--- /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_____________________________________________________________*)
+
+include "turing/universal/alphabet.ma".
+include "turing/mono.ma".
+
+(************************* turning DeqMove into a DeqSet **********************)
+definition move_eq ≝ λm1,m2:move.
+ match m1 with
+ [R ⇒ match m2 with [R ⇒ true | _ ⇒ false]
+ |L ⇒ match m2 with [L ⇒ true | _ ⇒ false]
+ |N ⇒ match m2 with [N ⇒ true | _ ⇒ false]].
+
+lemma move_eq_true:∀m1,m2.
+ move_eq m1 m2 = true ↔ m1 = m2.
+*
+ [* normalize [% #_ % |2,3: % #H destruct ]
+ |* normalize [1,3: % #H destruct |% #_ % ]
+ |* normalize [1,2: % #H destruct |% #_ % ]
+qed.
+
+definition DeqMove ≝ mk_DeqSet move move_eq move_eq_true.
+
+unification hint 0 ≔ ;
+ X ≟ DeqMove
+(* ---------------------------------------- *) ⊢
+ move ≡ carr X.
+
+unification hint 0 ≔ m1,m2;
+ X ≟ DeqMove
+(* ---------------------------------------- *) ⊢
+ move_eq m1 m2 ≡ eqb X m1 m2.
+
+
+(************************ turning DeqMove into a FinSet ***********************)
+definition move_enum ≝ [L;R;N].
+
+lemma move_enum_unique: uniqueb ? [L;R;N] = true.
+// qed.
+
+lemma move_enum_complete: ∀x:move. memb ? x [L;R;N] = true.
+* // qed.
+
+definition FinMove ≝
+ mk_FinSet DeqMove [L;R;N] move_enum_unique move_enum_complete.
+
+unification hint 0 ≔ ;
+ X ≟ FinMove
+(* ---------------------------------------- *) ⊢
+ move ≡ FinSetcarr X.
+
+(*************************** normal Turing Machines ***************************)
+
+(* A normal turing machine is just an ordinary machine where:
+ 1. the tape alphabet is bool
+ 2. the finite state are supposed to be an initial segment of the natural
+ numbers.
+ Formally, it is specified by a record with the number n of states, a proof
+ that n is positive, the transition function and the halting function.
+*)
+
+definition trans_source ≝ λn.FinProd (initN n) (FinOption FinBool).
+definition trans_target ≝ λn.FinProd (initN n) (FinOption (FinProd FinBool FinMove)).
+
+record normalTM : Type[0] ≝
+{ no_states : nat;
+ pos_no_states : (0 < no_states);
+ ntrans : trans_source no_states → trans_target no_states;
+ nhalt : initN no_states → bool
+}.
+
+(* A normal machine is just a special case of Turing Machine. *)
+
+definition normalTM_to_TM ≝ λM:normalTM.
+ mk_TM FinBool (initN (no_states M))
+ (ntrans M) (mk_Sig ?? 0 (pos_no_states M)) (nhalt M).
+
+coercion normalTM_to_TM.
+
+definition nconfig ≝ λn. config FinBool (initN n).
+
+(******************************** tuples **************************************)
+
+(* By general results on FinSets we know that there is every function f between
+two finite sets A and B can be described by means of a finite graph of pairs
+〈a,f a〉. Hence, the transition function of a normal turing machine can be
+described by a finite set of tuples 〈i,c〉,〈j,action〉〉 of the following type:
+ (Nat_to n × (option FinBool)) × (Nat_to n × (option (FinBool × move))).
+Unfortunately this description is not suitable for an Universal Machine, since
+such a machine must work with a fixed alphabet, while the size on n is unknown.
+Hence, we must pass from natural numbers to a representation for them on a
+finitary, e.g. binary, alphabet. In general, we shall associate
+to a pair 〈〈i,c〉,〈j,action〉〉 a tuples with the following syntactical structure
+ |w_ix,w_jy,z
+where
+1. "|" and "," are special characters used as delimiters;
+2. w_i and w_j are list of booleans representing the states $i$ and $j$;
+3. x is special symbol null if C=None and is a if c=Some a
+4. y and z are both null if action = None, and are equal to b,m' if
+ action = Some b,m;
+5. finally, m' = 0 if m = L, m' = 1 if m=R and m' = null if m = N
+
+As a minor, additional complication, we shall suppose that every characters is
+decorated by an additonal bit, normally set to false, to be used as a marker.
+*)
+
+definition mk_tuple ≝ λqin,cin,qout,cout,mv.
+ 〈bar,false〉::qin@cin::〈comma,false〉::qout@cout::〈comma,false〉::[mv].
+
+(* by definition, a tuple is not marked *)
+definition tuple_TM : nat → list STape → Prop ≝
+ λn,t.∃qin,cin,qout,cout,mv.
+ no_marks qin ∧ no_marks qout ∧
+ only_bits qin ∧ only_bits qout ∧
+ bit_or_null cin = true ∧ bit_or_null cout = true ∧ bit_or_null mv = true ∧
+ (cout = null → mv = null) ∧
+ |qin| = n ∧ |qout| = n ∧
+ t = mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉.
