(hd ? (ls1@b::ls2) (all_blanks …)) (tail ? (ls1@b::ls2)) rs j) ∧
(not_blank sig n i b = false) ∧
(hd (multi_sig sig n) (ls1@[b]) (all_blanks …) = a) ∧ (* not implied by the next fact *)
- (∀j.j ≤n → to_blank_i ?? j (ls1@b::ls2) = to_blank_i ?? j (a::ls)) ∧
+ (∀j.j < n → to_blank_i ?? j (ls1@b::ls2) = to_blank_i ?? j (a::ls)) ∧
t2 = midtape ? ls2 b ((reverse ? ls1)@rs)).
theorem sem_move_to_blank_L: ∀sig,n,i.
[ inl _ ⇒ false
| inr _ ⇒ true]].
-definition not_head ≝ λA,sig,n.λc:multi_sig (TA A sig) n.
+definition not_head ≝ λA,sig,n.λc:multi_sig (TA A sig) (S n).
¬(is_head A sig (nth n ? (vec … c) (blank ?))).
+lemma not_head_all_blanks : ∀A,sig,n.
+ not_head A sig n (all_blanks … (S n)) = true.
+#A #sig #n whd in ⊢ (??%?); >blank_all_blanks //
+qed.
+
definition no_head_in ≝ λA,sig,n,l.
- ∀x. mem ? x (trace (TA A sig) n n l) → is_head … x = false.
+ ∀x. mem ? x (trace (TA A sig) (S n) n l) → is_head … x = false.
(*
lemma not_head_true: ∀A,sig,n,c. not_head A sig n c = true →
is_head … (nth n ? (vec … c) (blank ?)) = false.
*)
+lemma hd_nil : ∀A,d. hd A [ ] d = d.
+// qed.
+
definition mtiL ≝ λA,sig,n,i.
- move_to_blank_L (TA A sig) n i ·
- shift_i_L (TA A sig) n i ·
+ move_to_blank_L (TA A sig) (S n) i ·
+ shift_i_L (TA A sig) (S n) i ·
move_until ? L (not_head A sig n).
definition Rmtil ≝ λA,sig,n,i,t1,t2.
∀ls,a,rs.
- t1 = midtape (MTA A sig n) ls a rs →
+ t1 = midtape (MTA A sig (S n)) ls a rs →
is_head A sig (nth n ? (vec … a) (blank ?)) = true →
- (∀i.regular_trace (TA A sig) n a ls rs i) →
+ (∀i.regular_trace (TA A sig) (S n) a ls rs i) →
(* next: we cannot be on rightof on trace i *)
(nth i ? (vec … a) (blank ?) = (blank ?)
→ nth i ? (vec … (hd ? rs (all_blanks …))) (blank ?) ≠ (blank ?)) →
no_head_in … ls →
no_head_in … rs →
(∃ls1,a1,rs1.
- t2 = midtape (MTA A sig n) ls1 a1 rs1 ∧
+ t2 = midtape (MTA A sig (S n)) ls1 a1 rs1 ∧
(∀i.regular_trace … a1 ls1 rs1 i) ∧
- (∀j. j ≤ n → j ≠ i → to_blank_i ? n j (a1::ls1) = to_blank_i ? n j (a::ls)) ∧
- (∀j. j ≤ n → j ≠ i → to_blank_i ? n j rs1 = to_blank_i ? n j rs) ∧
- (to_blank_i ? n i ls1 = to_blank_i ? n i (a::ls)) ∧
- (to_blank_i ? n i (a1::rs1)) = to_blank_i ? n i rs).
+ (∀j. j ≤ n → j ≠ i → to_blank_i ? (S n) j (a1::ls1) = to_blank_i ? (S n) j (a::ls)) ∧
+ (∀j. j ≤ n → j ≠ i → to_blank_i ? (S n) j rs1 = to_blank_i ? (S n) j rs) ∧
+ (to_blank_i ? (S n) i ls1 = to_blank_i ? (S n) i (a::ls)) ∧
+ (to_blank_i ? (S n) i (a1::rs1)) = to_blank_i ? (S n) i rs).
theorem sem_Rmtil: ∀A,sig,n,i. i < n → mtiL A sig n i ⊨ Rmtil A sig n i.
