--- /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_____________________________________________________________*)
+
+
+(* COMPARE BIT
+
+*)
+
+include "turing/universal/tuples.ma".
+
+definition write_states ≝ initN 2.
+
+definition write ≝ λalpha,c.
+ mk_TM alpha write_states
+ (λp.let 〈q,a〉 ≝ p in
+ match q with
+ [ O ⇒ 〈1,Some ? 〈c,N〉〉
+ | S _ ⇒ 〈1,None ?〉 ])
+ O (λx.x == 1).
+
+definition R_write ≝ λalpha,c,t1,t2.
+ ∀ls,x,rs.t1 = midtape alpha ls x rs → t2 = midtape alpha ls c rs.
+
+axiom sem_write : ∀alpha,c.Realize ? (write alpha c) (R_write alpha c).
+
+definition copy_step_subcase ≝
+ λalpha,c,elseM.ifTM ? (test_char ? (λx.x == 〈c,true〉))
+ (seq (FinProd alpha FinBool) (adv_mark_r …)
+ (seq ? (move_l …)
+ (seq ? (adv_to_mark_l … (is_marked alpha))
+ (seq ? (write ? 〈c,false〉)
+ (seq ? (move_r …)
+ (seq ? (mark …)
+ (seq ? (move_r …) (adv_to_mark_r … (is_marked alpha)))))))))
+ elseM tc_true.
+
+definition R_copy_step_subcase ≝
+ λalpha,c,RelseM,t1,t2.
+ ∀ls,x,rs.t1 = midtape (FinProd … alpha FinBool) ls 〈x,true〉 rs →
+ (x = c ∧
+ ∀a,l1,x0,a0,l2,l3. (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ ls = l1@〈a0,false〉::〈x0,true〉::l2 →
+ rs = 〈a,false〉::l3 →
+ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
+ (x ≠ c ∧ RelseM t1 t2).
+
+axiom sem_copy_step_subcase :
+ ∀alpha,c,elseM,RelseM.
+ Realize ? (copy_step_subcase alpha c elseM) (R_copy_step_subcase alpha c RelseM).
+
+(*
+if current = 0,tt
+ then advance_mark_r;
+ move_l;
+ advance_to_mark_l;
+ write(0,ff)
+ move_r;
+ mark;
+ move_r;
+ advance_to_mark_r;
+else if current = 1,tt
+ then advance_mark_r;
+ move_l;
+ advance_to_mark_l;
+ write(1,ff)
+ move_r;
+ mark;
+ move_r;
+ advance_to_mark_r;
+else nop
+*)
+
+definition copy_step ≝
+ ifTM ? (test_char STape (λc.is_bit (\fst c)))
+ (single_finalTM ? (copy_step_subcase FSUnialpha (bit false)
+ (copy_step_subcase FSUnialpha (bit true) (nop ?))))
+ (nop ?)
+ tc_true.
+
+definition R_copy_step_true ≝
+ λt1,t2.
+ ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls 〈c,true〉 rs →
+ ∃x. c = bit x ∧
+ (∀a,l1,c0,a0,l2,l3. (∀y.memb ? y l1 = true → is_marked ? y = false) →
+ ls = l1@〈a0,false〉::〈c0,true〉::l2 →
+ rs = 〈a,false〉::l3 →
+ t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3).
+
+definition R_copy_step_false ≝
+ λt1,t2.
+ ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
+ is_bit (\fst c) = false ∧ t2 = t1.
+
+axiom sem_comp_step :
+ accRealize ? copy_step (inr … (inl … (inr … 0))) R_copy_step_true R_copy_step_false.
+
+definition copy ≝ whileTM ? copy_step (inr … (inl … (inr … 0))).
+
+definition R_copy ≝ λt1,t2.
+ ∀ls,c,rs.t1 = midtape ? ls 〈c,true〉 rs →
+ (∀l1,d,l2,l3,l4.
+ 〈c,false〉::rs = l1@〈d,false〉::l2 → only_bits l1 → is_bit d = false →
+ ls = l3@l4@〈c0,true〉::l5 → |l4| = |l1@[〈d,false〉]|
+
+
+
+axiom no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
+(*
+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 bar_or_grid ≝ λc:STape.is_bar (\fst c) ∨ is_grid (\fst c).
+
+definition mark_next_tuple ≝
+ seq ? (adv_to_mark_r ? bar_or_grid)
+ (ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
+ (move_right_and_mark ?) (nop ?) 1).
+
+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 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 ?) 1) ????)
