+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/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 *********************************************************)
-
-(* Note: it does not hold replacing |L1| with |L2| *)
-lemma cpqs_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ➤* T2 →
- ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ➤* T2.
-#L1 #T1 #T2 #H elim H -L1 -T1 -T2
-[ //
-| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- lapply (ldrop_fwd_ldrop2_length … HLK1) #Hi
- lapply (ldrop_fwd_O1_length … HLK1) #H2i
- elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // -Hi
- <H2i -H2i <minus_plus_m_m /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_ldrop2_length … 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-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/ldrop_ldrop.ma".
-include "basic_2/restricted/cpqs.ma".
-
-(* CONTEXT-SENSITIVE RESTRICTED PARALLEL COMPUTATION FOR TERMS **************)
-
-(* Relocation properties ****************************************************)
-
-lemma cpqs_lift: l_liftable cpqs.
-#K #T1 #T2 #H elim H -K -T1 -T2
-[ #I #K #L #d #e #_ #U1 #H1 #U2 #H2
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
- elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
- elim (ldrop_trans_le … HLK … HKV) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K #Y #HKV #HVY #H destruct /3 width=8/
- | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=6/
- ]
-| #a #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5/
-| #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
-| #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #H #U2 #HTU2
- elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
- elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=5/
-| #K #V #T1 #T2 #_ #IHT12 #L #d #e #HLK #U1 #H #U2 #HTU2
- elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5/
-]
-qed.
-
-lemma cpqs_inv_lift1: l_deliftable_sn cpqs.
-#L #U1 #U2 #H elim H -L -U1 -U2
-[ * #L #i #K #d #e #_ #T1 #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
- elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
- elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
- | elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
- elim (lift_split … HVW2 d (i - e + 1)) -HVW2 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O /3 width=8/
- ]
-| #a #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
- elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5/
-| #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1) -V1
- elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5/
-| #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU1 (K.ⓓW1) … HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
- elim (lift_div_le … HU2 … HTU) -U // /3 width=5/
-| #L #V #U1 #U2 #_ #IHU12 #K #d #e #HLK #X #H
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3/
-]
-qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/lpss.ma".
-include "basic_2/restricted/cpqs.ma".
-
-(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
-
-definition lpqs: relation lenv ≝ lpx_sn cpqs.
-
-interpretation "restricted parallel computation (local environment, sn variant)"
- 'PRestStarSn L1 L2 = (lpqs L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpqs_inv_atom1: ∀L2. ⋆ ⊢ ➤* L2 → L2 = ⋆.
-/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-
-lemma lpqs_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ➤* L2 →
- ∃∃K2,V2. K1 ⊢ ➤* K2 & K1 ⊢ V1 ➤* V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-
-lemma lpqs_inv_atom2: ∀L1. L1 ⊢ ➤* ⋆ → L1 = ⋆.
-/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-
-lemma lpqs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ➤* K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ⊢ ➤* K2 & K1 ⊢ V1 ➤* V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpqs_refl: ∀L. L ⊢ ➤* L.
-/2 width=1 by lpx_sn_refl/ qed.
-
-lemma lpqs_append: ∀K1,K2. K1 ⊢ ➤* K2 → ∀L1,L2. L1 ⊢ ➤* L2 →
- L1 @@ K1 ⊢ ➤* L2 @@ K2.
-/3 width=1 by lpx_sn_append, cpqs_append/ qed.
-
-lemma lpss_lpqs: ∀L1,L2. L1 ⊢ ▶* L2 → L1 ⊢ ➤* L2.
-#L1 #L2 #H elim H -L1 -L2 // /3 width=1/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpqs_fwd_length: ∀L1,L2. L1 ⊢ ➤* L2 → |L1| = |L2|.
-/2 width=2 by lpx_sn_fwd_length/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpqs_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ➤* L →
- ∃∃K2,L2. K1 ⊢ ➤* K2 & L = K2 @@ L2.
-/2 width=2 by lpx_sn_fwd_append1/ qed-.
-
-lemma lpqs_fwd_append2: ∀L,K2,L2. L ⊢ ➤* K2 @@ L2 →
- ∃∃K1,L1. K1 ⊢ ➤* K2 & L = K1 @@ L1.