+
+(***************************** state encoding *********************************)
+(* p < n is represented with a list of bits of lenght n with the p-th bit from
+left set to 1. An additional intial bit is set to 1 if the state is final and
+to 0 otherwise. *)
+
+let rec to_bitlist n p: list bool ≝
+ match n with [ O ⇒ [ ] | S q ⇒ (eqb p q)::to_bitlist q p].
+
+let rec from_bitlist l ≝
+ match l with
+ [ nil ⇒ 0 (* assert false *)
+ | cons b tl ⇒ if b then |tl| else from_bitlist tl].
+
+lemma bitlist_length: ∀n,p.|to_bitlist n p| = n.
+#n elim n normalize //
+qed.
+
+lemma bitlist_inv1: ∀n,p.p<n → from_bitlist (to_bitlist n p) = p.
+#n elim n normalize -n
+ [#p #abs @False_ind /2/
+ |#n #Hind #p #lepn
+ cases (le_to_or_lt_eq … (le_S_S_to_le … lepn))
+ [#ltpn lapply (lt_to_not_eq … ltpn) #Hpn
+ >(not_eq_to_eqb_false … Hpn) normalize @Hind @ltpn
+ |#Heq >(eq_to_eqb_true … Heq) normalize <Heq //
+ ]
+ ]
+qed.
+
+lemma bitlist_lt: ∀l. 0 < |l| → from_bitlist l < |l|.
+#l elim l normalize // #b #tl #Hind cases b normalize //
+#Htl cases (le_to_or_lt_eq … (le_S_S_to_le … Htl)) -Htl #Htl
+ [@le_S_S @lt_to_le @Hind //
+ |cut (tl=[ ]) [/2 by append_l2_injective/] #eqtl >eqtl @le_n
+ ]
+qed.
+
+definition nat_of: ∀n. Nat_to n → nat.
+#n normalize * #p #_ @p
+qed.
+
+definition bits_of_state ≝ λn.λh:Nat_to n → bool.λs:Nat_to n.
+ h s::(to_bitlist n (nat_of n s)).
+
+definition m_bits_of_state ≝ λn.λh.λp.
+ map ? (unialpha×bool) (λx.〈bit x,false〉) (bits_of_state n h p).
+
+lemma no_marks_bits_of_state : ∀n,h,p. no_marks (m_bits_of_state n h p).
+#n #h #p #x whd in match (m_bits_of_state n h p);
+#H cases (orb_true_l … H) -H
+ [#H >(\P H) %
+ |elim (to_bitlist n (nat_of n p))
+ [whd in ⊢ ((??%?)→?); #H destruct
+ |#b #l #Hind #H cases (orb_true_l … H) -H #H
+ [>(\P H) % |@Hind @H]
+ ]
+ ]
+qed.
+
+lemma only_bits_bits_of_state : ∀n,h,p. only_bits (m_bits_of_state n h p).
+#n #h #p #x whd in match (m_bits_of_state n h p);
+#H cases (orb_true_l … H) -H
+ [#H >(\P H) %
+ |elim (to_bitlist n (nat_of n p))
+ [whd in ⊢ ((??%?)→?); #H destruct
+ |#b #l #Hind #H cases (orb_true_l … H) -H #H
+ [>(\P H) % |@Hind @H ]
+ ]
+ ]
+qed.
+
+(******************************** action encoding *****************************)
+definition low_action ≝ λaction.
+ match action with
+ [ None ⇒ 〈null,null〉
+ | Some act ⇒ let 〈na,m〉 ≝ act in
+ match m with
+ [ R ⇒ 〈bit na,bit true〉
+ | L ⇒ 〈bit na,bit false〉
+ | N ⇒ 〈bit na,null〉]
+ ].
+
+(******************************** tuple encoding ******************************)
+definition tuple_type ≝ λn.
+ (Nat_to n × (option FinBool)) × (Nat_to n × (option (FinBool × move))).
+
+definition tuple_encoding ≝ λn.λh:Nat_to n→bool.
+ λp:tuple_type n.
+ let 〈inp,outp〉 ≝ p in
+ let 〈q,a〉 ≝ inp in
+ let cin ≝ match a with [ None ⇒ null | Some b ⇒ bit b ] in
+ let 〈qn,action〉 ≝ outp in
+ let 〈cout,mv〉 ≝ low_action action in
+ let qin ≝ m_bits_of_state n h q in
+ let qout ≝ m_bits_of_state n h qn in
+ mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉.
+
+(*
+definition WFTuple_conditions ≝
+ λn,qin,cin,qout,cout,mv.
+ no_marks qin ∧ no_marks qout ∧ (* queste fuori ? *)
+ only_bits qin ∧ only_bits qout ∧
+ bit_or_null cin = true ∧ bit_or_null cout = true ∧ bit_or_null mv = true ∧
+ (cout = null → mv = null) ∧
+ |qin| = n ∧ |qout| = n. *)
+
+lemma is_tuple: ∀n,h,p. tuple_TM (S n) (tuple_encoding n h p).
