- MARK NEXT TUPLE machine
- (partially axiomatized)
-
- marks the first character after the first bar (rightwards)
- *)
-
-check unialpha
-
-axiom is_bar : FinProd … myalpha FinBool → bool.
-axiom is_grid : FinProd … myalpha FinBool → bool.
-definition bar_or_grid ≝ λc.is_bar c ∨ is_grid c.
-axiom bar : FinProd … myalpha FinBool.
-axiom grid : FinProd … myalpha FinBool.
-
-definition mark_next_tuple ≝
- seq ? (adv_to_mark_r ? bar_or_grid)
- (ifTM ? (test_char ? is_bar)
- (move_r_and_mark ?) (nop ?) 1).
-
-definition R_mark_next_tuple ≝
- λt1,t2.
- ∀ls,c,rs1,rs2.
- (* c non può essere un separatore ... speriamo *)
- t1 = midtape ? ls c (rs1@grid::rs2) →
- memb ? grid rs1 = false → bar_or_grid c = false →
- (∃rs3,rs4,d,b.rs1 = rs3 @ bar :: rs4 ∧
- memb ? bar rs3 = false ∧
- Some ? 〈d,b〉 = option_hd ? (rs4@grid::rs2) ∧
- t2 = midtape ? (bar::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@grid::rs2)))
- ∨
- (memb ? bar rs1 = false ∧
- t2 = midtape ? (reverse ? rs1@c::ls) grid 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 c = 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 ? is_bar) (mark ?) (nop ?) 1) ????)
-[@sem_if //
-| //
-|||#Hif cases (Hif intape) -Hif
- #j * #outc * #Hloop * #ta * #Hleft #Hright
- @(ex_intro ?? j) @ex_intro [|% [@Hloop] ]
- -Hloop
- #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hc
- cases (Hleft … Hrs)
- [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
- | * #_ #Hta cases (tech_split ? is_bar rs1)
- [ #H1 lapply (Hta rs1 grid rs2 (refl ??) ? ?)
- [ (* Hrs1, H1 *) @daemon
- | (* bar_or_grid grid = true *) @daemon
- | -Hta #Hta cases Hright
- [ * #tb * whd in ⊢ (%→?); #Hcurrent
- @False_ind cases(Hcurrent grid ?)
- [ #Hfalse (* grid is not a bar *) @daemon
- | >Hta % ]
- | * #tb * whd in ⊢ (%→?); #Hcurrent
- cases (Hcurrent grid ?)
- [ #_ #Htb whd in ⊢ (%→?); #Houtc
- %2 %
- [ (* H1 *) @daemon
- | >Houtc >Htb >Hta % ]
- | >Hta % ]
- ]
- ]
- | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
- % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
- lapply (Hta rs3 c0 (rs4@grid::rs2) ???)
- [ #x #Hrs3' (* Hrs1, Hrs3, Hsplit *) @daemon
- | (* bar → bar_or_grid *) @daemon
- | >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;
- [ >(eq_pair_fst_snd … grid)
- #Hta #Hsplit >(Htb … Hta)
- >(?:c0 = bar)
- [ @(ex_intro ?? (\fst grid)) @(ex_intro ?? (\snd grid))
- % [ % [ % [ (* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ]
- | (* Hc0 *) @daemon ]
- | #r5 #rs5 >(eq_pair_fst_snd … r5)
- #Hta #Hsplit >(Htb … Hta)
- >(?:c0 = bar)
- [ @(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.
\ No newline at end of file
+axiom append_eq_tech1 :
+ ∀A,l1,l2,l3,l4.l1@l2 = l3@l4 → |l1| < |l3| → ∃la:list A.l1@la = l3.
+axiom append_eq_tech2 :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false → ∃la:list A.l3 = l1@a::la.
+axiom list_decompose_cases :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → ∃la,lb:list A.l3 = la@a::lb ∨ l4 = la@a::lb.
+axiom list_decompose_l :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false →
+ ∃la,lb.l2 = la@lb ∧ l3 = l1@a::la.*)
+
+lemma list_decompose_r :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l3 = false →
+ ∃la,lb.l1 = la@lb ∧ l4 = lb@a::l2.
+#A #l1 #l2 #l3 generalize in match l1; generalize in match l2; elim l3
+ [normalize #l1 #l2 #l4 #a #H #_ @(ex_intro … []) @(ex_intro … l2) /2/
+ |#b #tl #Hind #l1 #l2 #l4 #a cases l2
+ [normalize #Heq destruct >(\b (refl … b)) normalize #Hfalse destruct
+ |#c #tl2 whd in ⊢ ((??%%)→?); #Heq destruct #Hmema
+ cases (Hind l1 tl2 l4 a ??)
