include "turing/while_machine.ma".
include "turing/if_machine.ma".
+include "turing/universal/alphabet.ma".
include "turing/universal/tests.ma".
(* ADVANCE TO MARK (right)
| >reverse_append #Htc @Htc ]
]
qed.
-
-inductive unialpha : Type[0] ≝
-| bit : bool → unialpha
-| comma : unialpha
-| bar : unialpha
-| grid : unialpha.
-
-definition unialpha_eq ≝
- λa1,a2.match a1 with
- [ bit x ⇒ match a2 with [ bit y ⇒ ¬ xorb x y | _ ⇒ false ]
- | comma ⇒ match a2 with [ comma ⇒ true | _ ⇒ false ]
- | bar ⇒ match a2 with [ bar ⇒ true | _ ⇒ false ]
- | grid ⇒ match a2 with [ grid ⇒ true | _ ⇒ false ] ].
-
-definition DeqUnialpha ≝ mk_DeqSet unialpha unialpha_eq ?.
-* [ #x * [ #y cases x cases y normalize % // #Hfalse destruct
- | *: normalize % #Hfalse destruct ]
- |*: * [1,5,9,13: #y ] normalize % #H1 destruct % ]
-qed.
-
-definition FSUnialpha ≝
- mk_FinSet DeqUnialpha [bit true;bit false;comma;bar;grid] ?.
-@daemon
-qed.
(*
MATCH AND ADVANCE(f)
ifTM ? (test_char ? (is_marked ?))
(single_finalTM … (comp_step_subcase FSUnialpha 〈bit false,true〉
(comp_step_subcase FSUnialpha 〈bit true,true〉
- (clear_mark …))))
+ (comp_step_subcase FSUnialpha 〈null,true〉
+ (clear_mark …)))))
(nop ?)
tc_true.
-definition is_bit ≝ λc.match c with [ bit _ ⇒ true | _ ⇒ false ].
-
definition R_comp_step_true ≝
λt1,t2.
∀l0,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) l0 c rs →
∃c'. c = 〈c',true〉 ∧
- ((is_bit c' = true ∧
+ ((bit_or_null c' = true ∧
∀a,l1,c0,a0,l2.
rs = 〈a,false〉::l1@〈c0,true〉::〈a0,false〉::l2 →
(∀c.memb ? c l1 = true → is_marked ? c = false) →
(c0 ≠ c' ∧
t2 = midtape (FinProd … FSUnialpha FinBool)
(reverse ? l1@〈a,false〉::〈c',true〉::l0) 〈c0,false〉 (〈a0,false〉::l2))) ∨
- (is_bit c' = false ∧ t2 = midtape ? l0 〈c',false〉 rs)).
+ (bit_or_null c' = false ∧ t2 = midtape ? l0 〈c',false〉 rs)).
definition R_comp_step_false ≝
λt1,t2.
cases (acc_sem_if … (sem_test_char ? (is_marked ?))
(sem_comp_step_subcase FSUnialpha 〈bit false,true〉 ??
(sem_comp_step_subcase FSUnialpha 〈bit true,true〉 ??
- (sem_clear_mark …)))
+ (sem_comp_step_subcase FSUnialpha 〈null,true〉 ??
+ (sem_clear_mark …))))
(sem_nop …) intape)
#k * #outc * * #Hloop #H1 #H2
@(ex_intro ?? k) @(ex_intro ?? outc) %
[ @sym_not_eq //
| @Houtc ]
]
- | * #Hc' whd in ⊢ (%→?); #Helse2 %2 %
- [ generalize in match Hc'; generalize in match Hc;
- cases c'
- [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
- | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
- |*: #_ #_ % ]
- | @(Helse2 … Hta)
+ | * #Hc' #Helse2 cases (Helse2 … Hta)
+ [ * #Hc'' #H1 % % [destruct (Hc'') % ]
+ #a #l1 #c0 #a0 #l2 #Hrs >Hrs in Hintape; #Hintape #Hl1
+ cases (H1 … Hl1 Hrs)
+ [ * #Htmp >Htmp -Htmp #Houtc % % // @Houtc
+ | * #Hneq #Houtc %2 %
+ [ @sym_not_eq //
+ | @Houtc ]
+ ]
+ | * #Hc'' whd in ⊢ (%→?); #Helse3 %2 %
+ [ generalize in match Hc''; generalize in match Hc'; generalize in match Hc;
+ cases c'
+ [ * [ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
+ | #Hfalse @False_ind @(absurd ?? Hfalse) % ]
+ | #_ #_ #Hfalse @False_ind @(absurd ?? Hfalse) %
+ |*: #_ #_ #_ % ]
+ | @(Helse3 … Hta)
+ ]
]
]
]
definition R_compare :=
λt1,t2.
∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
- (∀c'.is_bit c' = false → c = 〈c',true〉 → t2 = midtape ? ls 〈c',false〉 rs) ∧
+ (∀c'.bit_or_null c' = false → c = 〈c',true〉 → t2 = midtape ? ls 〈c',false〉 rs) ∧
(∀c'. c = 〈c',false〉 → t2 = t1) ∧
∀b,b0,bs,b0s,l1,l2.
|bs| = |b0s| →
- (∀c.memb (FinProd … FSUnialpha FinBool) c bs = true → is_bit (\fst c) = true) →
- (∀c.memb (FinProd … FSUnialpha FinBool) c b0s = true → is_bit (\fst c) = true) →
+ (∀c.memb (FinProd … FSUnialpha FinBool) c bs = true → bit_or_null (\fst c) = true) →
+ (∀c.memb (FinProd … FSUnialpha FinBool) c b0s = true → bit_or_null (\fst c) = true) →
(∀c.memb ? c bs = true → is_marked ? c = false) →
(∀c.memb ? c b0s = true → is_marked ? c = false) →
(∀c.memb ? c l1 = true → is_marked ? c = false) →
- c = 〈b,true〉 → is_bit b = true →
+ c = 〈b,true〉 → bit_or_null b = true →
rs = bs@〈grid,false〉::l1@〈b0,true〉::b0s@〈comma,false〉::l2 →
(〈b,true〉::bs = 〈b0,true〉::b0s ∧
t2 = midtape ? (reverse ? bs@〈b,false〉::ls)
]
| #tapea #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
whd in Hleft; #ls #c #rs #Htapea cases (Hleft … Htapea) -Hleft
- #c' * #Hc >Hc cases (true_or_false (is_bit c')) #Hc'
+ #c' * #Hc >Hc cases (true_or_false (bit_or_null c')) #Hc'
[2: *
[ * >Hc' #H @False_ind destruct (H)
| * #_ #Htapeb cases (IH … Htapeb) * #_ #H #_ %
| @Hl1 ]
| * #b' #bitb' * #b0' #bitb0' #bs' #b0s' #Hbs #Hb0s
generalize in match Hrs; >Hbs in ⊢ (%→?); >Hb0s in ⊢ (%→?);
- cut (is_bit b' = true ∧ is_bit b0' = true ∧
+ cut (bit_or_null b' = true ∧ bit_or_null b0' = true ∧
bitb' = false ∧ bitb0' = false)
[ % [ % [ % [ >Hbs in Hbs1; #Hbs1 @(Hbs1 〈b',bitb'〉) @memb_hd
| >Hb0s in Hb0s1; #Hb0s1 @(Hb0s1 〈b0',bitb0'〉) @memb_hd ]
]]]]]
qed.
-
-(*
-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)
- *)
-
-axiom myalpha : FinSet.
-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 sem_compare : Realize ? compare R_compare.
\ No newline at end of file
projects / helm.git / blobdiff