-/2 width=2 by lpx_sn_fwd_append2/ qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/fsup.ma".
-include "basic_2/restricted/lpqs_ldrop.ma".
-
-(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
-
-(* Main properties on context-sensitive rest parallel computation for terms *)
-
-fact cpqs_conf_lpqs_atom_atom:
- ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➤* T & L2 ⊢ ⓪{I} ➤* T.
-/2 width=3/ qed-.
-
-fact cpqs_conf_lpqs_atom_delta:
- ∀L0,i. (
- ∀L,T.♯{L, T} < ♯{L0, #i} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V2. K0 ⊢ V0 ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ #i ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpqs_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
-elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
-qed-.
-
-fact cpqs_conf_lpqs_delta_delta:
- ∀L0,i. (
- ∀L,T.♯{L, T} < ♯{L0, #i} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
- ∀V1. K0 ⊢ V0 ➤* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
- ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
- ∀V2. KX ⊢ VX ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
-#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
-elim (lpqs_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
-elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
-elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpqs_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
-lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_bind_bind:
- ∀a,I,L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➤* T1 →
- ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➤* T & L2 ⊢ ⓑ{a,I}V2.T2 ➤* T.
-#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
-qed-.
-
-fact cpqs_conf_lpqs_bind_zeta:
- ∀L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 →
- ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ +ⓓV1.T1 ➤* T & L2 ⊢ X2 ➤* T.
-#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_zeta_zeta:
- ∀L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
- ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ X1 ➤* T & L2 ⊢ X2 ➤* T.
-#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
-#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2
-elim (cpqs_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1
-elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2
-lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_flat_flat:
- ∀I,L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0 ⊢ T0 ➤* T1 →
- ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➤* T & L2 ⊢ ⓕ{I}V2.T2 ➤* T.
-#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) //
-elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
-qed-.
-
-fact cpqs_conf_lpqs_flat_tau:
- ∀L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀V1,T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ ⓝV1.T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #V0 #T0 #IH #V1 #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
-qed-.
-
-fact cpqs_conf_lpqs_tau_tau:
- ∀L0,V0,T0. (
- ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
- ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
- ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
- ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
- ) →
- ∀T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
- ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
- ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
-#L0 #V0 #T0 #IH #T1 #HT01
-#T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
-qed-.
-
-theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs cpqs.
-#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
-[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_atom1 … H1) -H1
- elim (cpqs_inv_atom1 … H2) -H2
- [ #H2 #H1 destruct
- /2 width=1 by cpqs_conf_lpqs_atom_atom/
- | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
- /3 width=10 by cpqs_conf_lpqs_atom_delta/
- | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
- /4 width=10 by ex2_commute, cpqs_conf_lpqs_atom_delta/
- | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
- * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
- /3 width=17 by cpqs_conf_lpqs_delta_delta/
- ]
-| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_bind1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #T1 #HT01 #HXT1 #H11 #H12
- ]
- elim (cpqs_inv_bind1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #T2 #HT02 #HXT2 #H21 #H22
- ] destruct
- [ /3 width=10 by cpqs_conf_lpqs_bind_bind/
- | /4 width=11 by ex2_commute, cpqs_conf_lpqs_bind_zeta/
- | /3 width=11 by cpqs_conf_lpqs_bind_zeta/
- | /3 width=12 by cpqs_conf_lpqs_zeta_zeta/
- ]
-| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
- elim (cpqs_inv_flat1 … H1) -H1 *
- [ #V1 #T1 #HV01 #HT01 #H1
- | #HX1 #H1
- ]
- elim (cpqs_inv_flat1 … H2) -H2 *
- [1,3: #V2 #T2 #HV02 #HT02 #H2
- |2,4: #HX2 #H2
- ] destruct
- [ /3 width=10 by cpqs_conf_lpqs_flat_flat/
- | /4 width=8 by ex2_commute, cpqs_conf_lpqs_flat_tau/
- | /3 width=8 by cpqs_conf_lpqs_flat_tau/
- | /3 width=7 by cpqs_conf_lpqs_tau_tau/
- ]
-]
-qed-.