+ [#l5 * #l6 * #eql #eql4
+ @(ex_intro … (b::l5)) @(ex_intro … l6) % /2/
+ |@e0
+ |cases (true_or_false (memb ? a tl)) [2://]
+ #H @False_ind @(absurd ?? not_eq_true_false)
+ <Hmema @sym_eq @memb_cons //
+ ]
+ ]
+ ]
+qed.
+
+(*axiom list_decompose_memb :
+ ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → memb A a l3 = true.*)
+
+lemma table_invert_r : ∀n,t,T.
+ tuple_TM n t → table_TM n (t@T) → table_TM n T.
+#n #t #T #Htuple #Htable inversion Htable
+[ cases Htuple #qin * #cin * #qout * #cout * #mv * #_ #Ht >Ht
+ normalize #Hfalse destruct (Hfalse)
+| #t0 #T0 #Htuple0 #Htable0 #_ #Heq
+ lapply (append_l2_injective ?????? Heq)
+ [ >(length_of_tuple … Htuple) >(length_of_tuple … Htuple0) % ]
+ -Heq #Heq destruct (Heq) // ]
+qed.
+
+lemma match_in_table_to_tuple :
+ ∀n,T,qin,cin,qout,cout,mv.
+ match_in_table n qin cin qout cout mv T → table_TM n T →
+ tuple_TM n (mk_tuple qin cin qout cout mv).
+#n #T #qin #cin #qout #cout #mv #Hmatch elim Hmatch
+[ //
+| #qin0 #cin0 #qout0 #cout0 #mv0 #tb #Htuple #Hmatch #IH #Htable
+ @IH @(table_invert_r ???? Htable) @Htuple
+]
+qed.
+
+lemma match_in_table_append :
+ ∀n,T,qin,cin,qout,cout,mv,t.
+ tuple_TM n t →
+ match_in_table n qin cin qout cout mv (t@T) →
+ t = mk_tuple qin cin qout cout mv ∨ match_in_table n qin cin qout cout mv T.
+#n #T #qin #cin #qout #cout #mv #t #Ht #Hmatch inversion Hmatch
+[ #T0 #H #H1 % >(append_l1_injective … H1) //
+ >(length_of_tuple … Ht) >(length_of_tuple … H) %
+| #qin0 #cin0 #qout0 #cout0 #mv0 #T0 #H #H1 #_ #H2 %2
+ >(append_l2_injective … H2) // >(length_of_tuple … Ht) >(length_of_tuple … H) %
+]
+qed.
+
+lemma generic_match_to_match_in_table_tech :
+ ∀n,t,T0,T1,T2.tuple_TM n t → table_TM n (T1@〈bar,false〉::T2) →
+ t@T0 = T1@〈bar,false〉::T2 → T1 = [] ∨ ∃T3.T1 = t@T3.
+#n #t #T0 #T1 #T2 #Ht cases T1
+[ #_ #_ % %
+| normalize #c #T1c #Htable #Heq %2
+ cases Ht in Heq; #qin * #cin * #qout * #cout * #mv **********
+ #Hqin1 #Hqout1 #Hqin2 #Hqout2 #Hcin #Hcout #Hmv #Hcoutmv #Hqinlen #Hqoutlen
+ #Heqt >Heqt whd in ⊢ (??%%→?); #Ht lapply (cons_injective_r ????? Ht)
+ #Ht' cases (list_decompose_r STape … (sym_eq … Ht') ?)
