--- /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/reduction.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 conv : T →T → Prop ≝
+ | cbeta: ∀P,M,N. conv (App (Lambda P M) N) (M[0 ≝ N])
+ | cdapp: ∀M,N. conv (App (D M) N) (D (App M N))
+ | cdlam: ∀M,N. conv (Lambda M (D N)) (D (Lambda M N))
+ | cappl: ∀M,M1,N. conv M M1 → conv (App M N) (App M1 N)
+ | cappr: ∀M,N,N1. conv N N1 → conv (App M N) (App M N1)
+ | claml: ∀M,M1,N. conv M M1 → conv (Lambda M N) (Lambda M1 N)
+ | clamr: ∀M,N,N1. conv N N1 → conv (Lambda M N) (Lambda M N1)
+ | cprodl: ∀M,M1,N. conv M M1 → conv (Prod M N) (Prod M1 N)
+ | cprodr: ∀M,N,N1. conv N N1 → conv (Prod M N) (Prod M N1)
+ | cd: ∀M,M1. conv (D M) (D M1).
+
+definition CO ≝ star … conv.
+
+lemma red_to_conv: ∀M,N. red M N → conv M N.
+#M #N #redMN (elim redMN) /2/
+qed.
+
+inductive d_eq : T →T → Prop ≝
+ | same: ∀M. d_eq M M
+ | ed: ∀M,M1. d_eq (D M) (D M1)
+ | eapp: ∀M1,M2,N1,N2. d_eq M1 M2 → d_eq N1 N2 →
+ d_eq (App M1 N1) (App M2 N2)
+ | elam: ∀M1,M2,N1,N2. d_eq M1 M2 → d_eq N1 N2 →
+ d_eq (Lambda M1 N1) (Lambda M2 N2)
+ | eprod: ∀M1,M2,N1,N2. d_eq M1 M2 → d_eq N1 N2 →
+ d_eq (Prod M1 N1) (Prod M2 N2).
+
+lemma conv_to_deq: ∀M,N. conv M N → red M N ∨ d_eq M N.
+#M #N #coMN (elim coMN)
+ [#P #B #C %1 //
+ |#M1 #N1 %1 //
+ |#M1 #N1 %1 //
+ |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
+ |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
+ |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
+ |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
+ |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
+ |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
+ |#P #M1 %2 //
+ ]
+qed.
+
+theorem main1: ∀M,N. CO M N →
+ ∃P,Q. star … red M P ∧ star … red N Q ∧ d_eq P Q.
+#M #N #coMN (elim coMN)
+ [#B #C #rMB #convBC * #P1 * #Q1 * * #redMP1 #redBQ1
+ #deqP1Q1 (cases (conv_to_deq … convBC))
+ [
+ |@(ex_intro … M) @(ex_intro … M) % // % //
+ ]
+
+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
+ |#M1 #N1 #eqH destruct
+ |#M1 #N1 #eqH destruct
+ |4,5,6,7,8,9:#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) ∨
+ (∃Q. N = D Q ∧ P = D (Lambda M Q)).
+#M #N #P #redMNP (inversion redMNP)
+ [#P1 #M1 #N1 #eqH destruct
+ |#M1 #N1 #eqH destruct
+ |#M1 #N1 #eqH destruct #eqP %2 (@(ex_intro … N1)) % //
+ |4,5,8,9:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
+ |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1 %1
+ (@(ex_intro … M1)) % //
+ |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1 %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: #M1 #N1 #eqH destruct
+ |4,5,6,7:#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_app : ∀M,N,P. red (App M N) P →
+ (∃M1,N1. M = (Lambda M1 N1) ∧ P = N1[0:=N]) ∨
+ (∃M1. M = (D M1) ∧ P = D (App M1 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 %1
+ @(ex_intro … P1) @(ex_intro … M1) % //
+ |#M1 #N1 #eqH destruct #eqP %1 %1 %2 /3/
+ |#M1 #N1 #eqH destruct
+ |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1 %2
+ (@(ex_intro … M1)) % //
+ |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %2
+ (@(ex_intro … N1)) % //
+ |6,7,8,9:#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 :#N #M #H destruct
+ |4,5,6,7,8,9: #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 :#N #M #H destruct
+ |4,5,6,7,8,9: #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 * #eqP #eqM1 >eqM1 >eqP normalize @rdapp
+ ]
+ |* #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/
+ ]
+ |* #M2 * #eqQ #eqM1 >eqM1 >eqQ normalize @rdlam
+ ]
+ |#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/
+ ]
+ |* #S * #eqQ #eqa >eqa @SN_d @(HindQ S) /3/
+ ]
+qed.
+
+(*
+lemma SH_Lambda: ∀N.SH N → ∀M.SH M → SN (Lambda N M).
+#N #snN (elim snN) #P #snP #HindP #M #snM (elim snM)
+#Q #snQ #HindQ % #a #redH (cases (red_lambda … redH))
+ [*
+ [* #S * #eqa #redPS >eqa @(HindP S ? Q ?) /2/
+ % /2/
+ |* #S * #eqa #redQS >eqa @(HindQ S) /2/
+ ]
+ |* #S * #eqQ #eqa >eqa @SN_d @(HindQ S) /3/
+ ]
+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 * #eqlam destruct
+ ]
+ |* #M2 * #eqQ #redlam >eqQ (cases (red_lambda …redlam))
+ [*
+ [* #M3 * #eqM2 #r2 >eqM2 @HindA // % /2/
+ |* #M3 * #eqM2 #r2 >eqM2 @HindC;
+ [%1 // |@(SN_step … snC1) /2/]
+ ]
+ |* #M3 * #eqC #eqM2 >eqM2 @SN_DAPP @HindC;
+ [%2 >eqC @inj //
+ |@(SN_subterm … snC1) >eqC normalize //
+ ]
+ ]
+ ]
+ |* #M2 * #eqQ #r2 >eqQ @HindB // @(SN_star … snC1)
+ @red_subst1 //
+ ]
+qed.
+
+
+
+
projects / helm.git / commitdiff