--- /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 "finite_lambda/terms_and_types.ma".
+
+(* some auxiliary lemmas *)
+
+lemma nth_to_default: ∀A,l,n,d.
+ |l| ≤ n → nth n A l d = d.
+#A #l elim l [//] #a #tl #Hind #n cases n
+ [#d normalize #H @False_ind @(absurd … H) @lt_to_not_le //
+ |#m #d normalize #H @Hind @le_S_S_to_le @H
+ ]
+qed.
+
+lemma mem_nth: ∀A,l,n,d.
+ n < |l| → mem ? (nth n A l d) l.
+#A #l elim l
+ [#n #d normalize #H @False_ind @(absurd … H) @lt_to_not_le //
+ |#a #tl #Hind * normalize
+ [#_ #_ %1 //| #m #d #HSS %2 @Hind @le_S_S_to_le @HSS]
+ ]
+qed.
+
+lemma nth_map: ∀A,B,l,f,n,d1,d2.
+ n < |l| → nth n B (map … f l) d1 = f (nth n A l d2).
+#n #B #l #f elim l
+ [#m #d1 #d2 normalize #H @False_ind @(absurd … H) @lt_to_not_le //
+ |#a #tl #Hind #m #d1 #d2 cases m normalize //
+ #m1 #H @Hind @le_S_S_to_le @H
+ ]
+qed.
+
+
+
+(* end of auxiliary lemmas *)
+
+let rec to_T O D ty on ty: FinSet_of_FType O D ty → T O D ≝
+ match ty return (λty.FinSet_of_FType O D ty → T O D) with
+ [atom o ⇒ λa.Val O D o a
+ |arrow ty1 ty2 ⇒ λa:FinFun ??.Vec O D ty1
+ (map ((FinSet_of_FType O D ty1)×(FinSet_of_FType O D ty2))
+ (T O D) (λp.to_T O D ty2 (snd … p)) (pi1 … a))
+ ]
+.
+
+lemma is_closed_to_T: ∀O,D,ty,a. is_closed O D 0 (to_T O D ty a).
+#O #D #ty elim ty //
+#ty1 #ty2 #Hind1 #Hind2 #a normalize @cvec #m #Hmem
+lapply (mem_map ????? Hmem) * #a1 * #H1 #H2 <H2 @Hind2
+qed.
+
+axiom inj_to_T: ∀O,D,ty,a1,a2. to_T O D ty a1 = to_T O D ty a2 → a1 = a2.
+(* complicata
+#O #D #ty elim ty
+ [#o normalize #a1 #a2 #H destruct //
+ |#ty1 #ty2 #Hind1 #Hind2 * #l1 #Hl1 * #l2 #Hl2 normalize #H destruct -H
+ cut (l1=l2) [2: #H generalize in match Hl1; >H //] -Hl1 -Hl2
+ lapply e0 -e0 lapply l2 -l2 elim l1
+ [#l2 cases l2 normalize [// |#a1 #tl1 #H destruct]
+ |#a1 #tl1 #Hind #l2 cases l2
+ [normalize #H destruct
+ |#a2 #tl2 normalize #H @eq_f2
+ [@Hind2 *)
+
+let rec assoc (A:FinSet) (B:Type[0]) (a:A) l1 l2 on l1 : option B ≝
+ match l1 with
+ [ nil ⇒ None ?
+ | cons hd1 tl1 ⇒ match l2 with
+ [ nil ⇒ None ?
+ | cons hd2 tl2 ⇒ if a==hd1 then Some ? hd2 else assoc A B a tl1 tl2
+ ]
+ ].
+
+lemma same_assoc: ∀A,B,a,l1,v1,v2,N,N1.
+ assoc A B a l1 (v1@N::v2) = Some ? N ∧ assoc A B a l1 (v1@N1::v2) = Some ? N1
+ ∨ assoc A B a l1 (v1@N::v2) = assoc A B a l1 (v1@N1::v2).
+#A #B #a #l1 #v1 #v2 #N #N1 lapply v1 -v1 elim l1
+ [#v1 %2 // |#hd #tl #Hind * normalize cases (a==hd) normalize /3/]
+qed.
+
+lemma assoc_to_mem: ∀A,B,a,l1,l2,b.
