+++ /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 "lambda/par_reduction.ma".
-include "basics/star.ma".
-
-(*
-inductive T : Type[0] ≝
- | Sort: nat → T
- | Rel: nat → T
- | App: T → T → T
- | Lambda: T → T → T (* type, body *)
- | Prod: T → T → T (* type, body *)
- | D: T →T
-. *)
-
-inductive red : T →T → Prop ≝
- | rbeta: ∀P,M,N. red (App (Lambda P M) N) (M[0 ≝ N])
- | rappl: ∀M,M1,N. red M M1 → red (App M N) (App M1 N)
- | rappr: ∀M,N,N1. red N N1 → red (App M N) (App M N1)
- | rlaml: ∀M,M1,N. red M M1 → red (Lambda M N) (Lambda M1 N)
- | rlamr: ∀M,N,N1. red N N1 → red(Lambda M N) (Lambda M N1)
- | rprodl: ∀M,M1,N. red M M1 → red (Prod M N) (Prod M1 N)
- | rprodr: ∀M,N,N1. red N N1 → red (Prod M N) (Prod M N1)
- | d: ∀M,M1. red M M1 → red (D M) (D M1).
-
-lemma red_to_pr: ∀M,N. red M N → pr M N.
-#M #N #redMN (elim redMN) /2/
-qed.
-
-lemma red_d : ∀M,P. red (D M) P → ∃N. P = D N ∧ red M N.
-#M #P #redMP (inversion redMP)
- [#P1 #M1 #N1 #eqH destruct
- |2,3,4,5,6,7:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
- |#Q1 #M1 #red1 #_ #eqH destruct #eqP @(ex_intro … M1) /2/
- ]
-qed.
-
-lemma red_lambda : ∀M,N,P. red (Lambda M N) P →
- (∃M1. P = (Lambda M1 N) ∧ red M M1) ∨
- (∃N1. P = (Lambda M N1) ∧ red N N1).
-#M #N #P #redMNP (inversion redMNP)
- [#P1 #M1 #N1 #eqH destruct
- |2,3,6,7:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
- |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1
- (@(ex_intro … M1)) % //
- |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %2
- (@(ex_intro … N1)) % //
- |#Q1 #M1 #red1 #_ #eqH destruct
- ]
-qed.
-
-lemma red_prod : ∀M,N,P. red (Prod M N) P →
- (∃M1. P = (Prod M1 N) ∧ red M M1) ∨
- (∃N1. P = (Prod M N1) ∧ red N N1).
-#M #N #P #redMNP (inversion redMNP)
- [#P1 #M1 #N1 #eqH destruct
- |2,3,4,5:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
- |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1
- (@(ex_intro … M1)) % //
- |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %2
- (@(ex_intro … N1)) % //
- |#Q1 #M1 #red1 #_ #eqH destruct
- ]
-qed.
-
-lemma red_app : ∀M,N,P. red (App M N) P →
- (∃M1,N1. M = (Lambda M1 N1) ∧ P = N1[0:=N]) ∨
- (∃M1. P = (App M1 N) ∧ red M M1) ∨
- (∃N1. P = (App M N1) ∧ red N N1).
-#M #N #P #redMNP (inversion redMNP)
- [#P1 #M1 #N1 #eqH destruct #eqP %1 %1
- @(ex_intro … P1) @(ex_intro … M1) % //
- |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1 %2
- (@(ex_intro … M1)) % //
- |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %2
- (@(ex_intro … N1)) % //
- |4,5,6,7:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
- |#Q1 #M1 #red1 #_ #eqH destruct
- ]
-qed.
-
-definition reduct ≝ λn,m. red m n.
-
-definition SN ≝ WF ? reduct.
-
-definition NF ≝ λM. ∀N. ¬ (reduct N M).
-
-theorem NF_to_SN: ∀M. NF M → SN M.
-#M #nfM % #a #red @False_ind /2/
-qed.
-
-lemma NF_Sort: ∀i. NF (Sort i).
-#i #N % #redN (inversion redN)
- [1: #P #N #M #H destruct
- |2,3,4,5,6,7: #N #M #P #_ #_ #H destruct
- |#M #N #_ #_ #H destruct
- ]
-qed.
