+++ /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 "static_2/static/rdeq_drops.ma".
-include "static_2/static/rdeq_fqup.ma".
-include "static_2/static/rdeq_rdeq.ma".
-include "basic_2/rt_transition/rpx_fsle.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-
-(* Properties with sort-irrelevant equivalence for local environments *******)
-
-lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 → ∀I,T.
- ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h,②{I}V.T] L & L ≛[V] L2.
-/3 width=5 by rpx_fsge_comp, rex_pair_sn_split/ qed-.
-
-lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → ∀I,V.
- ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L & L ≛[T] L2.
-/3 width=5 by rpx_fsge_comp, rex_flat_dx_split/ qed-.
-
-lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L2 → ∀p.
- ∃∃L,V. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[T] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V.
-/3 width=5 by rpx_fsge_comp, rex_bind_dx_split/ qed-.
-
-lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G,K1.ⓧ⦄ ⊢ ⬈[h,T] L2 → ∀p,I,V.
- ∃∃K2. ⦃G,K1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[T] L2.
-/3 width=5 by rpx_fsge_comp, rex_bind_dx_split_void/ qed-.
-
-lemma rpx_tdeq_conf: ∀h,G. s_r_confluent1 … cdeq (rpx h G).
-/2 width=5 by tdeq_rex_conf/ qed-.
-
-lemma rpx_tdeq_div: ∀h,T1,T2. T1 ≛ T2 →
- ∀G,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,T2] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T1] L2.
-/2 width=5 by tdeq_rex_div/ qed-.
-
-lemma cpx_tdeq_conf_rex: ∀h,G. R_confluent2_rex … (cpx h G) cdeq (cpx h G) cdeq.
-#h #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
-[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
- elim (tdeq_inv_sort1 … H0) -H0 #s1 #H destruct
- /3 width=3 by tdeq_sort, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(tdeq_inv_lref1 … H0) -H0
- elim (rpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
- elim (rdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
- elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
- elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
-| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(tdeq_inv_lref1 … H0) -H0
- elim (rpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
- elim (rdeq_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
- elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
- elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
-| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (rpx_inv_bind … H1) -H1 #HL01 #H1
- elim (rdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (rdeq_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -T0
- /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
-| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #HL01 #H1
- elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
-| #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #U2 #HV02 #HU02 #H destruct
- elim (rpx_inv_bind … H1) -H1 #HL01 #H1
- elim (rdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (rpx_inv_lifts_bi … H1 (Ⓣ) … HTU0) -H1 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H1
- lapply (rdeq_inv_lifts_bi … H2 (Ⓣ) … HTU0) -H2 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H2
- elim (tdeq_inv_lifts_sn … HU02 … HTU0) -U0 #T2 #HTU2 #HT02
- elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
- /3 width=5 by cpx_zeta, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #HL01 #H1
- elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=3 by cpx_eps, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
- elim (rpx_inv_flat … H1) -H1 #HL01 #H1
- elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
- /3 width=3 by cpx_ee, ex2_intro/
-| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
- elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
- elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
- elim (rdeq_inv_flat … H2) -H2 #H2LV0 #H2
- elim (rdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (rdeq_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
- elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
- elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
- elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
-| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
- elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
- elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
- elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
- elim (rdeq_inv_flat … H2) -H2 #H2LV0 #H2
- elim (rdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (rdeq_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
- elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
- elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
- elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- elim (tdeq_lifts_sn … HV1 … HVU1) -V1
- /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
-]
-qed-.
-
-lemma cpx_tdeq_conf: ∀h,G,L. ∀T0:term. ∀T1. ⦃G,L⦄ ⊢ T0 ⬈[h] T1 →
- ∀T2. T0 ≛ T2 →
- ∃∃T. T1 ≛ T & ⦃G,L⦄ ⊢ T2 ⬈[h] T.
-#h #G #L #T0 #T1 #HT01 #T2 #HT02
-elim (cpx_tdeq_conf_rex … HT01 … HT02 L … L) -HT01 -HT02
-/2 width=3 by rex_refl, ex2_intro/
-qed-.
-
-lemma tdeq_cpx_trans: ∀h,G,L,T2. ∀T0:term. T2 ≛ T0 →
- ∀T1. ⦃G,L⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G,L⦄ ⊢ T2 ⬈[h] T & T ≛ T1.
-#h #G #L #T2 #T0 #HT20 #T1 #HT01
-elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: cpx_lleq_conf *)
-lemma cpx_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. L0 ≛[T0] L2 →
- ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈[h] T & T1 ≛ T.
-#h #G #L0 #T0 #T1 #HT01 #L2 #HL02
-elim (cpx_tdeq_conf_rex … HT01 T0 … L0 … HL02) -HT01 -HL02
-/2 width=3 by rex_refl, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: lleq_cpx_trans *)
-lemma rdeq_cpx_trans: ∀h,G,L2,L0,T0. L2 ≛[T0] L0 →
- ∀T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈[h] T & T ≛ T1.
-#h #G #L2 #L0 #T0 #HL20 #T1 #HT01
-elim (cpx_rdeq_conf … HT01 L2) -HT01
-/3 width=3 by rdeq_sym, tdeq_sym, ex2_intro/
-qed-.
-
-lemma rpx_rdeq_conf: ∀h,G,T. confluent2 … (rpx h G T) (rdeq T).
-/3 width=6 by rpx_fsge_comp, rdeq_fsge_comp, cpx_tdeq_conf_rex, rex_conf/ qed-.
-
-lemma rdeq_rpx_trans: ∀h,G,T,L2,K2. ⦃G,L2⦄ ⊢ ⬈[h,T] K2 →
- ∀L1. L1 ≛[T] L2 →
- ∃∃K1. ⦃G,L1⦄ ⊢ ⬈[h,T] K1 & K1 ≛[T] K2.
-#h #G #T #L2 #K2 #HLK2 #L1 #HL12
-elim (rpx_rdeq_conf … HLK2 L1)
-/3 width=3 by rdeq_sym, ex2_intro/
-qed-.
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