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
- (move_right_and_mark ?) (nop ?) tc_true).
-
-definition R_mark_next_tuple ≝
- λt1,t2.
- ∀ls,c,rs1,rs2.
- (* c non può essere un separatore ... speriamo *)
- t1 = midtape STape ls c (rs1@〈grid,false〉::rs2) →
- no_marks rs1 → no_grids rs1 → bar_or_grid c = false →
- (∃rs3,rs4,d,b.rs1 = rs3 @ 〈bar,false〉 :: rs4 ∧
- no_bars rs3 ∧
- Some ? 〈d,b〉 = option_hd ? (rs4@〈grid,false〉::rs2) ∧
- t2 = midtape STape (〈bar,false〉::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2)))
- ∨
- (no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2).
-
-axiom daemon :∀P:Prop.P.
-
-axiom tech_split :
- ∀A:DeqSet.∀f,l.
- (∀x.memb A x l = true → f x = false) ∨
- (∃l1,c,l2.f c = true ∧ l = l1@c::l2 ∧ ∀x.memb ? x l1 = true → f x = false).
-(*#A #f #l elim l
-[ % #x normalize #Hfalse *)
-
-theorem sem_mark_next_tuple :
- Realize ? mark_next_tuple R_mark_next_tuple.
-#intape
-lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
- (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) tc_true) ????)
-[@sem_if [5: // |6: @sem_move_right_and_mark |7: // |*:skip]
-| //
-|||#Hif cases (Hif intape) -Hif
- #j * #outc * #Hloop * #ta * #Hleft #Hright
- @(ex_intro ?? j) @ex_intro [|% [@Hloop] ]
- -Hloop
- #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc
- cases (proj2 ?? Hleft … Hrs)
- [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
- | * * #_ #Hta #_ cases (tech_split STape (λc.is_bar (\fst c)) rs1)
- [ #H1 %2 % [@H1]
- lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
- [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
- | %
- | -Hta #Hta cases Hright
- [ * #tb * whd in ⊢ (%→?); * * #c1 * >Hta
- whd in ⊢ ((??%?)→?); #H destruct (H) whd in ⊢ ((??%?)→?); #H destruct
- | * #tb * whd in ⊢ (%→?); * #_ #Htb >Htb >Hta
- whd in ⊢ (%→?); #H @H
- ]
- ]
- |* #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
- % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
- lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???)
- [#x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
- #Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|]
- >(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 //
- |whd in ⊢ (??%?); >Hc0 %
- |>Hsplit >associative_append %
- ]-Hta #Hta
- cases Hright -Hright
- [* whd in ⊢ (%→?); #tb * * * #c1 * >Hta -Hta
- whd in ⊢ (??%?→?); #H destruct (H) #Hc1 #Htb
- whd in ⊢ (%→?); #Houtc
- cut (c1=〈bar,false〉)
- [lapply Hc1 lapply Hsplit cases c1 #c1l #c1r #Hsplit
- cases c1l normalize
- [#b #H destruct |2,3,5:#H destruct]
- #_ @eq_f @(Hrs1 … 〈c1l,c1r〉) >Hsplit @memb_append_l2 @memb_hd]
- #Hcut lapply Hsplit -Hsplit
- cases rs4 in Htb;
- [#Htb lapply(Houtc … Htb) -Houtc #Houtc #Hsplit
- @(ex_intro ?? grid) @(ex_intro ?? false) %
- [% [ % [<Hcut @Hsplit |@Hrs3 ] | % ]
- |>Houtc >Hcut %
- ]
- |* #r5l #r5r #rs5 #Htb
- lapply(Houtc … Htb) -Houtc #Houtc #Hsplit
- @(ex_intro ?? r5l) @(ex_intro ?? r5r) %
- [%[%[<Hcut @Hsplit| @Hrs3] | % ]
- |>Houtc >Hcut %
- ]
- ]
- |* whd in ⊢ (%→?); #tb * *
- #H @False_ind >Hta in H; #H lapply(H c0 (refl …))
- >Hc0 #H destruct
- ]
- ]
- ]
-]
-qed.
-
-definition init_current_on_match ≝
- move_l ? · adv_to_mark_l ? (λc:STape.is_grid (\fst c)) · move_r ? · mark ?.
-
-definition R_init_current_on_match ≝ λt1,t2.
- ∀l1,l2,c,rs. no_grids l1 → is_grid c = false →
- t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::l2) 〈grid,false〉 rs →
- t2 = midtape STape (〈grid,false〉::l2) 〈c,true〉 ((reverse ? l1)@〈grid,false〉::rs).
-
-lemma sem_init_current_on_match :
- Realize ? init_current_on_match R_init_current_on_match.
