+++ /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/rt_computation/cpme_aaa.ma".
-include "basic_2/rt_computation/cpre_cpre.ma".
-include "basic_2/rt_equivalence/cpes.ma".
-include "basic_2/dynamic/cnv_cpme.ma".
-
-(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-
-(* Properties with t-bound rt-equivalence for terms *************************)
-
-lemma cnv_cpes_dec (a) (h) (n1) (n2) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∀T2. ⦃G,L⦄ ⊢ T2 ![a,h] →
- Decidable … (⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2).
-#a #h #n1 #n2 #G #L #T1 #HT1 #T2 #HT2
-elim (cnv_fwd_aaa … HT1) #A1 #HA1
-elim (cnv_fwd_aaa … HT2) #A2 #HA2
-elim (cpme_total_aaa h n1 … HA1) -HA1 #U1 #HTU1
-elim (cpme_total_aaa h n2 … HA2) -HA2 #U2 #HTU2
-elim (eq_term_dec U1 U2) [ #H destruct | #HnU12 ]
-[ cases HTU1 -HTU1 #HTU1 #_
- cases HTU2 -HTU2 #HTU2 #_
- /3 width=3 by cpms_div, or_introl/
-| @or_intror * #T0 #HT10 #HT20
- lapply (cnv_cpme_cpms_conf … HT1 … HT10 … HTU1) -T1 #H1
- lapply (cnv_cpme_cpms_conf … HT2 … HT20 … HTU2) -T2 #H2
- /3 width=6 by cpre_mono/
-]
-qed-.
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
#a #h #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #HL1 #L2 #HL2
-lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
+lapply (cnv_inv_lref_pair … HT … HLK) -HT #HV
elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #HV1 #H destruct
elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[↑n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
#a #h #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #HL1 #L2 #HL2
-lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HW
+lapply (cnv_inv_lref_pair … HT … HLK) -HT #HW
elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK1 #HW1 #H destruct
elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
#K #V #HLK #Y #X #HLY #n1 #XV1 #HVX1 #n2 #XV2 #HVX2 #X1 #HXV1 #X2 #HXV2
#L1 #HL1 #L2 #HL2
lapply (drops_mono … HLY … HLK) -HLY #H destruct
-lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
+lapply (cnv_inv_lref_pair … HT … HLK) -HT #HV
elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #_ #H destruct
lapply (drops_isuni_fwd_drop2 … HLK1) -V1 // #HLK1
(* Properties with evaluation for t-bound rt-transition on terms ************)
+lemma cnv_cpme_trans (a) (h) (n) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T2 ![a,h].
+#a #h #n #G #L #T1 #HT1 #T2 * #HT12 #_
+/2 width=4 by cnv_cpms_trans/
+qed-.
+
lemma cnv_cpme_cpms_conf (a) (h) (n) (G) (L):
∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∀T1. ⦃G,L⦄ ⊢ T ➡*[n,h] T1 →
∀T2. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
--- /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/rt_computation/cpms_cnu.ma".
+include "basic_2/rt_computation/cpue.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with evaluation for t-unbound rt-transition on terms **********)
+
+lemma cnv_cpue_trans (a) (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T2 ![a,h].
+#a #h #G #L #T1 #HT1 #T2 * #n #HT12 #_
+/2 width=4 by cnv_cpms_trans/
+qed-.
+
+lemma cnv_cpue_cpms_conf (a) (h) (n) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] → ∀T1. ⦃G,L⦄ ⊢ T0 ➡*[n,h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T0 ⥲*[h] 𝐍⦃T2⦄ →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T⦄ & T2 ≅ T.
+#a #h #n1 #G #L #T0 #HT0 #T1 #HT01 #T2 * #n2 #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
+lapply (cpms_inv_cnu_sn … HT20 HT2) -HT20 #HT20
+/4 width=8 by cnu_tueq_trans, ex2_intro/
+qed-.