-
-theorem cpqs_conf: ∀L. confluent … (cpqs L).
-/2 width=6 by cpqs_conf_lpqs/ qed-.
-
-theorem cpqs_trans_lpqs: lpx_sn_transitive cpqs cpqs.
-#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * [|*]
-[ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct
- elim (cpqs_inv_atom1 … H1) -H1
- [ #H destruct
- elim (cpqs_inv_atom1 … HT2) -HT2
- [ #H destruct //
- | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
- elim (lpqs_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpqs_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) /3 width=9/
- ]
- | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
- elim (lpqs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpqs_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
- elim (cpqs_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) /3 width=9/
- ]
-| #a #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpqs_inv_bind1 … H1) -H1 *
- [ #V #T #HV1 #HT1 #H destruct
- elim (cpqs_inv_bind1 … H2) -H2 *
- [ #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
- | #T2 #HT2 #HXT2 #H1 #H2 destruct /4 width=5/
- ]
- | #Y1 #HTY1 #HXY1 #H11 #H12 destruct
- elim (lift_total X2 0 1) #Y2 #HXY2
- lapply (cpqs_lift … H2 (L2.ⓓV1) … HXY1 … HXY2) /2 width=1/ -X1 /4 width=5/
- ]
-| #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpqs_inv_flat1 … H1) -H1 *
- [ #V #T #HV1 #HT1 #H destruct
- elim (cpqs_inv_flat1 … H2) -H2 *
- [ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
- | #HX2 #H destruct /3 width=5/
- ]
- | #HX1 #H destruct /3 width=5/
-]
-qed-.
-
-theorem cpqs_trans: ∀L. Transitive … (cpqs L).
-/2 width=5 by cpqs_trans_lpqs/ qed-.
-
-(* Properties on context-sensitive rest. parallel computation for terms *****)
-
-lemma lpqs_cpqs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
- ∃∃T. L1 ⊢ T0 ➤* T & L1 ⊢ T1 ➤* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpqs_conf_lpqs … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
-qed-.
-
-lemma lpqs_cpqs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
- ∃∃T. L1 ⊢ T0 ➤* T & L0 ⊢ T1 ➤* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpqs_conf_lpqs … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
-qed-.
-
-lemma lpqs_cpqs_trans: ∀L1,L2. L1 ⊢ ➤* L2 →
- ∀T1,T2. L2 ⊢ T1 ➤* T2 → L1 ⊢ T1 ➤* T2.
-/2 width=5 by cpqs_trans_lpqs/ qed-.
-
-lemma fsup_cpqs_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➤* U2 →
- ∃∃L,U1. L1 ⊢ ➤* L & L ⊢ T1 ➤* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
-elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-elim (ldrop_lpqs_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
-lapply (cpqs_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
-qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/ldrop_lpx_sn.ma".
-include "basic_2/restricted/cpqs_lift.ma".
-include "basic_2/restricted/lpqs.ma".
-
-(* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
-
-(* Properies on local environment slicing ***********************************)
-
-lemma lpqs_ldrop_conf: dropable_sn lpqs.
-/3 width=5 by lpx_sn_deliftable_dropable, cpqs_inv_lift1/ qed-.
-
-lemma ldrop_lpqs_trans: dedropable_sn lpqs.
-/3 width=9 by lpx_sn_liftable_dedropable, cpqs_lift/ qed-.
-
-lemma lpqs_ldrop_trans_O1: dropable_dx lpqs.
-/2 width=3 by lpx_sn_dropable/ qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/restricted/lpqs_cpqs.ma".
-
-(* SN RESTRICTED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *****************)
-
-(* Main properties **********************************************************)
-
-theorem lpqs_conf: confluent … lpqs.
-/3 width=6 by lpx_sn_conf, cpqs_conf_lpqs/
-qed-.
-
-theorem lpqs_trans: Transitive … lpqs.
-/3 width=5 by lpx_sn_trans, cpqs_trans_lpqs/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T →
- ∃∃L2,T2. L @@ L1 ⊢ ➤* L @@ L2 & L @@ L1 ⊢ T1 ➤* T2 &
- T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1
-[ #L #T1 #T #HT1
- @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *)
-| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
- elim (cpqs_inv_bind1 … H) -H *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
- lapply (lpqs_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
- @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
-
-(* Note: includes: stass_refl *)
-definition sstas: ∀h. sd h → lenv → relation term ≝
- λh,g,L. star … (ssta_step h g L).
