+++ /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/reduction/llpr_llpr.ma".
-include "basic_2/computation/cprs_lift.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: pr3_t *)
-(* Basic_1: includes: pr1_t *)
-theorem cprs_trans: ∀G,L. Transitive … (cprs G L).
-normalize /2 width=3 by trans_TC/ qed-.
-
-(* Basic_1: was: pr3_confluence *)
-(* Basic_1: includes: pr1_confluence *)
-theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L).
-normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-.
-
-theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
-/3 width=5 by cprs_trans, cprs_bind_dx/
-qed.
-
-(* Basic_1: was: pr3_flat *)
-theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2.
-#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2
-/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/
-qed.
-
-theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/
-#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *)
-@(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/
-qed.
-
-theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/
-#V #V2 #_ #HV2 #IHV1
-@(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/
-qed.
-
-theorem cprs_theta_rc: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
- ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2
-/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/
-qed.
-
-theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
- ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
-/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was pr3_gen_appl *)
-lemma cprs_inv_appl1: ∀G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 &
- U2 = ⓐV2. T2
- | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
- ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
- | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
- ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
- ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
-#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
-#U #U2 #_ #HU2 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_appl1 … HU2) -HU2 *
- [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/
- | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
- lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12
- lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
- /5 width=5 by cprs_flat_dx, cpr_cprs, cprs_bind, lsubr_abst, ex2_3_intro, or3_intro1/
- | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
- @or3_intro2 @(ex4_5_intro … HV2 HT10) /3 width=3 by cprs_flat_sn, cprs_strap1, cpr_cprs, cprs_bind/ (**) (* full auto is too slow 11s *)
- ]
-| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/
-| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/
-]
-qed-.
-
-(* Properties concerning sn parallel reduction on local environments ********)
-
-(* Basic_1: was just: pr3_pr2_pr2_t *)
-(* Basic_1: includes: pr3_pr0_pr2_t *)
-lemma llpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (llpr G 0).
-#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
-[ /2 width=3 by/
-| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
- elim (llpr_inv_lref_ge_dx … HL12 … HLK2) -L2
- /5 width=7 by cprs_delta, cprs_strap2, llpr_cpr_conf/
-| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (llpr_inv_bind_O … HL12) -HL12 /4 width=1 by cprs_bind/
-| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 /3 width=1 by cprs_flat/
-| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_zeta/
-| #G #L2 #V2 #T1 #T2 #HT12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) /3 width=1 by cprs_tau/
-| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_beta/
-| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
- elim (llpr_inv_flat … HL12) -HL12 #HV1 #HL12
- elim (llpr_inv_bind_O … HL12) /3 width=3 by cprs_theta/
-]
-qed-.
-
-lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=9 by llpr_cpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma cprs_llpr_trans: ∀G. s_rs_transitive … (cpr G) (llpr G 0).
-/3 width=6 by llpr_cpr_trans, llpr_cpr_conf, s_r_trans_LTC1/ qed-.
-
-(* Basic_1: was: pr3_strip *)
-(* Basic_1: includes: pr1_strip *)
-lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
-normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
-
-lemma cprs_llpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #H @(cprs_ind_dx … T0 H) -T0 /2 width=3 by ex2_intro/
-#T0 #T #HT0 #_ #IHT1 #L1 #HL01
-elim (IHT1 … L1) /2 by llpr_cpr_conf/ -IHT1 #T2 #HT12 #HT2
-elim (llpr_cpr_conf_dx … HT0 … HL01) -L0 #T3 #HT03 #HT3
-elim (cprs_strip … HT2 … HT3) -T
-/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
-qed-.
-
-lemma cprs_llpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[T0, 0] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cprs_llpr_conf_dx … HT01 … HL01)
-/4 width=5 by cprs_llpr_trans, cprs_llpr_conf, ex2_intro/
-qed-.
-
-lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=9 by cprs_llpr_trans, cprs_bind_dx, llpr_bind_repl_O/ qed.
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