+
+(* Main properties with evaluation for t-unbound rt-transition on terms *****)
+
+theorem cnv_cpue_mono (a) (h) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] → ∀T1. ⦃G,L⦄ ⊢ T0 ⥲*[h] 𝐍⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T0 ⥲*[h] 𝐍⦃T2⦄ → T1 ≅ T2.
+#a #h #G #L #T0 #HT0 #T1 * #n1 #HT01 #HT1 #T2 * #n2 #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
+/3 width=8 by cpms_inv_cnu_sn, tueq_canc_dx/
+qed-.
]
qed-.
-(* Advanced forward lemmas **************************************************)
-
-lemma cnv_lref_fwd_drops (a) (h) (G):
- ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
- ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ![a, h].
-#a #h #o #i #L #H #I #K #V #HLK
+lemma cnv_inv_lref_pair (a) (h) (G):
+ ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
+ ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ![a, h].
+#a #h #G #i #L #H #I #K #V #HLK
elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #HX
lapply (drops_mono … HLY … HLK) -L #H destruct //
-qed-.
+qed-.
+
+lemma cnv_inv_lref_atom (a) (h) (b) (G):
+ ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
+ ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⊥.
+#a #h #b #G #i #L #H #Hi
+elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
+lapply (drops_gen b … HLY) -HLY #HLY
+lapply (drops_mono … HLY … Hi) -L #H destruct
+qed-.
+
+lemma cnv_inv_lref_unit (a) (h) (G):
+ ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
+ ∀I,K. ⬇*[i] L ≘ K.ⓤ{I} → ⊥.
+#a #h #G #i #L #H #I #K #HLK
+elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
+lapply (drops_mono … HLY … HLK) -L #H destruct
+qed-.
(* Properties with generic slicing for local environments *******************)
--- /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/rt_computation/cpue_csx.ma".
+include "basic_2/rt_equivalence/cpes_cprs.ma".
+include "basic_2/dynamic/cnv_cpue.ma".
+include "basic_2/dynamic/cnv_cpes.ma".
+include "basic_2/dynamic/cnv_preserve_cpes.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+lemma abst_dec (X): ∨∨ ∃∃p,W,T. X = ⓛ{p}W.T
+ | (∀p,W,T. X = ⓛ{p}W.T → ⊥).
+* [ #I | * [ #p * | #I ] #V #T ]
+[3: /3 width=4 by ex1_3_intro, or_introl/ ]
+@or_intror #q #W #U #H destruct
+qed-.
+
+(* main properties with evaluations for rt-transition on terms **************)
+
+theorem cnv_dec (a) (h) (G) (L) (T):
+ Decidable (⦃G,L⦄ ⊢ T ![a,h]).
+#a #h #G #L #T
+@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
+[ #s #HG #HL #HT destruct -IH
+ /2 width=1 by cnv_sort, or_introl/
+| #i #HG #HL #HT destruct
+ elim (drops_F_uni L i) [| * * ]
+ [ /3 width=8 by cnv_inv_lref_atom, or_intror/
+ | /3 width=9 by cnv_inv_lref_unit, or_intror/
+ | #I #V #K #HLK
+ elim (IH G K V) -IH [3: /2 width=2 by fqup_lref/ ]
