+(**************************************************************************)
+(* ___ *)
+(* ||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/rt_transition/cpr.ma".
+include "basic_2/rt_computation/fpbg_fqup.ma".
+include "basic_2/dynamic/cnv_fsb.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Inversion lemmas with degree-based equivalence for terms *****************)
+
+lemma cnv_cpr_tdeq_fwd_refl (a) (h) (o) (G) (L):
+ ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → T1 ≛[h,o] T2 →
+ ⦃G, L⦄ ⊢ T1 ![a,h] → T1 = T2.
+#a #h #o #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
+[ //
+| #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
+ lapply (tdeq_inv_lref1 … H1) -H1 #H destruct //
+| #I #G #K #T2 #X2 #i #_ #_ #_ #H1 #_ -a -I -G -K -T2
+ lapply (tdeq_inv_lref1 … H1) -H1 #H destruct //
+| #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
+ elim (tdeq_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
+ elim (cnv_inv_bind … H2) -H2 #HV1 #HT1
+ /3 width=3 by eq_f2/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
+ elim (tdeq_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
+ elim (cnv_fwd_flat … H2) -H2 #HV1 #HT1
+ /3 width=3 by eq_f2/
+| #G #K #V #T1 #X1 #X2 #HXT1 #HX12 #_ #H1 #H2
+ elim (cnv_fpbg_refl_false … o … H2) -a
+ @(fpbg_tdeq_div … H1) -H1
+ /3 width=9 by cpm_tdneq_cpm_fpbg, cpm_zeta, tdeq_lifts_inv_pair_sn/
+| #G #L #U #T1 #T2 #HT12 #_ #H1 #H2
+ elim (cnv_fpbg_refl_false … o … H2) -a
+ @(fpbg_tdeq_div … H1) -H1
+ /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_eps, tdeq_inv_pair_xy_y/
+| #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H1 #_
+ elim (tdeq_inv_pair … H1) -H1 #H #_ #_ destruct
+| #p #G #L #V1 #V2 #X2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H1 #_
+ elim (tdeq_inv_pair … H1) -H1 #H #_ #_ destruct
+]
+qed-.
+
+lemma cpm_tdeq_inv_bind (a) (h) (o) (n) (p) (I) (G) (L):
+ ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
+ ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛[h,o] X →
+ ∃∃T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓑ{p,I}V.T2.
+#a #h #o #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
+elim (cpm_inv_bind1 … H1) -H1 *
+[ #XV #T2 #HXV #HT12 #H destruct
+ elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
+ elim (cnv_inv_bind … H0) -H0 #HV #_
+ lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV -HV
+ /2 width=4 by ex3_intro/
+| #X1 #HXT1 #HX1 #H1 #H destruct
+ elim (cnv_fpbg_refl_false … o … H0) -a
+ @(fpbg_tdeq_div … H2) -H2
+ /3 width=9 by cpm_tdneq_cpm_fpbg, cpm_zeta, tdeq_lifts_inv_pair_sn/
+]
+qed-.
+
+lemma cpm_tdeq_inv_appl (a) (h) (o) (n) (G) (L):
+ ∀V,T1. ⦃G, L⦄ ⊢ ⓐV.T1 ![a,h] →
+ ∀X. ⦃G, L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛[h,o] X →
+ ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓐV.T2.
+#a #h #o #n #G #L #V #T1 #H0 #X #H1 #H2
+elim (cpm_inv_appl1 … H1) -H1 *
+[ #XV #T2 #HXV #HT12 #H destruct
+ elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
+ elim (cnv_inv_appl … H0) -H0 #m #q #W #U #_ #HV #_ #_ #_ -m -q -W -U
+ lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV -HV
+ /2 width=4 by ex3_intro/
+| #q #V2 #W1 #W2 #XT #T2 #_ #_ #_ #H1 #H destruct -H0
+ elim (tdeq_inv_pair … H2) -H2 #H #_ #_ destruct
+| #q #V2 #XV #W1 #W2 #XT #T2 #_ #_ #_ #_ #H1 #H destruct -H0
+ elim (tdeq_inv_pair … H2) -H2 #H #_ #_ destruct
+]
+qed-.
+
+lemma cpm_tdeq_inv_cast (a) (h) (o) (n) (G) (L):
+ ∀U1,T1. ⦃G, L⦄ ⊢ ⓝU1.T1 ![a,h] →
+ ∀X. ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛[h,o] X →
+ ∃∃U2,T2. ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛[h,o] U2 & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓝU2.T2.
+#a #h #o #n #G #L #U1 #T1 #H0 #X #H1 #H2
+elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
+[ #U2 #T2 #HU12 #HT12 #H destruct -H0
+ elim (tdeq_inv_pair … H2) -H2 #_ #H2U12 #H2T12
+ /2 width=7 by ex5_2_intro/
+| #HT1X
+ elim (cnv_fpbg_refl_false … o … H0) -a
+ @(fpbg_tdeq_div … H2) -H2
+ /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_eps, tdeq_inv_pair_xy_y/
+| #m #HU1X #H destruct
+ elim (cnv_fpbg_refl_false … o … H0) -a
+ @(fpbg_tdeq_div … H2) -H2
+ /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_ee, tdeq_inv_pair_xy_x/
+]
+qed-.
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