-#intape
-cases (sem_seq ????? (sem_move_l ?)
- (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (sem_seq ????? (sem_move_r ?) (sem_mark ?))) intape)
-#k * #outc * #Hloop #HR
-@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
-#l1 #l2 #c #rs #Hl1 #Hc #Hintape
-cases HR -HR #ta * whd in ⊢ (%→?); * #_ #Hta lapply (Hta … Hintape) -Hta -Hintape
-generalize in match Hl1; cases l1
- [#Hl1 whd in ⊢ ((???(??%%%))→?); #Hta
- * #tb * whd in ⊢ (%→?); * #_ #Htb cases (Htb … Hta) -Htb -Hta #_
- (* [* >Hc #Htemp destruct (Htemp) ] *)
- #Htb cases (Htb … Hc) -Htb #Htb #_
- lapply (Htb [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1)
- whd in ⊢ ((???(??%%%))→?); -Htb #Htb
- * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc
- whd in ⊢ ((???(??%%%))→?); #Htc
- whd in ⊢ (%→?); * #Houtc #_ lapply (Houtc … Htc) -Houtc #Houtc
- >Houtc %
- |#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Hta
- * #tb * whd in ⊢ (%→?); * #_ #Htb cases (Htb … Hta) -Htb
- #_ (* [* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)] *)
- #Htb cases (Htb ?) -Htb [2: @Htl @memb_hd]
- >append_cons #Htb #_ lapply (Htb … (refl ??) (refl …) ?)
- [#x #membx cases (memb_append … membx) -membx #membx
- [@Htl @memb_cons @membx | >(memb_single … membx) @Hc]]-Htb #Htb
- * #tc * whd in ⊢ (%→?); * #_ #Htc lapply (Htc … Htb) -Htb -Htc
- >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
- whd in ⊢ (%→?); * #Houtc #_ lapply (Houtc … Htc) -Houtc #Houtc
- >Houtc >reverse_cons >associative_append %
- ]
-qed.
-
-(*
-definition init_current_gen ≝
- seq ? (adv_to_mark_l ? (is_marked ?))
- (seq ? (clear_mark ?)
- (seq ? (move_l ?)
- (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (seq ? (move_r ?) (mark ?))))).
-
-definition R_init_current_gen ≝ λt1,t2.
- ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 →
- Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
- t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
- t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
- ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
-
-lemma sem_init_current_gen : Realize ? init_current_gen R_init_current_gen.
-#intape
-cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
- (sem_seq ????? (sem_clear_mark ?)
- (sem_seq ????? (sem_move_l ?)
- (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (sem_seq ????? (sem_move_r ?) (sem_mark ?))))) intape)
-#k * #outc * #Hloop #HR
-@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
-#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hintape
-cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape
- [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
-* #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] -Hta #Hta
-* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta #Htb
-* #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
-generalize in match Hc; generalize in match Hl2; cases l2
- [#_ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
- whd in ⊢ ((???(??%%%))→?); #Htc
- * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
- [2: * whd in ⊢ (??%?→?); #Htemp destruct (Htemp) ]
- * #_ #Htd >Htd in Htc; -Htd #Htd
- * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
- >reverse_append >reverse_cons
- whd in ⊢ ((???(??%%%))→?); #Hte
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
- >Houtc %
- |#d #tl #Htl #Hc0 whd in ⊢ ((???(??%%%))→?); #Htc
- * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
- [* >(Htl … (memb_hd …)) whd in ⊢ (??%?→?); #Htemp destruct (Htemp)]
- * #Hd #Htd lapply (Htd … (refl ??) (refl ??) ?)
- [#x #membx @Htl @memb_cons @membx] -Htd #Htd
- * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
- >reverse_append >reverse_cons >reverse_cons
- >reverse_cons in Hc0; >reverse_cons cases (reverse ? tl)
- [normalize in ⊢ (%→?); #Hc0 destruct (Hc0) #Hte
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
- >Houtc %
- |* #c2 #b2 #tl2 normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
- whd in ⊢ ((???(??%%%))→?); #Hte
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
- >Houtc >associative_append >associative_append >associative_append %
- ]
- ]
-qed.
-*)
-
-definition init_current ≝
- adv_to_mark_l ? (is_marked ?) ·clear_mark ? ·
- adv_to_mark_l ? (λc:STape.is_grid (\fst c)) · move_r ? · mark ?.
-
-definition R_init_current ≝ λt1,t2.
- ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false →
- Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
- t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
- t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
- ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
-
-lemma sem_init_current : Realize ? init_current R_init_current.
-#intape
-cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
- (sem_seq ????? (sem_clear_mark ?)