+ [ /3 width=5 by cnv_lref_drops, or_introl/
+ | /4 width=5 by cnv_inv_lref_pair, or_intror/
+ ]
+ ]
+| #l #HG #HL #HT destruct -IH
+ /3 width=6 by cnv_inv_gref, or_intror/
+| #p #I #V #T #HG #HL #HT destruct
+ elim (IH G L V) [| -IH | // ] #HV
+ [ elim (IH G (L.ⓑ{I}V) T) -IH [3: // ] #HT
+ [ /3 width=1 by cnv_bind, or_introl/ ]
+ ]
+ @or_intror #H
+ elim (cnv_inv_bind … H) -H /2 width=1 by/
+| #V #T #HG #HL #HT destruct
+ elim (IH G L V) [| -IH #HV | // ]
+ [ elim (IH G L T) -IH [| #HT #HV | // ]
+ [ cases a #HT #HV
+ [ elim (cnv_fwd_aaa … HT) #A #HA
+ elim (cpme_total_aaa h 1 … HA) #X
+ elim (abst_dec X) [ * ]
+ [ #p #W #U #H #HTU destruct
+ elim (cnv_cpes_dec … 1 0 … HV W) [ #HVW | #HnVW ]
+ [ elim HTU -HTU #HTU #_
+ /3 width=7 by cnv_appl_cpes, or_introl/
+ | @or_intror #H
+ elim (cnv_inv_appl_SO_cpes … H) -H #m1 #q #W0 #U0 #Hm1 #_ #_ #HVW0
+ >Hm1 -m1 [| // ] #HTU0
+ elim (cnv_cpme_cpms_conf … HT … HTU0 … HTU) -T #HU0 #_
+ elim (cpms_inv_abst_bi … HU0) -HU0 #_ #HW0 #_ -p -q -U0
+ /3 width=3 by cpes_cprs_trans/
+ | lapply (cnv_cpme_trans … HT … HTU) -T #H
+ elim (cnv_inv_bind … H) -H #HW #_ //
+ ]
+ | #HnTU #HTX
+ @or_intror #H
+ elim (cnv_inv_appl_SO_cpes … H) -H #m1 #q #W0 #U0 #Hm1 #_ #_ #_
+ >Hm1 -m1 [| // ] #HTU0
+ elim (cnv_cpme_cpms_conf … HT … HTU0 … HTX) -T #HX #_
+ elim (cpms_inv_abst_sn … HX) -HX #V0 #T0 #_ #_ #H destruct -W0 -U0
+ /2 width=4 by/
+ ]
+ | elim (cpue_total_csx h G L T)
+ [| /3 width=2 by cnv_fwd_fsb, fsb_inv_csx/ ] #X
+ elim (abst_dec X) [ * ]
+ [ #p #W #U #H #HTU destruct
+ elim (cnv_cpes_dec … 1 0 … HV W) [ #HVW | #HnVW ]
+ [ elim HTU -HTU #n #HTU #_
+ @or_introl @(cnv_appl_cpes … HV … HT … HVW … HTU) #H destruct
+ | @or_intror #H
+ elim (cnv_inv_appl_SO_cpes … H) -H #m1 #q #W0 #U0 #_ #_ #_ #HVW0 #HTU0
+ elim (cnv_cpue_cpms_conf … HT … HTU0 … HTU) -m1 -T #X * #m2 #HU0X #_ #HUX
+ elim (tueq_inv_bind1 … HUX) -HUX #X0 #_ #H destruct -U
+ elim (cpms_inv_abst_bi … HU0X) -HU0X #_ #HW0 #_ -p -q -m2 -U0 -X0
+ /3 width=3 by cpes_cprs_trans/
+ | lapply (cnv_cpue_trans … HT … HTU) -T #H
+ elim (cnv_inv_bind … H) -H #HW #_ //
+ ]
+ | #HnTU #HTX
+ @or_intror #H
+ elim (cnv_inv_appl_SO_cpes … H) -H #m1 #q #W0 #U0 #_ #_ #_ #_ #HTU0
+ elim (cnv_cpue_cpms_conf … HT … HTU0 … HTX) -m1 -T #X0 * #m2 #HUX0 #_ #HX0
+ elim (cpms_inv_abst_sn … HUX0) -HUX0 #V0 #T0 #_ #_ #H destruct -m2 -W0 -U0
+ elim (tueq_inv_bind2 … HX0) -HX0 #X0 #_ #H destruct
+ /2 width=4 by/
+ ]
+ ]
+ ]
+ ]
+ @or_intror #H
+ elim (cnv_inv_appl_SO … H) -H /2 width=1 by/
+| #U #T #HG #HL #HT destruct
+ elim (IH G L U) [| -IH | // ] #HU
+ [ elim (IH G L T) -IH [3: // ] #HT
+ [ elim (cnv_cpes_dec … 0 1 … HU … HT) #HUT
+ [ /3 width=1 by cnv_cast_cpes, or_introl/ ]
+ ]
+ ]
+ @or_intror #H
+ elim (cnv_inv_cast_cpes … H) -H /2 width=1 by/
+]
+qed-.