+#n #h * * #q #a * #qn #action
+@(ex_intro … (m_bits_of_state n h q))
+letin cin ≝ match a with [ None ⇒ null | Some b ⇒ bit b ]
+@(ex_intro … cin)
+@(ex_intro … (m_bits_of_state n h qn))
+letin cout ≝
+ match action with
+ [ None ⇒ null | Some act ⇒ bit (\fst act)]
+@(ex_intro … cout)
+letin mv ≝ match action with
+ [ None ⇒ null
+ | Some act ⇒
+ match \snd act with
+ [ R ⇒ bit true | L ⇒ bit false | N ⇒ null]
+ ]
+@(ex_intro … mv)
+%[%[%[%[%[%[%[% /3/
+ |whd in match cin ; cases a //]
+ |whd in match cout; cases action //]
+ |whd in match mv; cases action // * #b #m cases m //]
+ |whd in match cout; whd in match mv; cases action
+ [// | #act whd in ⊢ ((??%?)→?); #Hfalse destruct ]]
+ |>length_map normalize @eq_f //]
+ |>length_map normalize @eq_f //]
+ |normalize cases a cases action normalize //
+ [* #c #m cases m % |* #c #m #c1 cases m %]
+ ]
+qed.
+
+definition tuple_length ≝ λn.2*n+6.
+
+lemma length_of_tuple: ∀n,t. tuple_TM n t →
+ |t| = tuple_length n.
+#n #t * #qin * #cin * #qout * #cout * #mv *** #_ #Hqin #Hqout #eqt >eqt
+whd in match (mk_tuple ?????); normalize >length_append >Hqin -Hqin normalize
+>length_append normalize >Hqout -Hqout //
+qed.
+
+definition tuples_list ≝ λn.λh.map … (λp.tuple_encoding n h p).
+
+(******************* general properties of encoding of tuples *****************)
+
+lemma no_grids_in_tuple : ∀n,l.tuple_TM n l → no_grids l.
+#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
+#_ #_ #Hqin #Hqout #Hcin #Hcout #Hmv #_ #_ #_ #Hl >Hl
+#c #Hc cases (orb_true_l … Hc) -Hc #Hc
+[ >(\P Hc) %
+| cases (memb_append … Hc) -Hc #Hc
+[ @bit_not_grid @(Hqin … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| cases (memb_append …Hc) -Hc #Hc
+[ @bit_not_grid @(Hqout … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| >(memb_single … Hc) @bit_or_null_not_grid @Hmv
+]]]]]]
+qed.
+
+lemma no_marks_in_tuple : ∀n,l.tuple_TM n l → no_marks l.
+#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
+#Hqin #Hqout #_ #_ #_ #_ #_ #_ #_ #_ #Hl >Hl
+#c #Hc cases (orb_true_l … Hc) -Hc #Hc
+[ >(\P Hc) %
+| cases (memb_append … Hc) -Hc #Hc
+[ @(Hqin … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) %
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| cases (memb_append … Hc) -Hc #Hc
+[ @(Hqout … Hc)
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) %
+| cases (orb_true_l … Hc) -Hc #Hc
+[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
+| >(memb_single … Hc) %
+]]]]]]
+qed.
+
+
V_____________________________________________________________*)
-(* COMPARE BIT
+(****************************** table of tuples *******************************)
+include "turing/universal/normalTM.ma".
-*)
-
-include "turing/universal/marks.ma".
-
-definition STape ≝ FinProd … FSUnialpha FinBool.
-
-definition only_bits ≝ λl.
- ∀c.memb STape c l
- = true → is_bit (\fst c) = true.
-
-definition only_bits_or_nulls ≝ λl.
- ∀c.memb STape c l = true → bit_or_null (\fst c) = true.
-
-definition no_grids ≝ λl.
- ∀c.memb STape c l = true → is_grid (\fst c) = false.
-
-definition no_bars ≝ λl.
- ∀c.memb STape c l = true → is_bar (\fst c) = false.
-
-definition no_marks ≝ λl.
- ∀c.memb STape c l = true → is_marked ? c = false.
-
-lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false.
-* // normalize #H destruct
-qed.
-
-lemma bit_or_null_not_grid: ∀d. bit_or_null d = true → is_grid d = false.
-* // normalize #H destruct
-qed.
+(* a well formed table is a list of tuples *)
+
+inductive table_TM (n:nat) : list STape → Prop ≝
+| ttm_nil : table_TM n []
+| ttm_cons : ∀t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@T).
-lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false.
-* // normalize #H destruct
+lemma wftable: ∀n,h,l.table_TM (S n) (flatten ? (tuples_list n h l)).
+#n #h #l elim l // -l #a #tl #Hind
+whd in match (flatten … (tuples_list …));
+@ttm_cons //
qed.
-lemma bit_or_null_not_bar: ∀d. bit_or_null d = true → is_bar d = false.
-* // normalize #H destruct
+(*********************** general properties of tables *************************)
+lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
+#n #l #t elim t
+ [normalize #c #H destruct
+ |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_grids_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
+ cases (memb_append … Hx) -Hx #Hx
+ [ @(Ht1 … Hx)
+ | @(IH … Hx) ] ]
qed.