+ [ #la * #lb * #HT1c #HT0 %{lb} >HT1c @(eq_f2 ??? (append ?) (c::la)) //
+ >HT0 in Ht'; >HT1c >associative_append in ⊢ (???%→?); #Ht'
+ <(append_l1_injective_r … Ht') // <(cons_injective_l ????? Ht) %
+ |@(noteq_to_eqnot ? true) @(not_to_not … not_eq_true_false) #Hbar @sym_eq
+ cases (memb_append … Hbar) -Hbar #Hbar
+ [@(Hqin2 … Hbar)
+ |cases (orb_true_l … Hbar) -Hbar
+ [#Hbar lapply (\P Hbar) -Hbar #Hbar destruct (Hbar) @Hcin
+ |whd in ⊢ ((??%?)→?); #Hbar cases (memb_append … Hbar) -Hbar #Hbar
+ [@(Hqout2 … Hbar)
+ |cases (orb_true_l … Hbar) -Hbar
+ [#Hbar lapply (\P Hbar) -Hbar #Hbar destruct (Hbar) @Hcout
+ |#Hbar cases (orb_true_l … Hbar) -Hbar
+ [whd in ⊢ ((??%?)→?); #Hbar @Hbar
+ |#Hbar lapply (memb_single … Hbar) -Hbar #Hbar destruct (Hbar) @Hmv
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+lemma generic_match_to_match_in_table :
+ ∀n,T.table_TM n T →
+ ∀qin,cin,qout,cout,mv.|qin| = n → |qout| = n →
+ only_bits qin → only_bits qout →
+ bit_or_null (\fst cin) = true → bit_or_null (\fst cout) = true →
+ bit_or_null (\fst mv) = true →
+ ∀t1,t2.
+ T = (t1@〈bar,false〉::qin@cin::〈comma,false〉::qout@cout::〈comma,false〉::[mv])@t2 →
+ match_in_table n qin cin qout cout mv T.
+#n #T #Htable #qin #cin #qout #cout #mv #Hlenqin #Hlenqout
+#Hqinbits #Hqoutbits #Hcin #Hcout #Hmv
+elim Htable
+[ * [ #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+ | #c0 #t1 #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+| #tuple #T0 #H1 #Htable0#IH #t1 #t2 #HT cases H1 #qin0 * #cin0 * #qout0 * #cout0 * #mv0
+ * * * * * * * * * *
+ #Hqin0marks #Hqout0marks #Hqin0bits #Hqout0bits #Hcin0 #Hcout0 #Hmv0 #Hcout0mv0
+ #Hlenqin0 #Hlenqout0 #Htuple
+ lapply (generic_match_to_match_in_table_tech n ? T0 t1
+ (qin@cin::〈comma,false〉::qout@[cout;〈comma,false〉;mv]@t2) H1) #Htmp
+ >Htuple in H1; #H1
+ lapply (ttm_cons … T0 H1 Htable0) <Htuple in ⊢ (%→?); >HT
+ >associative_append normalize >associative_append normalize
+ >associative_append #Htable cases (Htmp Htable ?)
+ [ #Ht1 >Htuple in HT; >Ht1 normalize in ⊢ (??%%→?);
+ >associative_append >associative_append #HT
+ cut (qin0 = qin ∧ (〈cin0,false〉 = cin ∧ (qout0 = qout ∧
+ (〈cout0,false〉 = cout ∧ (〈mv0,false〉 = mv ∧ T0 = t2)))))
+ [ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (append_l1_injective … HT) [ >Hlenqin @Hlenqin0 ]
+ #Hqin % [ @Hqin ] -Hqin
+ lapply (append_l2_injective … HT) [ >Hlenqin @Hlenqin0 ] -HT #HT
+ lapply (cons_injective_l ????? HT) #Hcin % [ @Hcin ] -Hcin
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_r ????? HT) -HT
+ >associative_append >associative_append #HT
+ lapply (append_l1_injective … HT) [ >Hlenqout @Hlenqout0 ]
+ #Hqout % [ @Hqout ] -Hqout
+ lapply (append_l2_injective … HT) [ >Hlenqout @Hlenqout0 ] -HT normalize #HT
+ lapply (cons_injective_l ????? HT) #Hcout % [ @Hcout ] -Hcout
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_r ????? HT) -HT #HT
+ lapply (cons_injective_l ????? HT) #Hmv % [ @Hmv ] -Hmv
+ @(cons_injective_r ????? HT) ]
+ -HT * #Hqin * #Hcin * #Hqout * #Hcout * #Hmv #HT0
+ >(?:〈bar,false〉::qin0@(〈cin0,false〉::〈comma,false〉::qout0@
+ [〈cout0,false〉;〈comma,false〉;〈mv0,false〉])@T0 = tuple@T0)
+ [ >Htuple >Hqin >Hqout >Hcin >Hcout >Hmv % //
+ | >Htuple normalize >associative_append normalize >associative_append
+ normalize >associative_append % ]
+ | * #T3 #HT3 >HT3 in HT; >associative_append; >associative_append #HT
+ lapply (append_l2_injective … HT) // -HT #HT %2 //
+ @(IH T3 t2) >HT >associative_append %
+ |>HT >associative_append normalize >associative_append normalize
+ >associative_append % ]
+]
+qed.