#a #h #G #L1 #T #HT #L2 #H
@(lprs_ind_dx … H) -L2 /2 width=3 by cnv_lpr_trans/
qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cnv_inv_appl_SO (a) (h) (G) (L):
+ ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
+ ∃∃n,p,W0,U0. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
+ ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+* #h #G #L #V #T #H
+elim (cnv_inv_appl … H) -H [ * [| #n ] | #n ] #p #W #U #Ha #HV #HT #HVW #HTU
+[ elim (cnv_fwd_cpm_SO … (ⓛ{p}W.U))
+ [|*: /2 width=8 by cnv_cpms_trans/ ] #X #HU0
+ elim (cpm_inv_abst1 … HU0) #W0 #U0 #HW0 #_ #H0 destruct
+ lapply (cpms_step_dx … HVW … HW0) -HVW -HW0 #HVW0
+ lapply (cpms_step_dx … HTU … HU0) -HTU -HU0 #HTU0
+ /2 width=7 by ex5_4_intro/
+| lapply (Ha ?) -Ha [ // ] #Ha
+ lapply (le_n_O_to_eq n ?) [ /3 width=1 by le_S_S_to_le/ ] -Ha #H destruct
+ /2 width=7 by ex5_4_intro/
+| @(ex5_4_intro … HV HT HVW HTU) #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/rt_computation/cpme_aaa.ma".
+include "basic_2/rt_computation/cpre_cpre.ma".
+include "basic_2/rt_equivalence/cpes.ma".
+include "basic_2/dynamic/cnv_cpme.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with t-bound rt-equivalence for terms *************************)
+
+lemma cnv_cpes_dec (a) (h) (n1) (n2) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∀T2. ⦃G,L⦄ ⊢ T2 ![a,h] →
+ Decidable … (⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2).
+#a #h #n1 #n2 #G #L #T1 #HT1 #T2 #HT2
+elim (cnv_fwd_aaa … HT1) #A1 #HA1
+elim (cnv_fwd_aaa … HT2) #A2 #HA2
+elim (cpme_total_aaa h n1 … HA1) -HA1 #U1 #HTU1
+elim (cpme_total_aaa h n2 … HA2) -HA2 #U2 #HTU2
+elim (eq_term_dec U1 U2) [ #H destruct | #HnU12 ]
+[ cases HTU1 -HTU1 #HTU1 #_
+ cases HTU2 -HTU2 #HTU2 #_
+ /3 width=3 by cpms_div, or_introl/
+| @or_intror * #T0 #HT10 #HT20
+ lapply (cnv_cpme_cpms_conf … HT1 … HT10 … HTU1) -T1 #H1
+ lapply (cnv_cpme_cpms_conf … HT2 … HT20 … HTU2) -T2 #H2
+ /3 width=6 by cpre_mono/
+]
+qed-.
+
+(* Advanced inversion lemmas with t-bound rt-equivalence for terms **********)
+
+lemma cnv_inv_appl_SO_cpes (a) (h) (G) (L):
+ ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
+ ∃∃n,p,W,U. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
+ ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W.U.