-definition mk_tuple ≝ λqin,cin,qout,cout,mv.
- 〈bar,false〉 :: qin @ cin :: 〈comma,false〉:: qout @ cout :: 〈comma,false〉 :: [mv].
+lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
+#n #l #t elim t
+ [normalize #c #H destruct
+ |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_marks_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
+ cases (memb_append … Hx) -Hx #Hx
+ [ @(Ht1 … Hx)
+ | @(IH … Hx) ] ]
+qed.
-(* by definition, a tuple is not marked *)
-definition tuple_TM : nat → list STape → Prop ≝
- λn,t.∃qin,cin,qout,cout,mv.
- no_marks qin ∧ no_marks qout ∧
- only_bits qin ∧ only_bits qout ∧
- bit_or_null cin = true ∧ bit_or_null cout = true ∧ bit_or_null mv = true ∧
- (cout = null → mv = null) ∧
- |qin| = n ∧ |qout| = n ∧
- t = mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉.
-
-inductive table_TM (n:nat) : list STape → Prop ≝
-| ttm_nil : table_TM n []
-| ttm_cons : ∀t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@T).
+axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
+(************************** matching in a table *******************************)
inductive match_in_table (n:nat) (qin:list STape) (cin: STape)
(qout:list STape) (cout:STape) (mv:STape)
: list STape → Prop ≝
match_in_table n qin cin qout cout mv tb →
match_in_table n qin cin qout cout mv
(mk_tuple qin0 cin0 qout0 cout0 mv0@tb).
-
-axiom tuple_len : ∀n,t.tuple_TM n t → |t| = 2*n+6.
+
+lemma tuple_to_match: ∀n,h,l,qin,cin,qout,cout,mv,p.
+ p = mk_tuple qin cin qout cout mv
+ → mem ? p (tuples_list n h l) →
+ match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h l)).
+#n #h #l #qin #cin #qout #cout #mv #p
+#Hp elim l
+ [whd in ⊢ (%→?); @False_ind
+ |#p1 #tl #Hind *
+ [#H whd in match (tuples_list ???);
+ <H >Hp @mit_hd //
+ |#H whd in match (tuples_list ???);
+ cases (is_tuple n h p1) #qin1 * #cin1 * #qout1 * #cout1 * #mv1
+ * #_ #Htuplep1 >Htuplep1 @mit_tl // @Hind //
+ ]
+ ]
+qed.
+
+axiom match_decomp: ∀n,l,qin,cin,qout,cout,mv.
+ match_in_table (S n) qin cin qout cout mv l →
+ ∃l1,l2. l = l1@(mk_tuple qin cin qout cout mv)@l2 ∧
+ (∃q.|l1| = (tuple_length (S n))*q) ∧
+ tuple_TM (S n) (mk_tuple qin cin qout cout mv).
+
+axiom daemon: ∀P:Prop. P.
+
+lemma match_to_tuples_list: ∀n,h,l,qin,cin,qout,cout,mv.
+ match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h l)) →
+ ∃p. p = mk_tuple qin cin qout cout mv ∧ mem ? p (tuples_list n h l).
+#n #h #l #qin #cin #qout #cout #mv #Hmatch
+@(ex_intro … (mk_tuple qin cin qout cout mv)) % //
+cases (match_decomp … Hmatch) #l1 * #l2 * * #Hflat #Hlen #Htuple
+@(flatten_to_mem … Hflat … Hlen)
+ [//
+ |@daemon
+ |@(length_of_tuple … Htuple)
+ ]
+qed.
+
+lemma match_to_tuple: ∀n,h,l,qin,cin,qout,cout,mv.
+ match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h l)) →
+ ∃p. tuple_encoding n h p = mk_tuple qin cin qout cout mv ∧ mem ? p l.
+#n #h #l #qin #cin #qout #cout #mv #Hmatch
+cases (match_to_tuples_list … Hmatch)
+#p * #eqp #memb
+cases(mem_map … (λp.tuple_encoding n h p) … memb)
+#p1 * #Hmem #H @(ex_intro … p1) % /2/
+qed.
+
+lemma match_to_trans:
+ ∀n.∀trans: trans_source n → trans_target n.
+ ∀h,qin,cin,qout,cout,mv.
+ match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h (graph_enum ?? trans))) →
+ ∃s,t. tuple_encoding n h 〈s,t〉 = mk_tuple qin cin qout cout mv
+ ∧ trans s = t.
+#n #trans #h #qin #cin #qout #cout #mv #Hmatch
+cases (match_to_tuple … Hmatch) -Hmatch * #s #t * #Heq #Hmem
+@(ex_intro … s) @(ex_intro … t) % // @graph_enum_correct
+@mem_to_memb @Hmem
+qed.
+
+lemma trans_to_match:
+ ∀n.∀h.∀trans: trans_source n → trans_target n.