+ assoc A B a l1 l2 = Some ? b → mem ? b l2.
+#A #B #a #l1 elim l1
+ [#l2 #b normalize #H destruct
+ |#hd1 #tl1 #Hind *
+ [#b normalize #H destruct
+ |#hd2 #tl2 #b normalize cases (a==hd1) normalize
+ [#H %1 destruct //|#H %2 @Hind @H]
+ ]
+ ]
+qed.
+
+lemma assoc_to_mem2: ∀A,B,a,l1,l2,b.
+ assoc A B a l1 l2 = Some ? b → ∃l21,l22.l2=l21@b::l22.
+#A #B #a #l1 elim l1
+ [#l2 #b normalize #H destruct
+ |#hd1 #tl1 #Hind *
+ [#b normalize #H destruct
+ |#hd2 #tl2 #b normalize cases (a==hd1) normalize
+ [#H %{[]} %{tl2} destruct //
+ |#H lapply (Hind … H) * #la * #lb #H1
+ %{(hd2::la)} %{lb} >H1 //]
+ ]
+ ]
+qed.
+
+lemma assoc_map: ∀A,B,C,a,l1,l2,f,b.
+ assoc A B a l1 l2 = Some ? b → assoc A C a l1 (map ?? f l2) = Some ? (f b).
+#A #B #C #a #l1 elim l1
+ [#l2 #f #b normalize #H destruct
+ |#hd1 #tl1 #Hind *
+ [#f #b normalize #H destruct
+ |#hd2 #tl2 #f #b normalize cases (a==hd1) normalize
+ [#H destruct // |#H @(Hind … H)]
+ ]
+ ]
+qed.
+
+(*************************** One step reduction *******************************)
+
+inductive red (O:Type[0]) (D:O→FinSet) : T O D →T O D → Prop ≝
+ | (* we only allow beta on closed arguments *)
+ rbeta: ∀P,M,N. is_closed O D 0 N →
+ red O D (App O D (Lambda O D P M) N) (subst O D M 0 N)
+ | riota: ∀ty,v,a,M.
+ assoc ?? a (enum (FinSet_of_FType O D ty)) v = Some ? M →
+ red O D (App O D (Vec O D ty v) (to_T O D ty a)) M
+ | rappl: ∀M,M1,N. red O D M M1 → red O D (App O D M N) (App O D M1 N)
+ | rappr: ∀M,N,N1. red O D N N1 → red O D (App O D M N) (App O D M N1)
+ | rlam: ∀ty,N,N1. red O D N N1 → red O D (Lambda O D ty N) (Lambda O D ty N1)
+ | rmem: ∀ty,M. red O D (Lambda O D ty M)
+ (Vec O D ty (map ?? (λa. subst O D M 0 (to_T O D ty a))
+ (enum (FinSet_of_FType O D ty))))
+ | rvec: ∀ty,N,N1,v,v1. red O D N N1 →
+ red O D (Vec O D ty (v@N::v1)) (Vec O D ty (v@N1::v1)).
+
+(*********************************** inversion ********************************)
+lemma red_vec: ∀O,D,ty,v,M.
+ red O D (Vec O D ty v) M → ∃N,N1,v1,v2.
+ red O D N N1 ∧ v = v1@N::v2 ∧ M = Vec O D ty (v1@N1::v2).
+#O #D #ty #v #M #Hred inversion Hred
+ [#ty1 #M0 #N #Hc #H destruct
+ |#ty1 #v1 #a #M0 #_ #H destruct
+ |#M0 #M1 #N #_ #_ #H destruct
+ |#M0 #M1 #N #_ #_ #H destruct
+ |#ty1 #M #M1 #_ #_ #H destruct
+ |#ty1 #M0 #H destruct
+ |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct #_ %{N} %{N1} %{v1} %{v2} /3/
+ ]
+qed.
+
+lemma red_lambda: ∀O,D,ty,M,N.
+ red O D (Lambda O D ty M) N →
+ (∃M1. red O D M M1 ∧ N = (Lambda O D ty M1)) ∨
+ N = Vec O D ty (map ?? (λa. subst O D M 0 (to_T O D ty a))
+ (enum (FinSet_of_FType O D ty))).