#A #sig #n #i #lt_in
(* we start looking into Rmitl *)
#ls #a #rs #Htin (* tin is a midtape *)
#Hheada #Hreg #no_rightof #Hnohead_ls #Hnohead_rs
-cut (regular_i ? n (a::ls) i)
+cut (regular_i ? (S n) (a::ls) i)
[cases (Hreg i) * //
cases (true_or_false (nth i ? (vec … a) (blank ?) == (blank ?))) #Htest
[#_ @daemon (* absurd, since hd rs non e' blank *)
* #H3 @False_ind @(absurd (true=false)) [2://] <H3 @sym_eq
<(notb_notb true) @(eq_f … notb) @Hnohead_rs >H2 >trace_append @mem_append_l2
lapply Hb0 cases rs2
- [whd in match (hd ???); #H >H in H3; whd in match (not_head ????);
- >all_blank_n normalize -H #H destruct (H); @False_ind
+ [>hd_nil #H >H in H3; >not_head_all_blanks #Habs destruct (Habs)
|#c #r #H4 %1 >H4 //
]
|*
(* cut (trace sig n j (a1::ls20)=trace sig n j (ls1@b::ls2)) *)
* #ls10 * #a1 * #ls20 * * * #Hls20 #Ha1 #Hnh #Htout
cut (∀j.j ≠ i →
- trace ? n j (reverse (multi_sig (TA A sig) n) rs1@b::ls2) =
- trace ? n j (ls10@a1::ls20))
+ trace ? (S n) j (reverse (multi_sig (TA A sig) (S n)) rs1@b::ls2) =
+ trace ? (S n) j (ls10@a1::ls20))
[#j #ineqj >append_cons <reverse_cons >trace_def <map_append <reverse_map
- lapply (trace_shift_neq …lt_in ? (sym_not_eq … ineqj) … Hrss) [//] #Htr
+ lapply (trace_shift_neq … (le_S … lt_in) ? (sym_not_eq … ineqj) … Hrss) [//] #Htr
<(trace_def … (b::rs1)) <Htr >reverse_map >map_append @eq_f @Hls20 ]
#Htracej
- cut (trace ? n i (reverse (multi_sig (TA A sig) n) (rs1@[b0])@ls2) =
- trace ? n i (ls10@a1::ls20))
+ cut (trace ? (S n) i (reverse (multi_sig (TA A sig) (S n)) (rs1@[b0])@ls2) =
+ trace ? (S n) i (ls10@a1::ls20))
[>trace_def <map_append <reverse_map <map_append <(trace_def … [b0])
- cut (trace ? n i [b0] = [blank ?]) [@daemon] #Hcut >Hcut
- lapply (trace_shift … lt_in … Hrss) [//] whd in match (tail ??); #Htr <Htr
+ cut (trace ? (S n) i [b0] = [blank ?]) [@daemon] #Hcut >Hcut
+ lapply (trace_shift … (le_S … lt_in) … Hrss) [//] whd in match (tail ??); #Htr <Htr
>reverse_map >map_append <trace_def <Hls20 %
]
#Htracei
cut (∀j. j ≠ i →
- (trace ? n j (reverse (MTA A sig n) rs11) = trace ? n j ls10) ∧
- (trace ? n j (ls1@b::ls2) = trace ? n j (a1::ls20)))
+ (trace ? (S n) j (reverse (MTA A sig (S n)) rs11) = trace ? (S n) j ls10) ∧
+ (trace ? (S n) j (ls1@b::ls2) = trace ? (S n) j (a1::ls20)))
[@daemon (* si fa
#j #ineqj @(first_P_to_eq ? (λx. x ≠ head ?))