+ ∀inp,outp,qin,cin,qout,cout,mv. trans inp = outp →
+ tuple_encoding n h 〈inp,outp〉 = mk_tuple qin cin qout cout mv →
+ match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h (graph_enum ?? trans))).
+#n #h #trans #inp #outp #qin #cin #qout #cout #mv #Htrans #Htuple
+@(tuple_to_match … (refl…)) <Htuple @mem_map_forward
+@(memb_to_mem (FinProd (trans_source n) (trans_target n)))
+@graph_enum_complete //
+qed.
+
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 :
normalize #Hfalse destruct (Hfalse)
| #t0 #T0 #Htuple0 #Htable0 #_ #Heq
lapply (append_l2_injective ?????? Heq)
- [ >(tuple_len … Htuple) >(tuple_len … Htuple0) % ]
+ [ >(length_of_tuple … Htuple) >(length_of_tuple … Htuple0) % ]
-Heq #Heq destruct (Heq) // ]
qed.
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) //
- >(tuple_len … Ht) >(tuple_len … H) %
+ >(length_of_tuple … Ht) >(length_of_tuple … H) %
| #qin0 #cin0 #qout0 #cout0 #mv0 #T0 #H #H1 #_ #H2 %2
- >(append_l2_injective … H2) // >(tuple_len … Ht) >(tuple_len … H) %
+ >(append_l2_injective … H2) // >(length_of_tuple … Ht) >(length_of_tuple … H) %
]
qed.
]
qed.
-(*
-lemma table_invert_l : ∀n,T0,qin,cin,qout,cout,mv.
- table_TM n (mk_tuple qin cin qout cout mv@〈bar,false〉::T0) →
- tuple_TM n (mk_tuple qin cin qout cout mv).
-#n #T #qin #cin #qout #cout #mv #HT inversion HT
-[ change with (append ???) in ⊢ (??(??%?)?→?);cases qin [ #Hfalse | #t0 #ts0 #Hfalse] normalize in Hfalse; destruct (Hfalse)
-| #t0 #T0 #Ht0 #HT0 #_
-
-
-lemma table_invert_r : ∀n,T0,qin,cin,qout,cout,mv.
- table n (mk_tuple qin cin qout cout mv@〈bar,false〉::T0) → table n T0.
-*)
-
-lemma no_grids_in_tuple : ∀n,l.tuple_TM n l → no_grids l.
-#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
-#_ #_ #Hqin #Hqout #Hcin #Hcout #Hmv #_ #_ #_ #Hl >Hl
-#c #Hc cases (orb_true_l … Hc) -Hc #Hc
-[ >(\P Hc) %
-| cases (memb_append … Hc) -Hc #Hc
-[ @bit_not_grid @(Hqin … Hc)
-| cases (orb_true_l … Hc) -Hc #Hc
-[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
-| cases (orb_true_l … Hc) -Hc #Hc
-[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
-| cases (memb_append …Hc) -Hc #Hc
-[ @bit_not_grid @(Hqout … Hc)
-| cases (orb_true_l … Hc) -Hc #Hc
-[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
-| cases (orb_true_l … Hc) -Hc #Hc
-[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
-| >(memb_single … Hc) @bit_or_null_not_grid @Hmv
-]]]]]]
-qed.
-
-lemma no_marks_in_tuple : ∀n,l.tuple_TM n l → no_marks l.
-#n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
-#Hqin #Hqout #_ #_ #_ #_ #_ #_ #_ #_ #Hl >Hl
-#c #Hc cases (orb_true_l … Hc) -Hc #Hc
-[ >(\P Hc) %
-| cases (memb_append … Hc) -Hc #Hc
-[ @(Hqin … Hc)
-| cases (orb_true_l … Hc) -Hc #Hc
-[ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) %
-| cases (orb_true_l … Hc) -Hc #Hc
-[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
-| cases (memb_append … Hc) -Hc #Hc
-[ @(Hqout … Hc)
-| cases (orb_true_l … Hc) -Hc #Hc
-[ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) %
-| cases (orb_true_l … Hc) -Hc #Hc
-[ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
-| >(memb_single … Hc) %
-]]]]]]
-qed.
-
-lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
-#n #l #t elim t
- [normalize #c #H destruct
- |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_grids_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
- cases (memb_append … Hx) -Hx #Hx
- [ @(Ht1 … Hx)
- | @(IH … Hx) ] ]
-qed.
-
-lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
-#n #l #t elim t
- [normalize #c #H destruct
- |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_marks_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
- cases (memb_append … Hx) -Hx #Hx
- [ @(Ht1 … Hx)
- | @(IH … Hx) ] ]
-qed.
-
-axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
-
-(*
-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 ?) tc_true).
-
-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 daemon :∀P:Prop.P.
-
-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 ?) tc_true) ????)
-[@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 in ⊢ (%→?); #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_on_match ≝
- (seq ? (move_l ?)
- (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (seq ? (move_r ?) (mark ?)))).
-
-definition R_init_current_on_match ≝ λt1,t2.
- ∀l1,l2,c,rs. no_grids l1 → is_grid c = false →
- t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::l2) 〈grid,false〉 rs →
- t2 = midtape STape (〈grid,false〉::l2) 〈c,true〉 ((reverse ? l1)@〈grid,false〉::rs).