-
-lemma NF_Rel: ∀i. NF (Rel i).
-#i #N % #redN (inversion redN)
- [1: #P #N #M #H destruct
- |2,3,4,5,6,7: #N #M #P #_ #_ #H destruct
- |#M #N #_ #_ #H destruct
- ]
-qed.
-
-lemma red_subst : ∀N,M,M1,i. red M M1 → red M[i≝N] M1[i≝N].
-#N @Telim_size #P (cases P)
- [1,2:#j #Hind #M1 #i #r1 @False_ind /2/
- |#P #Q #Hind #M1 #i #r1 (cases (red_app … r1))
- [*
- [* #M2 * #N2 * #eqP #eqM1 >eqP normalize
- >eqM1 >(plus_n_O i) >(subst_lemma N2) <(plus_n_O i)
- (cut (i+1 =S i)) [//] #Hcut >Hcut @rbeta
- |* #M2 * #eqM1 #rP >eqM1 normalize @rappl @Hind /2/
- ]
- |* #N2 * #eqM1 #rQ >eqM1 normalize @rappr @Hind /2/
- ]
- |#P #Q #Hind #M1 #i #r1 (cases (red_lambda …r1))
- [* #P1 * #eqM1 #redP >eqM1 normalize @rlaml @Hind /2/
- |* #Q1 * #eqM1 #redP >eqM1 normalize @rlamr @Hind /2/
- ]
- |#P #Q #Hind #M1 #i #r1 (cases (red_prod …r1))
- [* #P1 * #eqM1 #redP >eqM1 normalize @rprodl @Hind /2/
- |* #P1 * #eqM1 #redP >eqM1 normalize @rprodr @Hind /2/
- ]
- |#P #Hind #M1 #i #r1 (cases (red_d …r1))
- #P1 * #eqM1 #redP >eqM1 normalize @d @Hind /2/
- ]
-qed.
-
-lemma red_lift: ∀N,N1,n. red N N1 → ∀k. red (lift N k n) (lift N1 k n).
-#N #N1 #n #r1 (elim r1) normalize /2/
-qed.
-
-(* star red *)
-lemma star_appl: ∀M,M1,N. star … red M M1 →
- star … red (App M N) (App M1 N).
-#M #M1 #N #star1 (elim star1) //
-#B #C #starMB #redBC #H @(inj … H) /2/
-qed.
-
-lemma star_appr: ∀M,N,N1. star … red N N1 →
- star … red (App M N) (App M N1).
-#M #N #N1 #star1 (elim star1) //
-#B #C #starMB #redBC #H @(inj … H) /2/
-qed.
-
-lemma star_app: ∀M,M1,N,N1. star … red M M1 → star … red N N1 →
- star … red (App M N) (App M1 N1).
-#M #M1 #N #N1 #redM #redN @(trans_star ??? (App M1 N)) /2/
-qed.
-
-lemma star_laml: ∀M,M1,N. star … red M M1 →
- star … red (Lambda M N) (Lambda M1 N).
-#M #M1 #N #star1 (elim star1) //
-#B #C #starMB #redBC #H @(inj … H) /2/
-qed.
-
-lemma star_lamr: ∀M,N,N1. star … red N N1 →
- star … red (Lambda M N) (Lambda M N1).
-#M #N #N1 #star1 (elim star1) //
-#B #C #starMB #redBC #H @(inj … H) /2/
-qed.
-
-lemma star_lam: ∀M,M1,N,N1. star … red M M1 → star … red N N1 →
- star … red (Lambda M N) (Lambda M1 N1).
-#M #M1 #N #N1 #redM #redN @(trans_star ??? (Lambda M1 N)) /2/
-qed.
-
-lemma star_prodl: ∀M,M1,N. star … red M M1 →
- star … red (Prod M N) (Prod M1 N).
-#M #M1 #N #star1 (elim star1) //
-#B #C #starMB #redBC #H @(inj … H) /2/
-qed.
-
-lemma star_prodr: ∀M,N,N1. star … red N N1 →
- star … red (Prod M N) (Prod M N1).
-#M #N #N1 #star1 (elim star1) //
-#B #C #starMB #redBC #H @(inj … H) /2/
-qed.