+#a #h #G #L #V #T #H
+elim (cnv_inv_appl_SO … H) -H #n #p #W #U #Hn #HV #HT #HVW #HTU
+/3 width=7 by cpms_div, ex5_4_intro/
+qed-.
elim (cnv_inv_bind … HWU) -HWU #HW #HU
elim (cnv_inv_bind … HWT) -HWT #_ #HT
elim (cpms_inv_abst_sn … HUX) -HUX #W0 #X0 #_ #HUX0 #H destruct
-elim (cpms_inv_abst_bi … HTX) -HTX #_ #HTX0 -W0
+elim (cpms_inv_abst_bi … HTX) -HTX #_ #_ #HTX0 -W0
/3 width=3 by cnv_cast, conj/
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 *)
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⥲* [ break term 46 h ] 𝐍 ⦃ break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedITEval $h $G $L $T1 $T2 }.
--- /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 FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⥲[ break term 46 h ] 𝐍⦃ break term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedITNormal $h $G $L $T }.
+++ /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 FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬈ [ break term 46 h ] 𝐖𝐇 ⦃ break term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'PRedTyWHead $h $G $L $T }.
--- /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/rt_transition/cnu_cnu.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Inversion lemmas with normal terms for t-unbound rt-transition ***********)
+
+lemma cpms_inv_cnu_sn (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄ → T1 ≅ T2.
+#h #n #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 //
+#n1 #n2 #T1 #T0 #HT10 #_ #IH #HT1
+/5 width=8 by cnu_tueq_trans, tueq_trans/
+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/rt_transition/cpr_tdeq.ma".
-include "basic_2/rt_transition/cwhx_drops.ma".
-include "basic_2/rt_transition/cwhx_rdeq.ma".
-include "basic_2/rt_computation/fsb_aaa.ma".
-include "basic_2/rt_computation/cpms_cpms.ma".
-include "basic_2/rt_computation/cpms_fpbg.ma".
-
-(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-
-(* Properties with whd normality for unbound rt-transition ******************)
-
-lemma aaa_cpms_cwhx (h) (G) (L):
- ∀T1,A. ⦃G,L⦄ ⊢ T1 ⁝ A →
- ∃∃n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ⬈[h] 𝐖𝐇⦃T2⦄.
-#h #G #L #T1 #A #H
-@(aaa_ind_fpbg h … H) -G -L -T1 -A
-#G #L #T1 #A * -G -L -T1 -A
-[ #G #L #s #_ /2 width=4 by cwhx_sort, ex2_2_intro/
-| * #G #K #V1 #A #_ #IH -A
- elim (IH … G K V1) [2,4: /3 width=1 by fpb_fpbg, fpb_fqu, fqu_lref_O/ ] -IH #n #V2 #HV12 #HV2
- elim (lifts_total … V2 (𝐔❴1❵)) #T2 #HVT2
- /5 width=10 by cpms_ell, cpms_delta, cwhx_lifts, drops_refl, drops_drop, ex2_2_intro/
-| #I #G #K #A #i #_ #IH -A
- elim (IH … G K (#i)) [| /3 width=1 by fpb_fpbg, fpb_fqu/ ] -IH #n #V2 #HV12 #HV2
- elim (lifts_total … V2 (𝐔❴1❵)) #T2 #HVT2
- /5 width=10 by cpms_lref, cwhx_lifts, drops_refl, drops_drop, ex2_2_intro/
-| * #G #L #V #T1 #B #A #_ #_ #IH -B -A
- [ elim (cpr_abbr_pos h G L V T1) #T0 #HT10 #HnT10
- elim (IH G L T0) -IH [| /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HnT10 #n #T2 #HT02 #HT2
- /3 width=5 by cpms_step_sn, ex2_2_intro/
