X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Ffinite_lambda%2Freduction.ma;fp=matita%2Fmatita%2Flib%2Ffinite_lambda%2Freduction.ma;h=98c56e10af81c2d187e44644193a89d09eb061ed;hb=d4a98ff73d3506f72dafd3b264ce5e22e29674f3;hp=0000000000000000000000000000000000000000;hpb=9c0398174ebfa6b483dbdd5c10e8b15e39067329;p=helm.git

diff --git a/matita/matita/lib/finite_lambda/reduction.ma b/matita/matita/lib/finite_lambda/reduction.ma
new file mode 100644
index 000000000..98c56e10a
--- /dev/null
+++ b/matita/matita/lib/finite_lambda/reduction.ma
@@ -0,0 +1,308 @@
+(*
+    ||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.
+   
+
+
+
+