- (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape)
-#k * #outc * #Hloop #HR
-@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop]
-cases HR -HR #ta * whd in ⊢ (%→?); * #_ #Hta
-* #tb * whd in ⊢ (%→?); * #_ #Htb
-* #tc * whd in ⊢ (%→?); * #_ #Htc
-* #td * whd in ⊢ (%→%→?); * #_ #Htd * #Houtc #_
-#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape
-cases (Hta … Hintape) #_ -Hta #Hta cases (Hta (refl …)) -Hta
-#Hta #_ lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
--Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) #_ #Htc
-cases (Htc Hc) -Htc #Htc #_ lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
--Htc #Htc lapply (Htd … Htc) -Htd
->reverse_append >reverse_cons
->reverse_cons in Hc0; cases (reverse … l2)
-[ normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
- #Htd >(Houtc … Htd) %
-| * #c2 #b2 #tl2 normalize in ⊢ (%→?);
- #Hc0 #Htd >(Houtc … Htd)
- whd in ⊢ (???%); destruct (Hc0)
- >associative_append >associative_append %
-]
-qed.
-
-definition match_tuple_step ≝
- ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c)))
- (single_finalTM ?
- (compare ·
- (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
- (nop ?)
- (mark_next_tuple ·
- (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
- (mark ?) (move_l ? · init_current) tc_true)) tc_true)))
- (nop ?) tc_true.
-
-definition R_match_tuple_step_true_new ≝ λt1,t2.
- ∃ls,cur,rs.t1 = midtape STape ls cur rs \wedge
- \fst cur ≠ grid ∧
- (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
- only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
- bit_or_null c = true → bit_or_null c1 = true →
- only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
- table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
- ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
- rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
- (* facciamo match *)
- (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
- (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
- ∨
- (* non facciamo match e marchiamo la prossima tupla *)
- (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
- ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
- (* condizioni su l5 l6 l7 *)
- t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
- (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
- l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
- ∨
- (* non facciamo match e non c'è una prossima tupla:
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
-
-(* universal version
-definition R_match_tuple_step_true ≝ λt1,t2.
- ∀ls,cur,rs.t1 = midtape STape ls cur rs →
- \fst cur ≠ grid ∧
- (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
- only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
- bit_or_null c = true → bit_or_null c1 = true →
- only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
- table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
- ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
- rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
- (* facciamo match *)
- (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
- (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
- ∨
- (* non facciamo match e marchiamo la prossima tupla *)
- (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
- ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
- (* condizioni su l5 l6 l7 *)
- t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
- (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
- l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
- ∨
- (* non facciamo match e non c'è una prossima tupla:
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
-*)
-
-definition R_match_tuple_step_true ≝ λt1,t2.
- ∃ls,cur,rs.t1 = midtape STape ls cur rs \wedge
- \fst cur ≠ grid ∧
- (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
- only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
- bit_or_null c = true → bit_or_null c1 = true →
- only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
- table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
- ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
- rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
- (* facciamo match *)
- (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
- (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
- ∨
- (* non facciamo match e marchiamo la prossima tupla *)
- (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
- ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
- (* condizioni su l5 l6 l7 *)
- t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
- (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
- l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
- ∨
- (* non facciamo match e non c'è una prossima tupla:
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
-
-definition R_match_tuple_step_false ≝ λt1,t2.
- ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
-
-include alias "basics/logic.ma".
-
-(*
-lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B.
- ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 →
- f x1 x2 x3 x4 = f y1 y2 y3 y4.
-//
-qed-. *)
-
-lemma some_option_hd: ∀A.∀l:list A.∀a.∃b.
- Some ? b = option_hd ? (l@[a]) .
-#A #l #a cases l normalize /2/
-qed.
-
-axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
- memb A x l1 = false → memb ? x l3 = false →
- l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
-
-axiom injective_append : ∀A,l.injective … (λx.append A x l).
-
-lemma sem_match_tuple_step:
- accRealize ? match_tuple_step (inr … (inl … (inr … start_nop)))
- R_match_tuple_step_true R_match_tuple_step_false.