-
-interpretation "iterated stratified static type assignment (term)"
- 'StaticTypeStar h g L T U = (sstas h g L T U).
-
-(* Basic eliminators ********************************************************)
-
-lemma sstas_ind: ∀h,g,L,T. ∀R:predicate term.
- R T → (
- ∀U1,U2,l. ⦃h, L⦄ ⊢ T •* [g] U1 → ⦃h, L⦄ ⊢ U1 •[g] ⦃l+1, U2⦄ →
- R U1 → R U2
- ) →
- ∀U. ⦃h, L⦄ ⊢ T •*[g] U → R U.
-#h #g #L #T #R #IH1 #IH2 #U #H elim H -U //
-#U1 #U2 #H * /2 width=5/
-qed-.
-
-lemma sstas_ind_dx: ∀h,g,L,U2. ∀R:predicate term.
- R U2 → (
- ∀T,U1,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U1⦄ → ⦃h, L⦄ ⊢ U1 •* [g] U2 →
- R U1 → R T
- ) →
- ∀T. ⦃h, L⦄ ⊢ T •*[g] U2 → R T.
-#h #g #L #U2 #R #IH1 #IH2 #T #H @(star_ind_l … T H) -T //
-#T #T0 * /2 width=5/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma sstas_refl: ∀h,g,L. reflexive … (sstas h g L).
-// qed.
-
-lemma ssta_sstas: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ → ⦃h, L⦄ ⊢ T •*[g] U.
-/3 width=2 by R_to_star, ex_intro/ qed. (**) (* auto fails without trace *)
-
-lemma sstas_strap1: ∀h,g,L,T1,T2,U2,l. ⦃h, L⦄ ⊢ T1 •*[g] T2 → ⦃h, L⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
- ⦃h, L⦄ ⊢ T1 •*[g] U2.
-/3 width=4 by sstep, ex_intro/ (**) (* auto fails without trace *)
-qed.
-
-lemma sstas_strap2: ∀h,g,L,T1,U1,U2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ⦃h, L⦄ ⊢ U1 •*[g] U2 →
- ⦃h, L⦄ ⊢ T1 •*[g] U2.
-/3 width=3 by star_compl, ex_intro/ (**) (* auto fails without trace *)
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma sstas_inv_bind1: ∀h,g,a,I,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •*[g] U →
- ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •*[g] Z & U = ⓑ{a,I}Y.Z.
-#h #g #a #I #L #Y #X #U #H @(sstas_ind … H) -U /2 width=3/
-#T #U #l #_ #HTU * #Z #HXZ #H destruct
-elim (ssta_inv_bind1 … HTU) -HTU #Z0 #HZ0 #H destruct /3 width=4/
-qed-.
-
-lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X •*[g] U →
- ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
-#h #g #L #Y #X #U #H @(sstas_ind … H) -U /2 width=3/
-#T #U #l #_ #HTU * #Z #HXZ #H destruct
-elim (ssta_inv_appl1 … HTU) -HTU #Z0 #HZ0 #H destruct /3 width=4/
-qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_aaa.ma".
-include "basic_2/unwind/sstas.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-lemma sstas_aaa: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
- ∀A. L ⊢ T ⁝ A → L ⊢ U ⁝ A.
-#h #g #L #T #U #H @(sstas_ind_dx … H) -T // /3 width=6/
-qed.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_lift.ma".
-include "basic_2/unwind/sstas.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
-
-(* Advanced forward lemmas **************************************************)
-
-lemma sstas_fwd_correct: ∀h,g,L,T1,U1,l1. ⦃h, L⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
- ∀T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 →
- ∃∃U2,l2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l2, U2⦄.
-#h #g #L #T1 #U1 #l1 #HTU1 #T2 #H @(sstas_ind … H) -T2 [ /2 width=3/ ] -HTU1
-#T #T2 #l #_ #HT2 * #U #l0 #_ -l0
-elim (ssta_fwd_correct … HT2) -T /2 width=3/
-qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma sstas_lift: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
- ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 •*[g] U2.
