From: Ferruccio Guidi Date: Mon, 26 Nov 2012 20:27:40 +0000 (+0000) Subject: we started the theory of delifting substitution ... X-Git-Tag: make_still_working~1439 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=8a5391064ec6e84333444453f836ef9cb91fee7b;p=helm.git we started the theory of delifting substitution ... --- diff --git a/matita/matita/contribs/lambda/dsubst.ma b/matita/matita/contribs/lambda/dsubst.ma new file mode 100644 index 000000000..4d686ef37 --- /dev/null +++ b/matita/matita/contribs/lambda/dsubst.ma @@ -0,0 +1,85 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "lift.ma". + +(* DELIFTING SUBSTITUTION ***************************************************) + +(* Policy: depth (level) metavariables: d, e (as for lift) *) +let rec dsubst C d M on M ≝ match M with +[ VRef i ⇒ tri … i d (#i) (↑[i] C) (#(i-1)) +| Abst A ⇒ 𝛌. (dsubst C (d+1) A) +| Appl B A ⇒ @ (dsubst C d B). (dsubst C d A) +]. + +interpretation "delifting substitution" + 'DSubst C d M = (dsubst C d M). + +(* Note: the notation with "/" does not work *) +notation "hvbox( [ d ⬐ break C ] break term 55 M )" + non associative with precedence 55 + for @{ 'DSubst $C $d $M }. + +notation > "hvbox( [ ⬐ C ] break term 55 M )" + non associative with precedence 55 + for @{ 'DSubst $C 0 $M }. + +lemma dsubst_vref_lt: ∀i,d,C. i < d → [ d ⬐ C ] #i = #i. +normalize /2 width=1/ +qed. + +lemma dsubst_vref_eq: ∀d,C. [ d ⬐ C ] #d = ↑[d]C. +normalize // +qed. + +lemma dsubst_vref_gt: ∀i,d,C. d < i → [ d ⬐ C ] #i = #(i-1). +normalize /2 width=1/ +qed. + +theorem dsubst_lift_le: ∀h,C,M,d1,d2. d2 ≤ d1 → + [ d2 ⬐ ↑[d1 - d2, h] C ] ↑[d1 + 1, h] M = ↑[d1, h] [ d2 ⬐ C ] M. +#h #C #M elim M -M +[ #i #d1 #d2 #Hd21 elim (lt_or_eq_or_gt i d2) #Hid2 + [ lapply (lt_to_le_to_lt … Hid2 Hd21) -Hd21 #Hid1 + >(dsubst_vref_lt … Hid2) >(lift_vref_lt … Hid1) >lift_vref_lt /2 width=1/ + | destruct >dsubst_vref_eq >lift_vref_lt /2 width=1/ + | >(dsubst_vref_gt … Hid2) -Hd21 elim (lt_or_ge (i-1) d1) #Hi1d1 + [ >(lift_vref_lt … Hi1d1) >lift_vref_lt /2 width=1/ + | lapply (ltn_to_ltO … Hid2) #Hi + >(lift_vref_ge … Hi1d1) >lift_vref_ge /2 width=1/ -Hi1d1 >plus_minus // /3 width=1/ + ] + ] +| normalize #A #IHA #d1 #d2 #Hd21 + lapply (IHA (d1+1) (d2+1) ?) -IHA /2 width=1/ +| normalize #B #A #IHB #IHA #d1 #d2 #Hd21 + >IHB -IHB // >IHA -IHA // +] +qed. + +theorem dsubst_lift_be: ∀h,C,M,d1,d2. d1 ≤ d2 → d2 ≤ d1 + h → + [ d2 ⬐ C ] ↑[d1, h + 1] M = ↑[d1, h] M. +#h #C #M elim M -M +[ #i #d1 #d2 #Hd12 #Hd21 elim (lt_or_ge i d1) #Hid1 + [ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 -Hd21 #Hid2 + >(lift_vref_lt … Hid1) >(lift_vref_lt … Hid1) /2 width=1/ + | lapply (transitive_le … (i+h) Hd21 ?) -Hd12 -Hd21 /2 width=1/ #Hd2 + >(lift_vref_ge … Hid1) >(lift_vref_ge … Hid1) -Hid1 + >dsubst_vref_gt // /2 width=1/ + ] +| normalize #A #IHA #d1 #d2 #Hd12 #Hd21 + >IHA -IHA // /2 width=1/ +| normalize #B #A #IHB #IHA #d1 #d2 #Hd12 #Hd21 + >IHB -IHB // >IHA -IHA // +] +qed. diff --git a/matita/matita/lib/arithmetics/nat.ma b/matita/matita/lib/arithmetics/nat.ma index 06266a4b0..edb99b8c0 100644 --- a/matita/matita/lib/arithmetics/nat.ma +++ b/matita/matita/lib/arithmetics/nat.ma @@ -645,6 +645,14 @@ theorem monotonic_lt_minus_l: ∀p,q,n. n ≤ q → q < p → q - n < p - n. @lt_plus_to_minus_r