-@(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
- (sem_seq … sem_compare
- (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
- (sem_nop …)
- (sem_seq … sem_mark_next_tuple
- (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
- (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
- (sem_nop ?) …)
-[(* is_grid: termination case *)
- 2:#t1 #t2 #t3 whd in ⊢ (%→?); * #Hc #H #H1 whd #ls #c #rs #Ht1 %
- [lapply(Hc c ?) [>Ht1 %] #Hgrid @injective_notb @Hgrid |>H1 @H]
-|#tapea #tapeout #tapeb whd in ⊢ (%→?); #Hcur
- * #tapec * whd in ⊢ (%→?); #Hcompare #Hor
- cases Hcur * #c * -Hcur #Hcur #Hgrid #Htapeb cases (current_to_midtape … Hcur)
- #ls * #rs #Htapea @(ex_intro … ls) @(ex_intro … c) @(ex_intro … rs) %
- [%[@Htapea | cases (true_or_false (\fst c == grid))
- [#eqc @False_ind >(\P eqc) in Hgrid; normalize #H destruct |#eqc @(\Pf eqc)]]]
- #ls0 #cur #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn
- #eqlen #Htable #Hls -Hcur #Hcur #Hrs >Htapea in Htapeb; >Hls >Hcur >Hrs #Htapeb
- cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
- cases (Hcompare cur c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (refl …) Hc ?)
- -Hcompare
- [* #Htemp destruct (Htemp) #Htapec %1 % % [%]
- >Htapec in Hor; -Htapec *
- [2: * #t3 * whd in ⊢ (%→?); * #H #_ @False_ind
- lapply (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
- |* #taped * whd in ⊢ (%→?); * #_
- #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped
- %
- ]
- |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
- cut (〈cur,false〉::l1 ≠ 〈c1,false〉::l3)
- [>H2 >H3 elim la
- [@(not_to_not …H1) normalize #H destruct (H) %
- |#x #tl @not_to_not normalize #H destruct //
- ]
- ] #Hnoteq
- cut (bit_or_null d' = true)
- [cases la in H3;
- [normalize in ⊢ (%→?); #H destruct //
- |#x #tl #H @(Hl3 〈d',false〉)
- normalize in H; destruct @memb_append_l2 @memb_hd
- ]
- ] #Hd'
- >Htapec in Hor; -Htapec *
- [* #taped * whd in ⊢ (%→?); * * #c0 * normalize in ⊢ (%→?);
- #Hdes destruct (Hdes) >(bit_or_null_not_grid ? Hd') #Htemp destruct (Htemp)
- |* #taped * whd in ⊢ (%→?); * #_ (* * #_ #H cases (H … (refl …)) -H #_ *)
- #Htaped * #tapee * whd in ⊢ (%→?); #Htapee
- <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
- cases (Htapee … Htaped ???) -Htaped -Htapee
- [* #rs3 * * (* we proceed by cases on rs4 *)
- [(* rs4 is empty : the case is absurd since the tape
- cannot end with a bar *)
- * #d * #b * * * #Heq1 @False_ind
- cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
- >Hcut in Htable; >H3 >associative_append
- normalize >Heq1 <associative_append >Hcut
- <associative_append #Htable @(absurd … Htable)
- @last_of_table
- |(* rs4 not empty *)
- * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
- cut (memb STape 〈d2,b2〉 (l2@〈c1,false〉::l3@〈comma,false〉::l4) = true)
- [@memb_append_l2
- cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
- >Hcut >H3 >associative_append @memb_append_l2
- @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
- cut (is_grid d2 = false)
- [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
- cut (b2 = false)
- [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
- >Hb2 in Heq1; #Heq1 -Hb2 -b2
- whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
- [(* we know current is not grid *)
- * #tapef * whd in ⊢ (%→?); * * #c0 *
- normalize in ⊢ (%→?); #Hdes destruct (Hdes) >Hd2
- #Htemp destruct (Htemp)
- |* #tapef * whd in ⊢ (%→?); * #_ #Htapef
- * #tapeg >Htapef -Htapef *
- (* move_l *)
- whd in ⊢ (%→?); * #_ #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
- >Htapeg -Htapeg
- (* init_current *)
- whd in ⊢ (%→?); #Htapeout
- cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
- * #c00 #b00 #Hoption
- lapply
- (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb))
- c' (reverse ? la) false ls0 bar (〈d2,true〉::rs3'@〈grid,false〉::rs0) c00 b00 ?????) -Htapeout
- [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
- >associative_append
- generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls0); #l
- whd in ⊢ (???(???%)); >associative_append >associative_append %
- |>reverse_cons @Hoption
- |cases la in H2;
- [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
- @bit_or_null_not_grid @Hc
- |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
- @bit_or_null_not_grid @(Hl1bitnull 〈c',false〉) @memb_append_l2 @memb_hd
- ]
- |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
- [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
- [#eqc0 >(\P eqc0) @Hc |@Hl1bitnull]
- |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
- @memb_append_l1 @daemon
- ]
- |@daemon] #Htapeout % %2 % //
- @(ex_intro … d2)
- cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
- [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
- [
- | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
- @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