- | elim (IH … G (L.ⓓV) T1) -IH [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_bind_dx/ ] #n #T2 #HT12 #HT2
- /3 width=5 by cwhx_ldef, cpms_bind_dx, ex2_2_intro/
- ]
-| #p #G #L #W #T1 #B #A #_ #_ #_ -B -A
- /2 width=5 by cwhx_abst, ex2_2_intro/
-| #G #L #V #T1 #B #A #_ #HT1 #IH
- elim (IH … G L T1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_flat_dx/ ] #n1 #T2 #HT12 #HT2
- elim (tdeq_dec T1 T2) [ -n1 #HT12 | -HT2 #HnT12 ]
- [ lapply (tdeq_cwhx_trans … HT2 … HT12) -T2
- @(insert_eq_0 … L) #Y @(insert_eq_0 … T1) #X * -Y -X
- [ #L0 #s0 #H1 #H2 destruct -IH
- lapply (aaa_inv_sort … HT1) -HT1 #H destruct
- | #p #L0 #W0 #T0 #H1 #H2 destruct -HT1
- elim (IH G L0 (ⓓ{p}ⓝW0.V.T0)) -IH [ /4 width=5 by cpms_step_sn, cpm_beta, ex2_2_intro/ ]
- @fpb_fpbg @fpb_cpx [ /2 width=1 by cpx_beta/ ] #H
- elim (tdeq_inv_pair … H) -H #H #_ #_ destruct
- | #L0 #V0 #T0 #_ #H1 #H2 destruct -HT1
- elim (lifts_total V (𝐔❴1❵)) #W #HVW
- elim (IH G L0 (-ⓓV0.ⓐW.T0)) -IH [ /4 width=5 by cpms_step_sn, cpm_theta, ex2_2_intro/ ]
- @fpb_fpbg @fpb_cpx [ /2 width=3 by cpx_theta/ ] #H
- elim (tdeq_inv_pair … H) -H #H #_ #_ destruct
- ]
- | elim (IH G L (ⓐV.T2)) -IH [ /4 width=5 by cpms_trans, cpms_appl_dx, ex2_2_intro/ ]
- @(cpms_tdneq_fwd_fpbg … n1) [ /2 width=3 by cpms_appl_dx/ ] #H
- elim (tdeq_inv_pair … H) -H #_ #_ #H /2 width=1 by/
- ]
-| #G #L #U #T1 #A #_ #_ #IH -A
- elim (IH … G L T1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_flat_dx/ ] #n #T2 #HT12 #HT2
- /3 width=4 by cpms_eps, ex2_2_intro/
-]
-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/notation/relations/prediteval_5.ma".
+include "basic_2/rt_transition/cnu.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* EVALUATION FOR T-UNBOUND RT-TRANSITION ON TERMS **************************)
+
+definition cpue (h) (G) (L): relation2 term term ≝
+ λT1,T2. ∃∃n. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄.
+
+interpretation "evaluation for t-unbound context-sensitive parallel rt-transition (term)"
+ 'PRedITEval h G L T1 T2 = (cpue h G L T1 T2).
--- /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/rt_transition/cpm_cpx.ma".
+include "basic_2/rt_transition/cnu_tdeq.ma".
+include "basic_2/rt_computation/csx.ma".
+include "basic_2/rt_computation/cpue.ma".
+
+(* EVALUATION FOR T-UNBOUND RT-TRANSITION ON TERMS **************************)
+
+(* Properties with strong normalization for unbound rt-transition for terms *)
+
+lemma cpue_total_csx (h) (G) (L):
+ ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G, L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄.
+#h #G #L #T1 #H
+@(csx_ind … H) -T1 #T1 #_ #IHT1
+elim (cnu_dec_tdeq h G L T1) [ /3 width=4 by ex2_intro, ex_intro/ ] *
+#n1 #T0 #HT10 #HnT10
+elim (IHT1 … HnT10) -IHT1 -HnT10 [| /2 width=2 by cpm_fwd_cpx/ ]
+#T2 * #n2 #HT02 #HT2 /4 width=7 by cpms_step_sn, ex2_intro, ex_intro/
+qed-.
qed-.
(* Basic_2A1: includes: cprs_inv_abst *)
-lemma cpms_inv_abst_bi (n) (h) (G) (L):
- ∀p,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[n, h] ⓛ{p}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2.