-
-lemma star_prod: ∀M,M1,N,N1. star … red M M1 → star … red N N1 →
- star … red (Prod M N) (Prod M1 N1).
-#M #M1 #N #N1 #redM #redN @(trans_star ??? (Prod M1 N)) /2/
-qed.
-
-lemma star_d: ∀M,M1. star … red M M1 →
- star … red (D M) (D M1).
-#M #M1 #redM (elim redM) // #B #C #starMB #redBC #H @(inj … H) /2/
-qed.
-
-lemma red_subst1 : ∀M,N,N1,i. red N N1 →
- (star … red) M[i≝N] M[i≝N1].
-#M (elim M)
- [//
- |#i #P #Q #n #r1 (cases (true_or_false (leb i n)))
- [#lein (cases (le_to_or_lt_eq i n (leb_true_to_le … lein)))
- [#ltin >(subst_rel1 … ltin) >(subst_rel1 … ltin) //
- |#eqin >eqin >subst_rel2 >subst_rel2 @R_to_star /2/
- ]
- |#lefalse (cut (n < i)) [@not_le_to_lt /2/] #ltni
- >(subst_rel3 … ltni) >(subst_rel3 … ltni) //
- ]
- |#P #Q #Hind1 #Hind2 #M1 #N1 #i #r1 normalize @star_app /2/
- |#P #Q #Hind1 #Hind2 #M1 #N1 #i #r1 normalize @star_lam /2/
- |#P #Q #Hind1 #Hind2 #M1 #N1 #i #r1 normalize @star_prod /2/
- |#P #Hind #M #N #i #r1 normalize @star_d /2/
- ]
-qed.
-
-lemma SN_d : ∀M. SN M → SN (D M).
-#M #snM (elim snM) #b #H #Hind % #a #redd (cases (red_d … redd))
-#Q * #eqa #redbQ >eqa @Hind //
-qed.
-
-lemma SN_step: ∀N. SN N → ∀M. reduct M N → SN M.
-#N * #b #H #M #red @H //.
-qed.
-
-lemma SN_star: ∀M,N. (star … red) N M → SN N → SN M.
-#M #N #rstar (elim rstar) //
-#Q #P #HbQ #redQP #snNQ #snN @(SN_step …redQP) /2/
-qed.
-
-lemma sub_red: ∀M,N.subterm N M → ∀N1.red N N1 →
-∃M1.subterm N1 M1 ∧ red M M1.
-#M #N #subN (elim subN) /4/
-(* trsansitive case *)
-#P #Q #S #subPQ #subQS #H1 #H2 #A #redP (cases (H1 ? redP))
-#B * #subA #redQ (cases (H2 ? redQ)) #C * #subBC #redSC
-@(ex_intro … C) /3/
-qed.
-
-axiom sub_star_red: ∀M,N.(star … subterm) N M → ∀N1.red N N1 →
-∃M1.subterm N1 M1 ∧ red M M1.
-
-lemma SN_subterm: ∀M. SN M → ∀N.subterm N M → SN N.
-#M #snM (elim snM) #M #snM #HindM #N #subNM % #N1 #redN
-(cases (sub_red … subNM ? redN)) #M1 *
-#subN1M1 #redMM1 @(HindM … redMM1) //
-qed.
-
-lemma SN_subterm_star: ∀M. SN M → ∀N.(star … subterm N M) → SN N.
-#M #snM #N #Hstar (cases (star_inv T subterm M N)) #_ #H
-lapply (H Hstar) #Hstari (elim Hstari) //
-#M #N #_ #subNM #snM @(SN_subterm …subNM) //
-qed.
-
-definition shrink ≝ λN,M. reduct N M ∨ (TC … subterm) N M.
-
-definition SH ≝ WF ? shrink.
-
-lemma SH_subterm: ∀M. SH M → ∀N.(star … subterm) N M → SH N.
-#M #snM (elim snM) #M
-#snM #HindM #N #subNM (cases (star_case ???? subNM))
- [#eqNM >eqNM % /2/
- |#subsNM % #N1 *
- [#redN (cases (sub_star_red … subNM ? redN)) #M1 *
- #subN1M1 #redMM1 @(HindM M1) /2/
- |#subN1 @(HindM N) /2/
- ]
- ]
-qed.