-
- cut (sublist … lc l3)
- [ #x #Hx cases la in H3;
- [ normalize #H3 destruct (H3) @Hx
- | #p #la' normalize #Hla' destruct (Hla')
- @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
- @daemon]
- * #rs32 #Hrs3
- (* cut
- (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
- [@daemon] #Hcut *)
- cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
- [ >Hrs3 in Heq1; @daemon ] #Hl4
- @(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4]
- >Htapeout @eq_f2
- [(* by Hoption, H2 *) @daemon
- |(*>Hrs3 *)>append_cons
- > (?:l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
- = (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
- [|>associative_append normalize
- >associative_append normalize
- >associative_append normalize
- >associative_append normalize
- % ]
- >reverse_append >reverse_append >reverse_cons
- >reverse_reverse >reverse_cons >reverse_reverse
- >reverse_append >reverse_append >reverse_cons
- >reverse_reverse >reverse_reverse >reverse_reverse
- >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3
- =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3)))
- [|>associative_append >associative_append
- >associative_append >associative_append >associative_append
- >associative_append % ]
- <H2 normalize in ⊢ (??%?); >Hrs3
- >associative_append >associative_append normalize
- >associative_append >associative_append
- @eq_f @eq_f @eq_f
- >(?:la@(〈d',false〉::lc@〈comma,false〉::rs32)@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 =
- (la@〈d',false〉::lc)@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 )
- [| >associative_append normalize >associative_append % ]
- <H3 %
- ]
- ]
- ]
- |* #Hnobars #Htapee >Htapee -Htapee *
- [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); * #_
- #Htapef >Htapef -Htapef
- whd in ⊢ (%→?); * #Htapeout #_ %2 %
- [% [//] whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
- | >(Htapeout … (refl …)) % ]
- |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?);
- * #Hc0 lapply(Hc0 … (refl … )) normalize in ⊢ (%→?); #Htemp destruct (Htemp)
- ]
- |(* no marks in table *)
- #x #membx @(no_marks_in_table … Htable)
- @memb_append_l2
- cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
- >H3 >associative_append @memb_append_l2 @memb_cons @membx
- |(* no grids in table *)
- #x #membx @(no_grids_in_table … Htable)
- @memb_append_l2
- cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
- >H3 >associative_append @memb_append_l2 @memb_cons @membx
- |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
- ]
- ]
- |#x #membx @(no_marks_in_table … Htable)
- @memb_append_l2 @memb_cons @memb_append_l1 @membx
- |#x #membx @(no_marks_in_table … Htable)
- @memb_append_l1 @membx
- |%
- ]
- ]
-qed.
-
-(*
- MATCH TUPLE
-
- scrolls through the tuples in the transition table until one matching the
- current configuration is found
-*)
-
-definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … start_nop))).
-
-lemma is_grid_true : ∀c.is_grid c = true → c = grid.
-* normalize [ #b ] #H // destruct (H)
-qed.
-
-(* possible variante ?
-definition weakR_match_tuple ≝ λt1,t2.
- (∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧
- (∀c,l1,c1,l2,l3,ls0,rs0,n.
- t1 = midtape STape (〈grid,false〉::ls0) 〈bit c,true〉 rs
- (l1@〈grid,false〉::l2@〈bit c1,true〉::l3@〈grid,false〉::rs0) →
- only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
- table_TM (S n) (l2@〈c1,false〉::l3) →
- (* facciamo match *)
- (∃l4,newc,mv,l5.
- 〈c1,false〉::l3 = l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
- (l2@l4@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
- 〈grid,false〉::rs0))
- ∨
- (* non facciamo match su nessuna tupla;
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (current ? t2 = Some ? 〈grid,true〉 ∧
- ∀l4,newc,mv,l5.
- 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
-*)
-
-definition R_match_tuple0 ≝ λt1,t2.
- ∀ls,cur,rs.
- t1 = midtape STape ls cur rs →
- (is_grid (\fst cur) = true → t2 = t1) ∧
- (∀c,l1,c1,l2,l3,ls0,rs0,n.
- ls = 〈grid,false〉::ls0 →
- cur = 〈c,true〉 →
- rs = l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0 →
- is_bit c = true → is_bit c1 = true →
- only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
- table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3) →
- (* facciamo match *)
- (∃l4,newc,mv,l5.
- 〈bar,false〉::〈c1,false〉::l3 = l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
- (l2@l4@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
- 〈grid,false〉::rs0))
- ∨
- (* non facciamo match su nessuna tupla;
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (current ? t2 = Some ? 〈grid,true〉 ∧
- ∀l4,newc,mv,l5.
- 〈bar,false〉::〈c1,false〉::l3 ≠ l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
-
-axiom table_bit_after_bar :
- ∀n,l1,c,l2.table_TM n (l1@〈bar,false〉::〈c,false〉::l2) → is_bit c = true.