-
-lemma sem_init_current_on_match :
- Realize ? init_current_on_match R_init_current_on_match.
-#intape
-cases (sem_seq ????? (sem_move_l ?)
- (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] -Hloop
-#l1 #l2 #c #rs #Hl1 #Hc #Hintape
-cases HR -HR #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape
-generalize in match Hl1; cases l1
- [#Hl1 whd in ⊢ ((???(??%%%))→?); #Hta
- * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Hta
- [* >Hc #Htemp destruct (Htemp) ]
- * #_ #Htc lapply (Htc [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1)
- whd in ⊢ ((???(??%%%))→?); -Htc #Htc
- * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htc -Htd
- whd in ⊢ ((???(??%%%))→?); #Htd
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -Houtc #Houtc
- >Houtc %
- |#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Hta
- * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb
- [* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)]
- * #Hd >append_cons #Htb lapply (Htb … (refl ??) (refl …) ?)
- [#x #membx cases (memb_append … membx) -membx #membx
- [@Htl @memb_cons @membx | >(memb_single … membx) @Hc]]-Htb #Htb
- * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htb -Htc
- >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc #Houtc
- >Houtc >reverse_cons >associative_append %
- ]
-qed.
-
-(*
-definition init_current_gen ≝
- seq ? (adv_to_mark_l ? (is_marked ?))
- (seq ? (clear_mark ?)
- (seq ? (move_l ?)
- (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (seq ? (move_r ?) (mark ?))))).
-
-definition R_init_current_gen ≝ λt1,t2.
- ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 →
- 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_gen : Realize ? init_current_gen R_init_current_gen.
-#intape
-cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
- (sem_seq ????? (sem_clear_mark ?)
- (sem_seq ????? (sem_move_l ?)
- (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] -Hloop
-#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hintape
-cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape
- [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
-* #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] -Hta #Hta
-* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta #Htb
-* #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
-generalize in match Hc; generalize in match Hl2; cases l2
- [#_ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
- whd in ⊢ ((???(??%%%))→?); #Htc
- * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
- [2: * whd in ⊢ (??%?→?); #Htemp destruct (Htemp) ]
- * #_ #Htd >Htd in Htc; -Htd #Htd
- * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
- >reverse_append >reverse_cons
- whd in ⊢ ((???(??%%%))→?); #Hte
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
- >Houtc %
- |#d #tl #Htl #Hc0 whd in ⊢ ((???(??%%%))→?); #Htc
- * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
- [* >(Htl … (memb_hd …)) whd in ⊢ (??%?→?); #Htemp destruct (Htemp)]
- * #Hd #Htd lapply (Htd … (refl ??) (refl ??) ?)
- [#x #membx @Htl @memb_cons @membx] -Htd #Htd
- * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
- >reverse_append >reverse_cons >reverse_cons
- >reverse_cons in Hc0; >reverse_cons cases (reverse ? tl)
- [normalize in ⊢ (%→?); #Hc0 destruct (Hc0) #Hte
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
- >Houtc %
- |* #c2 #b2 #tl2 normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
- whd in ⊢ ((???(??%%%))→?); #Hte
- whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
- >Houtc >associative_append >associative_append >associative_append %
- ]
- ]
-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,cur,rs.t1 = midtape STape ls cur rs →
- \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.
-
-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.
-
-axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
- memb A x l1 = false → memb ? x l3 = false →
- l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
-
-axiom injective_append : ∀A,l.injective … (λx.append A x l).
-
-lemma sem_match_tuple_step:
- accRealize ? match_tuple_step (inr … (inl … (inr … start_nop)))
- 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 ⊢ (%→?); #Hcur
- * #tapec * whd in ⊢ (%→?); #Hcompare #Hor
- #ls #cur #rs #Htapea >Htapea in Hcur; #Hcur cases (Hcur ? (refl ??))