-
-theorem SN_to_SH: ∀N. SN N → SH N.
-#N #snN (elim snN) (@Telim_size)
-#b #Hsize #snb #Hind % #a * /2/ #subab @Hsize;
- [(elim subab)
- [#c #subac @size_subterm //
- |#b #c #subab #subbc #sab @(transitive_lt … sab) @size_subterm //
- ]
- |@SN_step @(SN_subterm_star b);
- [% /2/ |@TC_to_star @subab] % @snb
- |#a1 #reda1 cases(sub_star_red b a ?? reda1);
- [#a2 * #suba1 #redba2 @(SH_subterm a2) /2/ |/2/ ]
- ]
-qed.
-
-lemma SH_to_SN: ∀N. SH N → SN N.
-@WF_antimonotonic /2/ qed.
-
-lemma SN_Lambda: ∀N.SN N → ∀M.SN M → SN (Lambda N M).
-#N #snN (elim snN) #P #shP #HindP #M #snM
-(* for M we proceed by induction on SH *)
-(lapply (SN_to_SH ? snM)) #shM (elim shM)
-#Q #shQ #HindQ % #a #redH (cases (red_lambda … redH))
- [* #S * #eqa #redPS >eqa @(HindP S ? Q ?) //
- @SH_to_SN % /2/
- |* #S * #eqa #redQS >eqa @(HindQ S) /2/
- ]
-qed.
-
-lemma SN_Prod: ∀N.SN N → ∀M.SN M → SN (Prod N M).
-#N #snN (elim snN) #P #shP #HindP #M #snM (elim snM)
-#Q #snQ #HindQ % #a #redH (cases (red_prod … redH))
- [* #S * #eqa #redPS >eqa @(HindP S ? Q ?) //
- % /2/
- |* #S * #eqa #redQS >eqa @(HindQ S) /2/
- ]
-qed.
-
-lemma SN_subst: ∀i,N,M.SN M[i ≝ N] → SN M.
-#i #N (cut (∀P.SN P → ∀M.P=M[i ≝ N] → SN M));
- [#P #H (elim H) #Q #snQ #Hind #M #eqM % #M1 #redM
- @(Hind M1[i:=N]) // >eqM /2/
- |#Hcut #M #snM @(Hcut … snM) //
-qed.
-
-(*
-lemma SN_DAPP: ∀N,M. SN (App M N) → SN (App (D M) N).
-cut (∀P. SN P → ∀M,N. P = App M N → SN (App (D M) N)); [|/2/]
-#P #snP (elim snP) #Q #snQ #Hind
-#M #N #eqQ % #A #rA (cases (red_app … rA))
- [*
- [*
- [* #M1 * #N1 * #eqH destruct
- |* #M1 * #eqH destruct #eqA >eqA @SN_d % @snQ
- ]
- |* #M1 * #eqA #red1 (cases (red_d …red1))
- #M2 * #eqM1 #r2 >eqA >eqM1 @(Hind (App M2 N)) /2/
- ]
- |* #M2 * #eqA >eqA #r2 @(Hind (App M M2)) /2/
- ]
-qed. *)
-
-lemma SN_APP: ∀P.SN P → ∀N. SN N → ∀M.
- SN M[0:=N] → SN (App (Lambda P M) N).
-#P #snP (elim snP) #A #snA #HindA
-#N #snN (elim snN) #B #snB #HindB
-#M #snM1 (cut (SH M)) [@SN_to_SH @(SN_subst … snM1)] #shM
-(generalize in match snM1) (elim shM)
-#C #shC #HindC #snC1 % #Q #redQ (cases (red_app … redQ))
- [*
- [* #M2 * #N2 * #eqlam destruct #eqQ //
- |* #M2 * #eqQ #redlam >eqQ (cases (red_lambda …redlam))
- [* #M3 * #eqM2 #r2 >eqM2 @HindA // % /2/
- |* #M3 * #eqM2 #r2 >eqM2 @HindC;
- [%1 // |@(SN_step … snC1) /2/]
- ]
- ]
- |* #M2 * #eqQ #r2 >eqQ @HindB // @(SN_star … snC1)
- @red_subst1 //
- ]
-qed.
-
-
-
-
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