+#O #D #ty #M #N #Hred inversion Hred
+ [#ty1 #M0 #N #Hc #H destruct
+ |#ty1 #v1 #a #M0 #_ #H destruct
+ |#M0 #M1 #N #_ #_ #H destruct
+ |#M0 #M1 #N #_ #_ #H destruct
+ |#ty1 #P #P1 #redP #_ #H #H1 destruct %1 %{P1} % //
+ |#ty1 #M0 #H destruct #_ %2 //
+ |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct
+ ]
+qed.
+
+lemma red_val: ∀O,D,ty,a,N.
+ red O D (Val O D ty a) N → False.
+#O #D #ty #M #N #Hred inversion Hred
+ [#ty1 #M0 #N #Hc #H destruct
+ |#ty1 #v1 #a #M0 #_ #H destruct
+ |#M0 #M1 #N #_ #_ #H destruct
+ |#M0 #M1 #N #_ #_ #H destruct
+ |#ty1 #N1 #N2 #_ #_ #H destruct
+ |#ty1 #M0 #H destruct #_
+ |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct
+ ]
+qed.
+
+lemma red_rel: ∀O,D,n,N.
+ red O D (Rel O D n) N → False.
+#O #D #n #N #Hred inversion Hred
+ [#ty1 #M0 #N #Hc #H destruct
+ |#ty1 #v1 #a #M0 #_ #H destruct
+ |#M0 #M1 #N #_ #_ #H destruct
+ |#M0 #M1 #N #_ #_ #H destruct
+ |#ty1 #N1 #N2 #_ #_ #H destruct
+ |#ty1 #M0 #H destruct #_
+ |#ty1 #N #N1 #v1 #v2 #Hred1 #_ #H destruct
+ ]
+qed.
+
+(*************************** multi step reduction *****************************)
+lemma star_red_appl: ∀O,D,M,M1,N. star ? (red O D) M M1 →
+ star ? (red O D) (App O D M N) (App O D M1 N).
+#O #D #M #N #N1 #H elim H //
+#P #Q #Hind #HPQ #Happ %1[|@Happ] @rappl @HPQ
+qed.
+
+lemma star_red_appr: ∀O,D,M,N,N1. star ? (red O D) N N1 →
+ star ? (red O D) (App O D M N) (App O D M N1).
+#O #D #M #N #N1 #H elim H //
+#P #Q #Hind #HPQ #Happ %1[|@Happ] @rappr @HPQ
+qed.
+
+lemma star_red_vec: ∀O,D,ty,N,N1,v1,v2. star ? (red O D) N N1 →
+ star ? (red O D) (Vec O D ty (v1@N::v2)) (Vec O D ty (v1@N1::v2)).
+#O #D #ty #N #N1 #v1 #v2 #H elim H //
+#P #Q #Hind #HPQ #Hvec %1[|@Hvec] @rvec @HPQ
+qed.
+
+lemma star_red_vec1: ∀O,D,ty,v1,v2,v. |v1| = |v2| →
+ (∀n,M. n < |v1| → star ? (red O D) (nth n ? v1 M) (nth n ? v2 M)) →
+ star ? (red O D) (Vec O D ty (v@v1)) (Vec O D ty (v@v2)).
+#O #D #ty #v1 elim v1
+ [#v2 #v normalize #Hv2 >(lenght_to_nil … (sym_eq … Hv2)) normalize //
+ |#N1 #tl1 #Hind * [normalize #v #H destruct] #N2 #tl2 #v normalize #HS
+ #H @(trans_star … (Vec O D ty (v@N2::tl1)))
+ [@star_red_vec @(H 0 N1) @le_S_S //
+ |>append_cons >(append_cons ??? tl2) @(Hind… (injective_S … HS))
+ #n #M #H1 @(H (S n)) @le_S_S @H1
+ ]
+ ]
+qed.
+
+lemma star_red_vec2: ∀O,D,ty,v1,v2. |v1| = |v2| →
+ (∀n,M. n < |v1| → star ? (red O D) (nth n ? v1 M) (nth n ? v2 M)) →
+ star ? (red O D) (Vec O D ty v1) (Vec O D ty v2).
+#O #D #ty #v1 #v2 @(star_red_vec1 … [ ])
+qed.