-
-lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple0.
-#intape #k #outc #Hloop
-lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
-* #ta * #Hstar @(star_ind_l ??????? Hstar)
-[ whd in ⊢ (%→?); #Hleft
- #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
- [ #_ @Houtc
- | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs
- >Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?);
- #Hc destruct (Hc)
- ]
-| (* in the interesting case, we execute a true iteration, then we restart the
- while cycle, finally we end with a false iteration *)
- #tc #td whd in ⊢ (%→?); #Htc
- #Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
- #ls #cur #rs #Htb %
- [ (* cur can't be true because we assume at least one iteration *)
- #Hcur cases Htc #ls' * #c' * #rs' * * >Htb #Hdes destruct (Hdes)
- #Hfalse @False_ind @(absurd … (is_grid_true … Hcur) Hfalse)
- | (* current and a tuple are marked *)
- #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs #Hc #Hc1 #Hl1bitnull #Hl1marks
- #Hl1len #Htable
- cases Htc #ls' * #c' * #rs' * * >Htb #Hdes destruct (Hdes)
- -Htb * #_ #Htc
- (* expose the marked tuple in table *)
- cut (∃la,lb,mv,lc.l3 = la@〈comma,false〉::lb@〈comma,false〉::mv::lc ∧
- S n = |la| ∧ only_bits_or_nulls la)
- [@daemon] * #la * #lb * #mv * #lc * * #Hl3 #Hlalen #Hlabitnull
- >Hl3 in Htable; >append_cons #Htable
- >(?: l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0
- = (l2@[〈bar,false〉])@〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv::
- lc)@〈grid,false〉::rs0) in Hrs;
- [| >associative_append normalize >Hl3
- >associative_append normalize % ] #Hrs
- cases (Htc ????????? Hl1bitnull Hl1marks ?? Hlabitnull Hl1len ? Htable Hls Hcur Hrs)
- [5: <Hl1len @Hlalen
- |4: whd in ⊢ (??%?); >Hc1 %
- |3: whd in ⊢ (??%?); >Hc %
- |-Htc *
- [ (* case 1: match successful *)
- * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) %
- [%
- | cases (IH … Htc) -IH #Houtc #_ >(Houtc (refl ??))
- >Htc @eq_f normalize >associative_append normalize
- >associative_append normalize %
- ]
- | (* case 2: tuples don't match, we still have other tuples to try *)
- * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc
- cases (IH ??? … Htc) -IH #_ #IH
- (* by induction hypothesis *)
- lapply (IH ? l1 c2 (l2@〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5) l6 ? rs0 n (refl ??) (refl ??) ???????)
- [ generalize in match Htable;
- >associative_append normalize
- >associative_append normalize >Heqlblc
- >associative_append normalize //
- | @Hl1len
- | @Hl1marks
- | @Hl1bitnull
- | (*???*) @daemon
- | @Hc
- | >associative_append normalize
- >associative_append normalize
- >associative_append %
- |-IH *
- [ (* the while finally matches a tuple *)
- * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc %
- >Heqlblc @(ex_intro ?? (〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5@l7))
- %{newc} %{mv0} %{l8} %
- [ normalize >Hl7l8 >associative_append normalize
- >associative_append %
- | >Houtc @eq_f >associative_append normalize
- >associative_append normalize >associative_append
- normalize >associative_append %
- ]
- | (* the while fails finding a tuple: there are no matches in the whole table *)
- * #Houtc #Hdiff1 %2 %
- [ @Houtc
- | #l50 #newc #mv0 #l51 >Heqlblc
- @daemon
- ]
- ]
- ]
- ]
- | (* match failed and there is no next tuple: the next while cycle will just exit *)
- * * #Hdiff #Hnobars generalize in match (refl ? tc);
- cases tc in ⊢ (???% → %);
- [ #_ @daemon (* long normalize *) (*
- normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
- *)
- |2,3: #x #xs #_ @daemon (* long normalize *) (*
- normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) *) ]
- #ls1 #cur1 #rs1 #Htc @daemon (* long normalize *) (*
- normalize in ⊢ (??%?→?); #Hcur1
- cases (IH … Htc) -IH #IH #_ %2 %
- [ destruct (Hcur1) >IH [ >Htc % | % ]
- | #l4 #newc #mv0 #l5
- (* no_bars except the first one, where the tuple does not match ⇒
- no match *)
- @daemon
- ] *)
- ]
- ]
-qed.
-
-(* termination *)
-lemma WF_mts_niltape:
- WF ? (inv ? R_match_tuple_step_true) (niltape (FinProd FSUnialpha FinBool)).