+[@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 (Hleft … Hrs)
+ [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
+ | * #_ #Hta cases (tech_split STape (λc.is_bar (\fst c)) rs1)
+ [ #H1 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
+ [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
+ | %
+ | -Hta #Hta cases Hright
+ [ * #tb * whd in ⊢ (%→?); #Hcurrent
+ @False_ind cases (Hcurrent 〈grid,false〉 ?)
+ [ normalize #Hfalse destruct (Hfalse)
+ | >Hta % ]
+ | * #tb * whd in ⊢ (%→?); #Hcurrent
+ cases (Hcurrent 〈grid,false〉 ?)
+ [ #_ #Htb whd in ⊢ (%→?); #Houtc
+ %2 %
+ [ @H1
+ | >Houtc >Htb >Hta % ]
+ | >Hta % ]
+ ]
+ ]
+ | * #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
+ [ * #tb * whd in ⊢ (%→?); #Hta'
+ whd in ⊢ (%→?); #Htb
+ cases (Hta' c0 ?)
+ [ #_ #Htb' >Htb' in Htb; #Htb
+ generalize in match Hsplit; -Hsplit
+ cases rs4 in Hta;
+ [ #Hta #Hsplit >(Htb … Hta)
+ >(?:c0 = 〈bar,false〉)
+ [ @(ex_intro ?? grid) @(ex_intro ?? false)
+ % [ % [ %
+ [(* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ]
+ | (* Hc0 *) @daemon ]
+ | #r5 #rs5 >(eq_pair_fst_snd … r5)
+ #Hta #Hsplit >(Htb … Hta)
+ >(?:c0 = 〈bar,false〉)
+ [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5))
+ % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ]
+ | % ] | (* Hc0 *) @daemon ] ] | >Hta % ]
+ | * #tb * whd in ⊢ (%→?); #Hta'
+ whd in ⊢ (%→?); #Htb
+ cases (Hta' c0 ?)
+ [ #Hfalse @False_ind >Hfalse in Hc0;
+ #Hc0 destruct (Hc0)
+ | >Hta % ]
+]]]]
+qed.
+
+definition init_current ≝
+ seq ? (adv_to_mark_l ? (is_marked ?))
+ (seq ? (clear_mark ?)
+ (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (seq ? (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) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
+-Hta * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
+-Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) [ >Hc -Hc * #Hc destruct (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 ?
+ (seq ? compare
+ (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
+ (nop ?)
+ (seq ? mark_next_tuple
+ (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
+ (mark ?) (seq ? (move_l ?) init_current) tc_true)) tc_true)))
+ (nop ?) tc_true.
+
+definition R_match_tuple_step_true ≝ λt1,t2.
+ ∀ls,c,l1,l2,c1,l3,l4,rs,n.
+ is_bit c = true → only_bits l1 → no_grids l2 → is_bit c1 = true →
+ only_bits l3 → n = |l1| → |l1| = |l3| →
+ table_TM (S n) (〈c1,true〉::l3@〈comma,false〉::l4) →
+ t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
+ (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs) →
+ (* facciamo match *)
+ (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
+ t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
+ (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs))
+ ∨
+ (* non facciamo match e marchiamo la prossima tupla *)
+ ((〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
+ ∃c2,l5,l6,l7.l4 = l5@〈bar,false〉::〈c2,false〉::l6@〈comma,false〉::l7 ∧
+ (* condizioni su l5 l6 l7 *)
+ t2 = midtape STape (〈grid,false〉::ls) 〈c,true〉
+ (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::
+ l5@〈bar,false〉::〈c2,true〉::l6@〈comma,false〉::l7))
+ ∨
+ (* 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.
+
+lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false.
+* // normalize #H destruct
+qed.
+
+lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false.
+* // normalize #H destruct
+qed.
+
+axiom sem_match_tuple_step:
+ accRealize ? match_tuple_step (inr … (inl … (inr … 0)))
+ 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 ⊢ (%→?); #H #H1 whd #ls #c #rs #Ht1
+ cases (H c ?) [2: >Ht1 %] #Hgrid #Heq %
+ [@injective_notb @Hgrid | <Heq @H1]
+|#tapea #tapeout #tapeb whd in ⊢ (%→?); #Htapea
+ * #tapec * #Hcompare #Hor
+ #ls #c #l1 #l2 #c1 #l3 #l4 #rs #n #Hc #Hl1 #Hl2 #Hc1 #Hl3 #eqn
+ #eqlen #Htable #Htapea1 cases (Htapea 〈c,true〉 ?) >Htapea1 [2:%]
+ #notgridc -Htapea -Htapea1 -tapea #Htapeb
+ cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
+ cases (Hcompare c c1 l1 l3 (l2@[〈bar,false〉]) (l4@〈grid,false〉::rs) eqlen … (refl …) Hc ?)