[lapply (Htracej … ineqj) >trace_def in ⊢ (%→?); <map_append
>trace_def in ⊢ (%→?); <map_append #H @H
| *) ] #H2
- cut ((trace ? n i (b0::reverse ? rs11) = trace ? n i (ls10@[a1])) ∧
- (trace ? n i (ls1@ls2) = trace ? n i ls20))
+ cut ((trace ? (S n) i (b0::reverse ? rs11) = trace ? (S n) i (ls10@[a1])) ∧
+ (trace ? (S n) i (ls1@ls2) = trace ? (S n) i ls20))
[>H1 in Htracei; >reverse_append >reverse_single >reverse_append
>reverse_reverse >associative_append >associative_append
@daemon
] #H3
cut (∀j. j ≠ i →
- trace ? n j (reverse (MTA A sig n) ls10@rs2) = trace ? n j rs)
- [#j #jneqi @(injective_append_l … (trace ? n j (reverse ? ls1)))
+ trace ? (S n) j (reverse ? ls10@rs2) = trace ? (S n) j rs)
+ [#j #jneqi @(injective_append_l … (trace ? (S n) j (reverse ? ls1)))
>map_append >map_append >Hrs1 >H1 >associative_append
<map_append <map_append in ⊢ (???%); @eq_f
<map_append <map_append @eq_f2 // @sym_eq
<(reverse_reverse … rs11) <reverse_map <reverse_map in ⊢ (???%);
@eq_f @(proj1 … (H2 j jneqi))] #Hrs_j
- %{ls20} %{a1} %{(reverse ? (b0::ls10)@tail (MTA A sig n) rs2)}
+ %{ls20} %{a1} %{(reverse ? (b0::ls10)@tail ? rs2)}
%[%[%[%[%[@Htout
|#j cases (decidable_eq_nat j i)
[#eqji >eqji (* by cases wether a1 is blank *)
|@sym_eq @Hrs_j //
]
]]
- |#j #lejn #jneqi <(Hls1 … lejn)
+ |#j #lejn #jneqi <(Hls1 … (le_S_S … lejn))
>to_blank_i_def >to_blank_i_def @eq_f @sym_eq @(proj2 … (H2 j jneqi))]
|#j #lejn #jneqi >reverse_cons >associative_append >Hb0
<to_blank_hd_cons >to_blank_i_def >to_blank_i_def @eq_f @Hrs_j //]
- |<(Hls1 i) [2:@lt_to_le //]
+ |<(Hls1 i) [2:@le_S //]
lapply (all_blank_after_blank … reg_ls1_i)
[@(\P ?) @daemon] #allb_ls2
whd in match (to_blank_i ????); <(proj2 … H3)
@daemon ]
|>reverse_cons >associative_append
- cut (to_blank_i ? n i rs = to_blank_i ? n i (rs11@[b0])) [@daemon]
+ cut (to_blank_i ? (S n) i rs = to_blank_i ? (S n) i (rs11@[b0])) [@daemon]
#Hcut >Hcut >(to_blank_i_chop … b0 (a1::reverse …ls10)) [2: @Hb0blank]
>to_blank_i_def >to_blank_i_def @eq_f
>trace_def >trace_def @injective_reverse >reverse_map >reverse_cons
]
|(*we do not find the head: this is absurd *)
* #b1 * #lss * * #H2 @False_ind
- cut (∀x0. mem ? x0 (trace ? n n (b0::reverse ? rss@ls2)) → is_head … x0 = false)
+ cut (∀x0. mem ? x0 (trace ? (S n) n (b0::reverse ? rss@ls2)) → is_head … x0 = false)
[@daemon] -H2 #H2
- lapply (trace_shift_neq ? n i n … lt_in … Hrss)
+ lapply (trace_shift_neq ? (S n) i n … (le_S … lt_in) … Hrss)
[@lt_to_not_eq @lt_in | // ]
#H3 @(absurd
- (is_head … (nth n ? (vec … (hd ? (ls1@[b]) (all_blanks … n))) (blank ?)) = true))
+ (is_head … (nth n ? (vec … (hd ? (ls1@[b]) (all_blanks … (S n)))) (blank ?)) = true))
[>Hhead //
|@eqnot_to_noteq @H2 >trace_def %2 <map_append @mem_append_l1 <reverse_map <trace_def
>H3 >H1 >trace_def >reverse_map >reverse_cons >reverse_append
projects / helm.git / blobdiff