-@wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
-qed.
-
-lemma WF_mts_rightof:
- ∀a,ls. WF ? (inv ? R_match_tuple_step_true) (rightof (FinProd FSUnialpha FinBool) a ls).
-#a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
-qed.
-
-lemma WF_mts_leftof:
- ∀a,ls. WF ? (inv ? R_match_tuple_step_true) (leftof (FinProd FSUnialpha FinBool) a ls).
-#a #ls @wf #t1 whd in ⊢ (%→?); * #ls * #c * #rs * * #H destruct
-qed.
-
-lemma WF_cst_midtape_grid:
- ∀ls,b,rs. WF ? (inv ? R_match_tuple_step_true)
- (midtape (FinProd … FSUnialpha FinBool) ls 〈grid,b〉 rs).
-#ls #b #rs @wf #t1 whd in ⊢ (%→?); * #ls' * #c' * #rs' * * #H destruct
-* #Hfalse @False_ind @Hfalse %
-qed.
-
-definition Pre_match_tuple ≝ λt.
- ∃ls,cur,rs. t = midtape STape ls cur rs ∧
- (is_grid (\fst cur) = true ∨
- (∃ls0,c,l1,l2,c1,l3,l4,rs0,n.
- only_bits_or_nulls l1 ∧ no_marks l1 ∧
- bit_or_null c = true ∧ bit_or_null c1 = true ∧
- only_bits_or_nulls l3 ∧ S n = |l1| ∧|l1| = |l3| ∧
- table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) ∧
- ls = 〈grid,false〉::ls0 ∧ cur = 〈c,true〉 ∧
- rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0)).
-
-lemma acc_Realize_to_acc_GRealize: ∀sig,M.∀q:states sig M.∀P,R1,R2.
- M ⊨ [q:R1,R2] → accGRealize sig M q P R1 R2.
-#alpha #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.
-
-
-lemma terminate_match_tuple:
- ∀t. Pre_match_tuple t → Terminate ? match_tuple t.
-#t #HPre
-@(terminate_while_guarded ???
- Pre_match_tuple …
- (acc_Realize_to_acc_GRealize ??? Pre_match_tuple … sem_match_tuple_step)
- … HPre) [%]
- [-HPre -t #t1 #t2 #HPre cases HPre #ls * * #curl #curr * #rs * #Ht1 *
- [(* absurd case *)
- #Hgrid * #ls1 * #cur1 * #rs1 * * >Ht1 #Hdes destruct (Hdes)
- #Habs @False_ind @(absurd ?? Habs) @(is_grid_true … Hgrid)
- |* #ls0 * #c * #l1 * #l2 * #c1 * #l3 * #l4 * #rs0 * #n
- * * * * * * * * * *
- #Hl1 #Hmarksl1 #Hc #Hc1 #Hl3 #lenl1 #eqlen #Htable #Hls #Hcur #Hrs
- * #ls1 * #cur1 * #rs1 * * >Ht1 #Hdes destruct (Hdes) #Hdes #H
- lapply (H … Hl1 Hmarksl1 Hc Hc1 Hl3 lenl1 eqlen Htable Hls Hcur Hrs)
- -H *
- [* [ * #Hdes #Ht2 >Ht2
- @ex_intro [2:@ex_intro [2: @ex_intro [2: % [%]|]|]|]
- %1 %
- |* #test * #c2 * #l5 * #l6 * #Hl4 #Ht2
- cut (∃l7,l8. l6 = l7@〈comma,false 〉::l8 ∧ |l7| = |l1|) [@daemon]
- * #l7 * #l8 * #Hl6 #eqlen1
- @ex_intro [2:@ex_intro [2: @ex_intro [2: % [@Ht2]|]|]|] %2
- @(ex_intro … ls0) @(ex_intro … c) @(ex_intro … l1)
- @(ex_intro … (l2@〈c1,false〉::l3@〈comma,false〉::l5@[〈bar,false〉]))
- @(ex_intro … c2) @(ex_intro … l7) @(ex_intro … l8)
- @(ex_intro … rs0) @(ex_intro … n)
- % [2: >Hl6 >associative_append >associative_append @eq_f @eq_f @eq_f
- @eq_f >associative_append @eq_f @eq_f >associative_append % ]
- % [2: %] % [2: %] % [2:@daemon] % [2: @sym_eq @eqlen1]
- % [2: @lenl1] % [2: #x #memx @daemon]
- % [2: @daemon] % [2: @Hc] % [2: @Hmarksl1] @Hl1
- ]
- |* * #_ #_ #H cases (current_to_midtape … H) #ls * #rs #Ht1
- >Ht1 @ex_intro [2:@ex_intro [2: @ex_intro [2: % [%]|]|]|] %1 %
- ]
- ]
- |cases HPre -HPre #ls * * #curl #curr * #rs * #Ht *
