+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 ➤ * break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PRestStar $L $T1 $T2 }.
-
-include "basic_2/substitution/cpss.ma".
-
-(* CONTEXT-SENSITIVE RESTRICTED PARALLEL COMPUTATION FOR TERMS **************)
-
-inductive cpqs: lenv → relation term ≝
-| cpqs_atom : ∀I,L. cpqs L (⓪{I}) (⓪{I})
-| cpqs_delta: ∀L,K,V,V2,W2,i.
- ⇩[0, i] L ≡ K. ⓓV → cpqs K V V2 →
- ⇧[0, i + 1] V2 ≡ W2 → cpqs L (#i) W2
-| cpqs_bind : ∀a,I,L,V1,V2,T1,T2.
- cpqs L V1 V2 → cpqs (L. ⓑ{I} V1) T1 T2 →
- cpqs L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| cpqs_flat : ∀I,L,V1,V2,T1,T2.
- cpqs L V1 V2 → cpqs L T1 T2 →
- cpqs L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-| cpqs_zeta : ∀L,V,T1,T,T2. cpqs (L.ⓓV) T1 T →
- ⇧[0, 1] T2 ≡ T → cpqs L (+ⓓV. T1) T2
-| cpqs_tau : ∀L,V,T1,T2. cpqs L T1 T2 → cpqs L (ⓝV. T1) T2
-.
-
-interpretation "context-sensitive restricted parallel computation (term)"
- 'PRestStar L T1 T2 = (cpqs L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpqs_lsubr_trans: lsub_trans … cpqs lsubr.
-#L1 #T1 #T2 #H elim H -L1 -T1 -T2
-[ //
-| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
-| /4 width=1/
-|4,6: /3 width=1/
-| /4 width=3/
-]
-qed-.
-
-lemma cpss_cpqs: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➤* T2.
-#L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
-qed.
-
-lemma cpqs_refl: ∀T,L. L ⊢ T ➤* T.
-/2 width=1/ qed.
-
-lemma cpqs_delift: ∀L,K,V,T1,d. ⇩[0, d] L ≡ (K. ⓓV) →
- ∃∃T2,T. L ⊢ T1 ➤* T2 & ⇧[d, 1] T ≡ T2.
-#L #K #V #T1 #d #HLK
-elim (cpss_delift … T1 … HLK) -HLK /3 width=4/
-qed-.
-
-lemma cpqs_append: l_appendable_sn … cpqs.
-#K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/
-#K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpqs_delta … (L@@K0) V1 … HVW2) //
-@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cpqs_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➤* T2 → ∀I. T1 = ⓪{I} →
- T2 = ⓪{I} ∨
- ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ➤* V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
- I = LRef i.
-#L #T1 #T2 * -L -T1 -T2
-[ #I #L #J #H destruct /2 width=1/
-| #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #L #V #T1 #T #T2 #_ #_ #J #H destruct
-| #L #V #T1 #T2 #_ #J #H destruct
-]
-qed-.
-
-lemma cpqs_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➤* T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ➤* V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
- I = LRef i.
-/2 width=3 by cpqs_inv_atom1_aux/ qed-.
-
-lemma cpqs_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➤* T2 → T2 = ⋆k.
-#L #T2 #k #H
-elim (cpqs_inv_atom1 … H) -H //
-* #K #V #V2 #i #_ #_ #_ #H destruct
-qed-.
-
-lemma cpqs_inv_lref1: ∀L,T2,i. L ⊢ #i ➤* T2 →
- T2 = #i ∨
- ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
- K ⊢ V ➤* V2 &
- ⇧[O, i + 1] V2 ≡ T2.
-#L #T2 #i #H
-elim (cpqs_inv_atom1 … H) -H /2 width=1/
-* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
-qed-.
-
-lemma cpqs_inv_gref1: ∀L,T2,p. L ⊢ §p ➤* T2 → T2 = §p.
-#L #T2 #p #H
-elim (cpqs_inv_atom1 … H) -H //
-* #K #V #V2 #i #_ #_ #_ #H destruct
-qed-.
-
-fact cpqs_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➤* U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 &
- L. ⓑ{I} V1 ⊢ T1 ➤* T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
-#L #U1 #U2 * -L -U1 -U2
-[ #I #L #b #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /3 width=5/
-| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
-| #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W1 #U1 #H destruct /3 width=3/
-| #L #V #T1 #T2 #_ #b #J #W1 #U1 #H destruct
-]
-qed-.
-
-lemma cpqs_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ➤* U2 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 &
- L. ⓑ{I} V1 ⊢ T1 ➤* T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
-/2 width=3 by cpqs_inv_bind1_aux/ qed-.
-
-lemma cpqs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a} V1. T1 ➤* U2 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 &
- L. ⓓ V1 ⊢ T1 ➤* T2 &
- U2 = ⓓ{a} V2. T2
- ) ∨
- ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true.
-#a #L #V1 #T1 #U2 #H
-elim (cpqs_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
-qed-.
-
-lemma cpqs_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a} V1. T1 ➤* U2 →
- ∃∃V2,T2. L ⊢ V1 ➤* V2 &
- L. ⓛ V1 ⊢ T1 ➤* T2 &
- U2 = ⓛ{a} V2. T2.
-#a #L #V1 #T1 #U2 #H
-elim (cpqs_inv_bind1 … H) -H *
-[ /3 width=5/
-| #T #_ #_ #_ #H destruct
-]
-qed-.
-
-fact cpqs_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ➤* U2 →
- ∀I,V1,T1. U1 = ⓕ{I} V1. T1 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
- U2 = ⓕ{I} V2. T2
- ) ∨
- (L ⊢ T1 ➤* U2 ∧ I = Cast).
-#L #U1 #U2 * -L -U1 -U2
-[ #I #L #J #W1 #U1 #H destruct
-| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
-| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
-| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=5/
-| #L #V #T1 #T #T2 #_ #_ #J #W1 #U1 #H destruct
-| #L #V #T1 #T2 #HT12 #J #W1 #U1 #H destruct /3 width=1/
-]
-qed-.
-
-lemma cpqs_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ➤* U2 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
- U2 = ⓕ{I} V2. T2
- ) ∨
- (L ⊢ T1 ➤* U2 ∧ I = Cast).
-/2 width=3 by cpqs_inv_flat1_aux/ qed-.
-
-lemma cpqs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐ V1. T1 ➤* U2 →
- ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
- U2 = ⓐ V2. T2.
-#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 … H) -H *
-[ /3 width=5/
-| #_ #H destruct
-]
-qed-.
-
-lemma cpqs_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝ V1. T1 ➤* U2 → (
- ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
- U2 = ⓝ V2. T2
- ) ∨
- L ⊢ T1 ➤* U2.
-#L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 … H) -H * /2 width=1/ /3 width=5/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X
- >shift_append_assoc normalize #H
- elim (cpqs_inv_bind1 … H) -H *
- [ #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
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