- -Hcur #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 (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
- cases (Hcompare c c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (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
- %
- ]
- |* #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
- cut (bit_or_null 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 …)) >(bit_or_null_not_grid ? 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
- cases (Htapee … Htaped ???) -Htaped -Htapee
- [* #rs3 * * (* we proceed by cases on rs4 *)
- [(* rs4 is empty : the case is absurd since the tape
- cannot end with a bar *)
- * #d * #b * * * #Heq1 @False_ind
- cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
- >Hcut in Htable; >H3 >associative_append
- normalize >Heq1 <associative_append >Hcut
- <associative_append #Htable @(absurd … Htable)
- @last_of_table
- |(* rs4 not empty *)
- * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
- cut (memb STape 〈d2,b2〉 (l2@〈c1,false〉::l3@〈comma,false〉::l4) = true)
- [@memb_append_l2
- cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
- >Hcut >H3 >associative_append @memb_append_l2
- @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
- cut (is_grid d2 = false)
- [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
- cut (b2 = false)
- [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
- >Hb2 in Heq1; #Heq1 -Hb2 -b2
- whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
- [(* we know current is not grid *)
- * #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp)
- |* #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) #_ -Htapef #Htapef
- * #tapeg >Htapef -Htapef *
- (* move_l *)
- whd in ⊢ (%→?);
- #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
- >Htapeg -Htapeg
- (* init_current *)
- whd in ⊢ (%→?); #Htapeout
- cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
- * #c00 #b00 #Hoption
- lapply
- (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb))
- c' (reverse ? la) false ls0 bar (〈d2,true〉::rs3'@〈grid,false〉::rs0) c00 b00 ?????) -Htapeout
- [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
- >associative_append
- generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls0); #l
- whd in ⊢ (???(???%)); >associative_append >associative_append %
- |>reverse_cons @Hoption
- |cases la in H2;
- [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
- @bit_or_null_not_grid @Hc
- |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
- @bit_or_null_not_grid @(Hl1bitnull 〈c',false〉) @memb_append_l2 @memb_hd
- ]
- |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
- [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
- [#eqc0 >(\P eqc0) @Hc |@Hl1bitnull]
- |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
- @memb_append_l1 @daemon
- ]
- |@daemon] #Htapeout % %2 % //
- @(ex_intro … d2)
- cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
- [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
- [
- | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
- @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
-
- cut (sublist … lc l3)
- [ #x #Hx cases la in H3;
- [ normalize #H3 destruct (H3) @Hx
- | #p #la' normalize #Hla' destruct (Hla')
- @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
- @daemon]
- * #rs32 #Hrs3
- (* cut
- (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
- [@daemon] #Hcut *)
- cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
- [ >Hrs3 in Heq1; @daemon ] #Hl4
- @(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4]
- >Htapeout @eq_f2
- [(* by Hoption, H2 *) @daemon
- |(*>Hrs3 *)>append_cons
- > (?:l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
- = (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
- [|>associative_append normalize
- >associative_append normalize
- >associative_append normalize
- >associative_append normalize
- % ]
- >reverse_append >reverse_append >reverse_cons
- >reverse_reverse >reverse_cons >reverse_reverse
- >reverse_append >reverse_append >reverse_cons
- >reverse_reverse >reverse_reverse >reverse_reverse
- >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3
- =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3)))
- [|>associative_append >associative_append
- >associative_append >associative_append >associative_append
- >associative_append % ]
- <H2 normalize in ⊢ (??%?); >Hrs3
- >associative_append >associative_append normalize
- >associative_append >associative_append
- @eq_f @eq_f @eq_f
- >(?:la@(〈d',false〉::lc@〈comma,false〉::rs32)@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 =
- (la@〈d',false〉::lc)@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 )
- [| >associative_append normalize >associative_append % ]
- <H3 %
- ]
- ]
- ]
- |* #Hnobars #Htapee >Htapee -Htapee *
- [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
- whd in ⊢ (%→?); #Htapeout %2 %
- [% [//] whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
- | >(Htapeout … (refl …)) % ]
- |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) -Htapef
- whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
- ]
- |(* no marks in table *)
- #x #membx @(no_marks_in_table … Htable)
- @memb_append_l2
- cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
- >H3 >associative_append @memb_append_l2 @memb_cons @membx
- |(* no grids in table *)
- #x #membx @(no_grids_in_table … Htable)
- @memb_append_l2
- cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
- >H3 >associative_append @memb_append_l2 @memb_cons @membx
- |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
- ]
- ]
- |#x #membx @(no_marks_in_table … Htable)
- @memb_append_l2 @memb_cons @memb_append_l1 @membx
- |#x #membx @(no_marks_in_table … Htable)
- @memb_append_l1 @membx
- |%
- ]
- ]
-qed.
-
-(*
- 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 … start_nop))).
-
-lemma is_grid_true : ∀c.is_grid c = true → c = grid.
-* normalize [ #b ] #H // destruct (H)
-qed.
-
-(* possible variante ?
-definition weakR_match_tuple ≝ λt1,t2.
- (∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧
- (∀c,l1,c1,l2,l3,ls0,rs0,n.
- t1 = midtape STape (〈grid,false〉::ls0) 〈bit c,true〉 rs
- (l1@〈grid,false〉::l2@〈bit c1,true〉::l3@〈grid,false〉::rs0) →
- only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
- table_TM (S n) (l2@〈c1,false〉::l3) →
- (* facciamo match *)
- (∃l4,newc,mv,l5.
- 〈c1,false〉::l3 = l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
- (l2@l4@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
- 〈grid,false〉::rs0))
- ∨
- (* 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〉 ∧
- ∀l4,newc,mv,l5.
- 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
-*)
-
-definition R_match_tuple0 ≝ λt1,t2.
- ∀ls,cur,rs.
- t1 = midtape STape ls cur rs →
- (is_grid (\fst cur) = true → t2 = t1) ∧
- (∀c,l1,c1,l2,l3,ls0,rs0,n.
- ls = 〈grid,false〉::ls0 →
- cur = 〈c,true〉 →
- rs = l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0 →
- is_bit c = true → is_bit c1 = true →
- only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
- table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3) →
- (* facciamo match *)
- (∃l4,newc,mv,l5.