- [#Hgrid >Ht >(is_grid_true … Hgrid) @WF_cst_midtape_grid
- |* #ls0 * #c * #l1 * #l2 * #c1 * #l3 * #l4
- cut (∃len. |l4| = len) [/2/] * #lenl4
- lapply l4 lapply l3 lapply c1 lapply l2 lapply l1 lapply c lapply ls0 lapply Ht
- lapply curr lapply curl lapply ls lapply rs lapply t -l4 -l3 -l2 -l1 -c1 -curr -curl -ls -t
- -c -ls0 -rs
- (* by induction on the length of l4 *)
- @(nat_elim1 lenl4)
- #len #Hind #t #rs #ls #cl #cr #Ht #ls0 #c #l1 #l2 #c1 #l3 #l4 #Hlen
- * #rs0 * #n * * * * * * * * * *
- #Hl1 #Hmarksl1 #Hc #Hc1 #Hl3 #lenl1 #eqlen #Htable #Hls #Hcur #Hrs
- % #t1 >Ht whd in ⊢ (%→?); * #ls1 * #cur * #rs1 * * #Hdes destruct (Hdes)
- #Hgrid #H lapply (H … Hl1 Hmarksl1 Hc Hc1 Hl3 lenl1 eqlen Htable Hls Hcur Hrs)
- -H *
- [* [ * #Hdes destruct (Hdes) #Ht1 >Ht1 @WF_cst_midtape_grid
- | * #_ * #c2 * #l5 * #l6 * #Hl4 #Ht1
- cut (∃l7,l8. l6 = l7@〈comma,false 〉::l8 ∧ |l7| = |l1|) [@daemon]
- * #l7 * #l8 * #Hl6 #eqlen1
- @(Hind … Ht1 ls0 c l1 (l2@〈c1,false〉::l3@〈comma,false〉::l5@[〈bar,false〉]) c2 l7 l8 … (refl …))
- [<Hlen >Hl4 >Hl6 >length_append normalize in match (length … (cons …));
- >length_append normalize in match (length … (cons …)); <plus_n_Sm
- @le_S_S @daemon
- |@(ex_intro … rs0) @(ex_intro … n) %
- [2: >Hl6 >associative_append >associative_append @eq_f @eq_f @eq_f
- @eq_f >associative_append @eq_f @eq_f >associative_append % ]
- % [2: %] % [2: %] % [2:@daemon] % [2: @sym_eq @eqlen1]
- % [2: @lenl1] % [2: #x #memx @daemon]
- % [2: @daemon] % [2: @Hc] % [2: @Hmarksl1] @Hl1
- ]
- ]
- |* * #_ #_ #H cases (current_to_midtape … H) #ls * #rs #Ht1
- >Ht1 //
- ]
- ]
-qed.
-
-definition R_match_tuple ≝ λt1,t2.
- ∀ls,c,l1,c1,l2,rs,n.
- is_bit c = true → is_bit c1 = true →
- only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
- table_TM (S n) (〈bar,false〉::〈c1,false〉::l2) →
- t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
- (l1@〈grid,false〉::〈bar,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
- (* facciamo match *)
- (∃l3,newc,mv,l4.
- 〈bar,false〉::〈c1,false〉::l2 = l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
- (l3@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l4@〈grid,false〉::rs))
- ∨
- (* non facciamo match su nessuna tupla;
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (current ? t2 = Some ? 〈grid,true〉 ∧
- ∀l3,newc,mv,l4.
- 〈bar,false〉::〈c1,false〉::l2 ≠ l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4).
-
-lemma sem_match_tuple0 : GRealize ? match_tuple Pre_match_tuple R_match_tuple0.
-@WRealize_to_GRealize [@terminate_match_tuple | @wsem_match_tuple]
-qed.
-
-lemma sem_match_tuple : GRealize ? match_tuple Pre_match_tuple R_match_tuple.
-generalize in match sem_match_tuple0; @GRealize_to_GRealize
-#t1 #t2 #HR #ls #c #l1 #c1 #l2 #rs #n #Hc #Hc1 #Hl1bitsnulls #Hl1marks #Hl1len #Htable #Ht1
-cases (HR … Ht1) -HR #_ #HR
-@(HR ??? [] … (refl ??) (refl ??) (refl ??) Hc Hc1 Hl1bitsnulls Hl1marks
- Hl1len Htable)
-qed.
\ No newline at end of file
projects / helm.git / blobdiff