-
-(**************************** Vector Operations *******************************)
-
-let rec resize A (l:list A) i d on i ≝
- match i with
- [ O ⇒ [ ]
- | S j ⇒ (hd ? l d)::resize A (tail ? l) j d
- ].
-
-lemma length_resize : ∀A,l,i,d. |resize A l i d| = i.
-#A #l #i lapply l -l elim i
- [#l #d %
- |#m #Hind #l cases l
- [#d normalize @eq_f @Hind
- |#a #tl #d normalize @eq_f @Hind
- ]
- ]
-qed.
-
-lemma resize_id : ∀A,n,l,d. |l| = n →
- resize A l n d = l.
-#A #n elim n
- [#l #d #H >(lenght_to_nil … H) //
- |#m #Hind * #d normalize
- [#H destruct |#a #tl #H @eq_f @Hind @injective_S // ]
- ]
-qed.
-
-definition resize_vec :∀A,n.Vector A n → ∀i.A→Vector A i.
-#A #n #a #i #d @(mk_Vector A i (resize A a i d) ) @length_resize
-qed.
-
-axiom nth_resize_vec :∀A,n.∀v:Vector A n.∀d,i,j. i < j →i < n →
- nth i ? (resize_vec A n v j d) d = nth i ? v d.
-
-lemma resize_vec_id : ∀A,n.∀v:Vector A n.∀d.
- resize_vec A n v n d = v.
-#A #n #v #d @(eq_vec … d) #i #ltin @nth_resize_vec //
-qed.
-
-definition vec_single: ∀A,a. Vector A 1 ≝ λA,a.
- mk_Vector A 1 [a] (refl ??).
-
-definition vec_cons_right : ∀A.∀a:A.∀n. Vector A n → Vector A (S n).
-#A #a #n #v >(plus_n_O n) >plus_n_Sm @(vec_append … v (vec_single A a))
->length_append >(len A n v) //
-qed.
-
-lemma eq_cons_right : ∀A,a1,a2.∀n.∀v1,v2:Vector A n.
- a1 = a2 → v1 = v2 → vec_cons_right A a1 n v1 = vec_cons_right A a2 n v2.
-// qed.
-
-axiom nth_cons_right: ∀A.∀a:A.∀n.∀v:Vector A n.∀d.
- nth n ? (vec_cons_right A a n v) d = a.
-(*
-#A #a #n elim n
- [#v #d >(vector_nil … v) //
- |#m #Hind #v #d >(vec_expand … v) whd in ⊢ (??%?);
- whd in match (vec_cons_right ????);
-*)
-
-lemma get_moves_cons_right: ∀Q,sig,n,q,moves,a.
- get_moves Q sig (S n)
- (vec_cons_right ? (Some ? (inl ?? 〈q,moves〉)) (S n) a) = moves.
-#Q #sig #n #q #moves #a whd in ⊢(??%?); >nth_cons_right //
-qed.
-
-axiom resize_cons_right: ∀A.∀a:A.∀n.∀v:Vector A n.∀d.
- resize_vec ? (S n) (vec_cons_right A a n v) n d = v.