notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}.
interpretation "conditional realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
+(*************************** guarded realizablity *****************************)
+definition GRealize ≝ λsig.λM:TM sig.λPre:tape sig →Prop.λR:relation (tape sig).
+∀t.Pre t → ∃i.∃outc.
+ loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
+
+definition accGRealize ≝ λsig.λM:TM sig.λacc:states sig M.
+λPre: tape sig → Prop.λRtrue,Rfalse.
+∀t.Pre t → ∃i.∃outc.
+ loopM sig M i (initc sig M t) = Some ? outc ∧
+ (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
+ (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
+
+lemma WRealize_to_GRealize : ∀sig.∀M: TM sig.∀Pre,R.
+ (∀t.Pre t → M ↓ t) → M ⊫ R → GRealize sig M Pre R.
+#sig #M #Pre #R #HT #HW #t #HPre cases (HT … t HPre) #i * #outc #Hloop
+@(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
+qed.
+
+
(******************************** monotonicity ********************************)
lemma Realize_to_Realize : ∀alpha,M,R1,R2.
R1 ⊆ R2 → Realize alpha M R1 → Realize alpha M R2.
@Hsub @(HR1 … i) @Hloop
qed.
+lemma GRealize_to_GRealize : ∀alpha,M,P,R1,R2.
+ R1 ⊆ R2 → GRealize alpha M P R1 → GRealize alpha M P R2.
+#alpha #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.
+
lemma acc_Realize_to_acc_Realize: ∀sig,M.∀q:states sig M.∀R1,R2,R3,R4.
R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
#alpha #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
% [@Hloop |@Hsub @Houtc]
qed.
+
+(* composition with guards *)
+theorem sem_seq_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2.
+ GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 →
+ (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) →
+ GRealize sig (M1 · M2) Pre1 (R1 ∘ R2).
+#sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #HGR1 #HGR2 #Hinv #t1 #HPre1
+cases (HGR1 t1 HPre1) #k1 * #outc1 * #Hloop1 #HM1
+cases (HGR2 (ctape sig (states ? M1) 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 (states sig M1) (states sig M2))
+ (step sig M1) (step sig (seq sig M1 M2))
+ (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_seq_liftL //
+ | #x <p_halt_liftL %
+ |6:cases outc1 #s1 #t1 %
+ |7:@(loop_lift … (initc ?? (ctape … 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 M1 M2 ??)
+ [ whd in ⊢ (??%?); whd in ⊢ (???%);
+ @config_eq whd in ⊢ (???%); //
+ | @(loop_Some ?????? Hloop10) ]
+ ]
+| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+ % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
+]
+qed.
+
+theorem sem_seq_app_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2,R3.
+ GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 →
+ (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → R1 ∘ R2 ⊆ R3 →
+ GRealize sig (M1 · M2) Pre1 R3.
+#sig #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.
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.
(mark ?) (move_l ? · init_current) tc_true)) tc_true)))
(nop ?) tc_true.
+(* universal version
definition R_match_tuple_step_true ≝ λt1,t2.
∀ls,cur,rs.t1 = midtape STape ls cur rs →
\fst cur ≠ grid ∧
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.
[lapply(Hc c ?) [>Ht1 %] #Hgrid @injective_notb @Hgrid |>H1 @H]
|#tapea #tapeout #tapeb whd in ⊢ (%→?); #Hcur
* #tapec * whd in ⊢ (%→?); #Hcompare #Hor
- #ls #cur #rs #Htapea >Htapea in Hcur; * * #c *
- normalize in ⊢ (%→?); #Hdes destruct (Hdes) #Hcur #Htapeb %
- [ % #Hfalse >Hfalse in Hcur; normalize #Hfalse1 destruct (Hfalse1)]
- #ls0 #c #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn
- #eqlen #Htable #Hls #Hcur #Hrs -Htapea >Hls in Htapeb; >Hcur >Hrs #Htapeb
+ 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 c c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (refl …) Hc ?)
+ 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 *
%
]
|* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
- cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3)
+ cut (〈cur,false〉::l1 ≠ 〈c1,false〉::l3)
[>H2 >H3 elim la
- [@(not_to_not …H1) normalize #H destruct %
+ [@(not_to_not …H1) normalize #H destruct (H) %
|#x #tl @not_to_not normalize #H destruct //
]
] #Hnoteq
#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 … Htb) * #Hfalse @False_ind @Hfalse @(is_grid_true … Hcur)
+ #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 … Htb) -Htc -Htb * #_ #Htc
+ #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)
]
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 →
∀l3,newc,mv,l4.
〈bar,false〉::〈c1,false〉::l2 ≠ l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4).
-(* we still haven't proved termination *)
-axiom sem_match_tuple0 : Realize ? match_tuple R_match_tuple0.
+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 : Realize ? match_tuple R_match_tuple.
-generalize in match sem_match_tuple0; @Realize_to_Realize
+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
projects / helm.git / commitdiff