inductive drops (s:bool): list2 nat nat → relation lenv ≝
| drops_nil : ∀L. drops s (◊) L L
| drops_cons: ∀L1,L,L2,des,d,e.
- drops s des L1 L â\86\92 â\87©[s, d, e] L ≡ L2 → drops s ({d, e} @ des) L1 L2
+ drops s des L1 L â\86\92 â¬\87[s, d, e] L ≡ L2 → drops s ({d, e} @ des) L1 L2
.
interpretation "iterated slicing (local environment) abstract"
*)
definition l_liftable1: relation2 lenv term → predicate bool ≝
- λR,s. â\88\80K,T. R K T â\86\92 â\88\80L,d,e. â\87©[s, d, e] L ≡ K →
- â\88\80U. â\87§[d, e] T ≡ U → R L U.
+ λR,s. â\88\80K,T. R K T â\86\92 â\88\80L,d,e. â¬\87[s, d, e] L ≡ K →
+ â\88\80U. â¬\86[d, e] T ≡ U → R L U.
definition l_liftables1: relation2 lenv term → predicate bool ≝
- λR,s. â\88\80L,K,des. â\87©*[s, des] L ≡ K →
- â\88\80T,U. â\87§*[des] T ≡ U → R K T → R L U.
+ λR,s. â\88\80L,K,des. â¬\87*[s, des] L ≡ K →
+ â\88\80T,U. â¬\86*[des] T ≡ U → R K T → R L U.
definition l_liftables1_all: relation2 lenv term → predicate bool ≝
- λR,s. â\88\80L,K,des. â\87©*[s, des] L ≡ K →
- â\88\80Ts,Us. â\87§*[des] Ts ≡ Us →
+ λR,s. â\88\80L,K,des. â¬\87*[s, des] L ≡ K →
+ â\88\80Ts,Us. â¬\86*[des] Ts ≡ Us →
all … (R K) Ts → all … (R L) Us.
(* Basic inversion lemmas ***************************************************)
-fact drops_inv_nil_aux: â\88\80L1,L2,s,des. â\87©*[s, des] L1 ≡ L2 → des = ◊ → L1 = L2.
+fact drops_inv_nil_aux: â\88\80L1,L2,s,des. â¬\87*[s, des] L1 ≡ L2 → des = ◊ → L1 = L2.
#L1 #L2 #s #des * -L1 -L2 -des //
#L1 #L #L2 #d #e #des #_ #_ #H destruct
qed-.
(* Basic_1: was: drop1_gen_pnil *)
-lemma drops_inv_nil: â\88\80L1,L2,s. â\87©*[s, ◊] L1 ≡ L2 → L1 = L2.
+lemma drops_inv_nil: â\88\80L1,L2,s. â¬\87*[s, ◊] L1 ≡ L2 → L1 = L2.
/2 width=4 by drops_inv_nil_aux/ qed-.
-fact drops_inv_cons_aux: â\88\80L1,L2,s,des. â\87©*[s, des] L1 ≡ L2 →
+fact drops_inv_cons_aux: â\88\80L1,L2,s,des. â¬\87*[s, des] L1 ≡ L2 →
∀d,e,tl. des = {d, e} @ tl →
- â\88\83â\88\83L. â\87©*[s, tl] L1 â\89¡ L & â\87©[s, d, e] L ≡ L2.
+ â\88\83â\88\83L. â¬\87*[s, tl] L1 â\89¡ L & â¬\87[s, d, e] L ≡ L2.
#L1 #L2 #s #des * -L1 -L2 -des
[ #L #d #e #tl #H destruct
| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
qed-.
(* Basic_1: was: drop1_gen_pcons *)
-lemma drops_inv_cons: â\88\80L1,L2,s,d,e,des. â\87©*[s, {d, e} @ des] L1 ≡ L2 →
- â\88\83â\88\83L. â\87©*[s, des] L1 â\89¡ L & â\87©[s, d, e] L ≡ L2.
+lemma drops_inv_cons: â\88\80L1,L2,s,d,e,des. â¬\87*[s, {d, e} @ des] L1 ≡ L2 →
+ â\88\83â\88\83L. â¬\87*[s, des] L1 â\89¡ L & â¬\87[s, d, e] L ≡ L2.
/2 width=3 by drops_inv_cons_aux/ qed-.
lemma drops_inv_skip2: ∀I,s,des,des2,i. des ▭ i ≡ des2 →
- â\88\80L1,K2,V2. â\87©*[s, des2] L1 ≡ K2. ⓑ{I} V2 →
+ â\88\80L1,K2,V2. â¬\87*[s, des2] L1 ≡ K2. ⓑ{I} V2 →
∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
- â\87©*[s, des1] K1 ≡ K2 &
- â\87§*[des1] V2 ≡ V1 &
+ â¬\87*[s, des1] K1 ≡ K2 &
+ â¬\86*[des1] V2 ≡ V1 &
L1 = K1. ⓑ{I} V1.
#I #s #des #des2 #i #H elim H -des -des2 -i
[ #i #L1 #K2 #V2 #H
(* Basic properties *********************************************************)
(* Basic_1: was: drop1_skip_bind *)
-lemma drops_skip: â\88\80L1,L2,s,des. â\87©*[s, des] L1 â\89¡ L2 â\86\92 â\88\80V1,V2. â\87§*[des] V2 ≡ V1 →
- â\88\80I. â\87©*[s, des + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
+lemma drops_skip: â\88\80L1,L2,s,des. â¬\87*[s, des] L1 â\89¡ L2 â\86\92 â\88\80V1,V2. â¬\86*[des] V2 ≡ V1 →
+ â\88\80I. â¬\87*[s, des + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
#L1 #L2 #s #des #H elim H -L1 -L2 -des
[ #L #V1 #V2 #HV12 #I
>(lifts_inv_nil … HV12) -HV12 //