+
+lemma star_red_lambda: ∀O,D,ty,N,N1. star ? (red O D) N N1 →
+ star ? (red O D) (Lambda O D ty N) (Lambda O D ty N1).
+#O #D #ty #N #N1 #H elim H //
+#P #Q #Hind #HPQ #Hlam %1[|@Hlam] @rlam @HPQ
+qed.
+
+(************************ reduction and substitution **************************)
+
+lemma red_star_subst : ∀O,D,M,N,N1,i.
+ star ? (red O D) N N1 → star ? (red O D) (subst O D M i N) (subst O D M i N1).
+#O #D #M #N #N1 #i #Hred lapply i -i @(T_elim … M) normalize
+ [#o #a #i //
+ |#i #n cases (leb n i) normalize // cases (eqb n i) normalize //
+ |#P #Q #HindP #HindQ #n normalize
+ @(trans_star … (App O D (subst O D P n N1) (subst O D Q n N)))
+ [@star_red_appl @HindP |@star_red_appr @HindQ]
+ |#ty #P #HindP #i @star_red_lambda @HindP
+ |#ty #v #Hindv #i @star_red_vec2 [>length_map >length_map //]
+ #j #Q inversion v [#_ normalize //] #a #tl #_ #Hv
+ cases (true_or_false (leb (S j) (|a::tl|))) #Hcase
+ [lapply (leb_true_to_le … Hcase) -Hcase #Hcase
+ >(nth_map ?????? a Hcase) >(nth_map ?????? a Hcase) #_ @Hindv >Hv @mem_nth //
+ |>nth_to_default
+ [2:>length_map @le_S_S_to_le @not_le_to_lt @leb_false_to_not_le //]
+ >nth_to_default
+ [2:>length_map @le_S_S_to_le @not_le_to_lt @leb_false_to_not_le //] //
+ ]
+ ]
+qed.
+
+lemma red_star_subst2 : ∀O,D,M,M1,N,i. is_closed O D 0 N →
+ red O D M M1 → star ? (red O D) (subst O D M i N) (subst O D M1 i N).
+#O #D #M #M1 #N #i #HNc #Hred lapply i -i elim Hred
+ [#ty #P #Q #HQc #i normalize @starl_to_star @sstepl
+ [|@rbeta >(subst_closed … HQc) //] >(subst_closed … HQc) //
+ lapply (subst_lemma ?? P ?? i 0 (is_closed_mono … HQc) HNc) //
+ <plus_n_Sm <plus_n_O #H <H //
+ |#ty #v #a #P #HP #i normalize >(subst_closed … (le_O_n …)) //
+ @R_to_star @riota @assoc_map @HP
+ |#P #P1 #Q #Hred #Hind #i normalize @star_red_appl @Hind
+ |#P #P1 #Q #Hred #Hind #i normalize @star_red_appr @Hind
+ |#ty #P #P1 #Hred #Hind #i normalize @star_red_lambda @Hind
+ |#ty #P #i normalize @starl_to_star @sstepl [|@rmem]
+ @star_to_starl @star_red_vec2 [>length_map >length_map >length_map //]
+ #n #Q >length_map #H
+ cut (∃a:(FinSet_of_FType O D ty).True)
+ [lapply H -H lapply (enum_complete (FinSet_of_FType O D ty))
+ cases (enum (FinSet_of_FType O D ty))
+ [#x normalize #H @False_ind @(absurd … H) @lt_to_not_le //
+ |#x #l #_ #_ %{x} //
+ ]
+ ] * #a #_
+ >(nth_map ?????? a H) >(nth_map ?????? Q) [2:>length_map @H]
+ >(nth_map ?????? a H)
+ lapply (subst_lemma O D P (to_T O D ty
+ (nth n (FinSet_of_FType O D ty) (enum (FinSet_of_FType O D ty)) a))
+ N i 0 (is_closed_mono … (is_closed_to_T …)) HNc) // <plus_n_O #H1 >H1
+ <plus_n_Sm <plus_n_O //
+ |#ty #P #Q #v #v1 #Hred #Hind #n normalize
+ <map_append <map_append @star_red_vec @Hind
+ ]
+qed.
+
+
+
+
+
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