- 〈bar,false〉::〈c1,false〉::l3 = l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
- (l2@l4@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
- 〈grid,false〉::rs0))
- ∨
- (* 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〉 ∧
- ∀l4,newc,mv,l5.
- 〈bar,false〉::〈c1,false〉::l3 ≠ l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
-
-axiom table_bit_after_bar :
- ∀n,l1,c,l2.table_TM n (l1@〈bar,false〉::〈c,false〉::l2) → is_bit c = true.
-
-lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple0.
-#intape #k #outc #Hloop
-lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
-* #ta * #Hstar @(star_ind_l ??????? Hstar)
-[ #tb whd in ⊢ (%→?); #Hleft
- #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
- [ #_ @Houtc
- | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs
- >Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?);
- #Hc destruct (Hc)
- ]
-| (* in the interesting case, we execute a true iteration, then we restart the
- while cycle, finally we end with a false iteration *)
- #tb #tc #td whd in ⊢ (%→?); #Htc
- #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)
- | (* 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
- (* 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)
- [@daemon] * #la * #lb * #mv * #lc * * #Hl3 #Hlalen #Hlabitnull
- >Hl3 in Htable; >append_cons #Htable
- >(?: l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0
- = (l2@[〈bar,false〉])@〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv::
- lc)@〈grid,false〉::rs0) in Hrs;
- [| >associative_append normalize >Hl3
- >associative_append normalize % ] #Hrs
- cases (Htc ????????? Hl1bitnull Hl1marks ?? Hlabitnull Hl1len ? Htable Hls Hcur Hrs)
- [5: <Hl1len @Hlalen
- |4: whd in ⊢ (??%?); >Hc1 %
- |3: whd in ⊢ (??%?); >Hc %
- |-Htc *
- [ (* case 1: match successful *)
- * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) %
- [%
- | cases (IH … Htc) -IH #Houtc #_ >(Houtc (refl ??))
- >Htc @eq_f normalize >associative_append normalize
- >associative_append normalize %
- ]
- | (* case 2: tuples don't match, we still have other tuples to try *)
- * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc
- cases (IH ??? … Htc) -IH #_ #IH
- (* by induction hypothesis *)
- lapply (IH ? l1 c2 (l2@〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5) l6 ? rs0 n (refl ??) (refl ??) ???????)
- [ generalize in match Htable;
- >associative_append normalize
- >associative_append normalize >Heqlblc
- >associative_append normalize //
- | @Hl1len
- | @Hl1marks
- | @Hl1bitnull
- | (*???*) @daemon
- | @Hc
- | >associative_append normalize
- >associative_append normalize
- >associative_append %
- |-IH *
- [ (* the while finally matches a tuple *)
- * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc %
- >Heqlblc @(ex_intro ?? (〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5@l7))
- %{newc} %{mv0} %{l8} %
- [ normalize >Hl7l8 >associative_append normalize
- >associative_append %
- | >Houtc @eq_f >associative_append normalize
- >associative_append normalize >associative_append
- normalize >associative_append %
- ]
- | (* the while fails finding a tuple: there are no matches in the whole table *)
- * #Houtc #Hdiff1 %2 %
- [ @Houtc
- | #l50 #newc #mv0 #l51 >Heqlblc
- @daemon
- ]
- ]
- ]
- ]
- | (* match failed and there is no next tuple: the next while cycle will just exit *)
- * * #Hdiff #Hnobars generalize in match (refl ? tc);
- cases tc in ⊢ (???% → %);
- [ #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
- |2,3: #x #xs #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ]
- #ls1 #cur1 #rs1 #Htc normalize in ⊢ (??%?→?); #Hcur1
- cases (IH … Htc) -IH #IH #_ %2 %
- [ destruct (Hcur1) >IH [ >Htc % | % ]
- | #l4 #newc #mv0 #l5
- (* no_bars except the first one, where the tuple does not match ⇒
- no match *)
- @daemon
- ]
- ]
- ]
-qed.
-
-definition R_match_tuple ≝ λt1,t2.
- ∀ls,c,l1,c1,l2,rs,n.
- is_bit c = true → is_bit c1 = true →
- only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
- table_TM (S n) (〈bar,false〉::〈c1,false〉::l2) →
- t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
- (l1@〈grid,false〉::〈bar,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
- (* facciamo match *)
- (∃l3,newc,mv,l4.
- 〈bar,false〉::〈c1,false〉::l2 = l3@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l4 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
- (l3@〈bar,false〉::〈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.
- 〈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.
-
-axiom Realize_to_Realize :
- ∀alpha,M,R1,R2.(∀t1,t2.R1 t1 t2 → R2 t1 t2) → Realize alpha M R1 → Realize alpha M R2.
-
-lemma sem_match_tuple : Realize ? match_tuple R_match_tuple.
-generalize in match sem_match_tuple0; @Realize_to_Realize
-#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
- Hl1len Htable)
-qed.
\ No newline at end of file
projects / helm.git / commitdiff