-#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1
-[ #L2 #d #e #HL21 #X #HX #U2 #HU12
- >(lift_mono … HX … HU12) -X //
-| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL21 #T2 #HT02 #U2 #HU12
- elim (lift_total U0 d e) /3 width=10/
-]
-qed.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma sstas_inv_lift1: ∀h,g,L2,T2,U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 →
- ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
- ∃∃U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 & ⇧[d, e] U1 ≡ U2.
-#h #g #L2 #T2 #U2 #H @(sstas_ind_dx … H) -T2 /2 width=3/
-#T0 #U0 #l0 #HTU0 #_ #IHU01 #L1 #d #e #HL21 #U1 #HU12
-elim (ssta_inv_lift1 … HTU0 … HL21 … HU12) -HTU0 -HU12 #U #HU1 #HU0
-elim (IHU01 … HL21 … HU0) -IHU01 -HL21 -U0 /3 width=4/
-qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_lpss.ma".
-include "basic_2/unwind/sstas.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
-
-(* Properties about sn parallel unfold **************************************)
-
-lemma sstas_tpss_lpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
- ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ▶* U2.
-#h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1 /2 width=3/
-#T0 #U0 #l0 #HTU0 #_ #IHU01 #T #HT0 #L2 #HL12
-elim (ssta_tpss_lpss_conf … HTU0 … HT0 … HL12) -HTU0 -HT0 #U #HTU #HU0
-elim (IHU01 … HU0 … HL12) -IHU01 -U0 -HL12 /3 width=4/
-qed-.
-
-lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
- ∀T2. L ⊢ T1 ▶* T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* U2.
-/2 width=3 by sstas_tpss_lpss_conf/ qed-.
-
-lemma sstas_lpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
- ∀L2. L1 ⊢ ▶* L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L1 ⊢ U1 ▶* U2.
-/2 width=3 by sstas_tpss_lpss_conf/ qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_ssta.ma".
-include "basic_2/unwind/sstas.ma".
-
-(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma sstas_inv_O: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
- ∀T0. ⦃h, L⦄ ⊢ T •[g] ⦃0, T0⦄ → U = T.
-#h #g #L #T #U #H @(sstas_ind_dx … H) -T //
-#T0 #U0 #l0 #HTU0 #_ #_ #T1 #HT01
-elim (ssta_mono … HTU0 … HT01) <plus_n_Sm #H destruct
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma sstas_strip: ∀h,g,L,T,U1. ⦃h, L⦄ ⊢ T •*[g] U1 →
- ∀U2,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U2⦄ →
- T = U1 ∨ ⦃h, L⦄ ⊢ U2 •*[g] U1.
-#h #g #L #T #U1 #H1 @(sstas_ind_dx … H1) -T /2 width=1/
-#T #U #l0 #HTU #HU1 #_ #U2 #l #H2
-elim (ssta_mono … H2 … HTU) -H2 -HTU #H1 #H2 destruct /2 width=1/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem sstas_trans: ∀h,g,L,T1,U. ⦃h, L⦄ ⊢ T1 •*[g] U →
- ∀T2. ⦃h, L⦄ ⊢ U •*[g] T2 → ⦃h, L⦄ ⊢ T1 •*[g] T2.
-/2 width=3/ qed-.
-
-theorem sstas_conf: ∀h,g,L,T,U1. ⦃h, L⦄ ⊢ T •*[g] U1 →
- ∀U2. ⦃h, L⦄ ⊢ T •*[g] U2 →
- ⦃h, L⦄ ⊢ U1 •*[g] U2 ∨ ⦃h, L⦄ ⊢ U2 •*[g] U1.
-#h #g #L #T #U1 #H1 @(sstas_ind_dx … H1) -T /2 width=1/
-#T #U #l #HTU #HU1 #IHU1 #U2 #H2
-elim (sstas_strip … H2 … HTU) #H destruct
-[ -H2 -IHU1 /3 width=4/
-| -T /2 width=1/
-]
-qed-.
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