+ -Hcompare
+ [* #Htemp destruct (Htemp) #Htapec %1 % [%]
+ >Htapec in Hor; -Htapec *
+ [2: * #t3 * whd in ⊢ (%→?); #H @False_ind
+ cases (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
+ |* #taped * whd in ⊢ (%→?); #Htaped cases (Htaped ? (refl …)) -Htaped *
+ #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped >associative_append
+ %
+ ]
+ |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
+ cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3)
+ [>H2 >H3 elim la
+ [@(not_to_not …H1) normalize #H destruct %
+ |#x #tl @not_to_not normalize #H destruct //
+ ]
+ ] #Hnoteq %2
+ cut (is_bit 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 ⊢ (%→?); #H @False_ind
+ cases (H … (refl …)) >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
+ lapply (Htapee … Htaped ???) -Htaped -Htapee
+ [whd in ⊢ (??%?); >(bit_not_grid … Hd') >(bit_not_bar … Hd') %
+ |#x #Hx cases (memb_append … Hx)
+ [-Hx #Hx @bit_not_grid @Hl3 cases la in H3; normalize
+ [#H3 destruct (H3) @Hx | #y #tl #H3 destruct (H3)
+ @memb_append_l2 @memb_cons @Hx ]
+ |-Hx #Hx @(no_grids_in_table … Htable)
+ @memb_cons @memb_append_l2 @Hx
+ ]
+ |@daemon (* TODO *)
+ |*
+ [* #rs3 * * (* we proceed by cases on rs4 *)
+ [* #d * #b * * * #Heq1 #Hnobars
+ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
+ #Htapee *
+ [* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef
+ cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
+ whd in ⊢ (%→?); #H lapply (H … ???? (refl …)) #Htapeout
+ %1 %
+ [ //| @daemon]
+ | >Htapeout %
+ ]
+ |* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef
+ cases (Htapef … (refl …)) whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
+ ]
+ |* #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
+ cut (is_grid d2 = false) [@daemon (* ??? *)] #Hd2
+ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
+ [* #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp)
+ |* #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) #_ -Htapef #Htapef
+ * #tapeg >Htapef -Htapef * whd in ⊢ (%→?);
+ #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
+ >Htapeg -Htapeg whd in ⊢ (%→?); #Htapeout
+ %1 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
+ * #c00 #b00 #Hoption
+ lapply
+ (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? (l2@[〈bar,false〉])@(〈grid,false〉::reverse ? lb))
+ c' (reverse ? la) false ls bar (〈d2,true〉::rs3'@〈grid,false〉::rs) c00 b00 ?????) -Htapeout
+ [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
+ >associative_append
+ generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls); #l
+ whd in ⊢ (???(???%)); >associative_append >associative_append
+ %
+ |@daemon
+ |@daemon
+ |@daemon
+ |@daemon
+ |@daemon
+ ]
+ ]
+ ]
+ |* #Hnobars #Htapee >Htapee -Htapee *
+ [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
+ whd in ⊢ (%→?); #Htapeout %2
+ >(Htapeout … (refl …)) %
+ [ %
+ [ @daemon
+ | @daemon
+ ]
+ | %
+ ]
+ |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) -Htapef
+ whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
+ ]
+ |
+
+
+
+
+
+
+ ????? (refl …) Hc ?) -Hcompare
+ #Hcompare
+ is_bit c = true → only_bits l1 → no_grids l2 → is_bit c1 = true →
+ only_bits l3 → n = |l2| → |l2| = |l3| →
+ table_TM (S n) (〈c1,true〉::l3@〈comma,false〉::l4) →#x
+
+ #intape
+cases
+ (acc_sem_if … (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 ?) intape)
+#k * #outc * * #Hloop #H1 #H2
+@(ex_intro ?? k) @(ex_intro ?? outc) %
+[ % [@Hloop ] ] -Hloop
+ *)
+
+(*
+ 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 … 0))).
+
+definition R_match_tuple ≝ λt1,t2.
+ ∀ls,c,l1,c1,l2,rs,n.
+ is_bit c = true → only_bits l1 → is_bit c1 = true → n = |l1| →
+ table_TM (S n) (〈c1,true〉::l2) →
+ t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
+ (l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
+ (* facciamo match *)
+ (∃l3,newc,mv,l4.
+ 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4 ∧
+ t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
+ (l3@〈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.
+ 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4).
projects / helm.git / commitdiff