-#n #h #G #L #p #W1 #W2 #T1 #T2 #H
+lemma cpms_inv_abst_bi (n) (h) (p1) (p2) (G) (L):
+ ∀W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ➡*[n, h] ⓛ{p2}W2.T2 →
+ ∧∧ p1 = p2 & ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2.
+#n #h #p1 #p2 #G #L #W1 #W2 #T1 #T2 #H
elim (cpms_inv_abst_sn … H) -H #W #T #HW1 #HT1 #H destruct
-/2 width=1 by conj/
+/2 width=1 by and3_intro/
qed-.
(* Basic_1: was pr3_gen_abbr *)
--- /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/rt_computation/cpre_csx.ma".
+include "basic_2/rt_computation/cpre_cpre.ma".
+include "basic_2/rt_equivalence/cpcs_cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL R-EQUIVALENCE FOR TERMS ***********************)
+
+(* Properties with strongly normalizing terms for unbound rt-transition *****)
+
+(* Basic_1: was: cpcs_dec *)
+lemma csx_cpcs_dec (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∀T2. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄ →
+ Decidable … (⦃G,L⦄ ⊢ T1 ⬌*[h] T2).
+#h #G #L #T1 #HT1 #T2 #HT2
+elim (cpre_total_csx … HT1) -HT1 #U1 #HTU1
+elim (cpre_total_csx … HT2) -HT2 #U2 #HTU2
+elim (eq_term_dec U1 U2) [ #H destruct | #HnU12 ]
+[ cases HTU1 -HTU1 #HTU1 #_
+ cases HTU2 -HTU2 #HTU2 #_
+ /3 width=3 by cprs_div, or_introl/
+| @or_intror #H
+ elim (cpcs_inv_cprs … H) -H #T0 #HT10 #HT20
+ lapply (cpre_cprs_conf … HT10 … HTU1) -T1 #H1
+ lapply (cpre_cprs_conf … HT20 … HTU2) -T2 #H2
+ /3 width=6 by cpre_mono/
+]
+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/rt_computation/cprs_cprs.ma".
+include "basic_2/rt_equivalence/cpes.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-EQUIVALENCE FOR TERMS **************)
+
+(* Properties with context-sensitive parallel r-computation on terms ********)
+
+lemma cpes_cprs_trans (h) (n) (G) (L) (T0):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ⬌*[h,n,0] T0 →
+ ∀T2. ⦃G,L⦄ ⊢ T0 ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h,n,0] T2.
+#h #n #G #L #T0 #T1 * #T #HT1 #HT0 #T2 #HT02
+elim (cprs_conf … HT0 … HT02) -T0 #T0 #HT0 #HT20
+/3 width=3 by cpms_div, cpms_cprs_trans/
+qed-.
elim (tdeq_inv_pair … H) -H #H destruct
| @or_intror @(ex2_intro … (ⓓ{p}ⓝW1.V1.U1)) [ /2 width=1 by cpm_beta/ ] #H
elim (tdeq_inv_pair … H) -H #H destruct
- ]
+ ]
| #T2 #HT12 #HnT12 #_
@or_intror @(ex2_intro … (ⓐV1.T2)) [ /2 width=1 by cpm_appl/ ] #H
elim (tdeq_inv_pair … H) -H /2 width=1 by/
--- /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/notation/relations/preditnormal_4.ma".
+include "static_2/syntax/tueq.ma".
+include "basic_2/rt_transition/cpm.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+definition cnu (h) (G) (L): predicate term ≝
+ λT1. ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≅ T2.
+
+interpretation
+ "normality for t-unbound context-sensitive parallel rt-transition (term)"
+ 'PRedITNormal h G L T = (cnu h G L T).
+
+(* Basic properties *********************************************************)
+
+lemma cnu_sort (h) (G) (L): ∀s. ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃⋆s⦄.
+#h #G #L #s1 #n #X #H
+elim (cpm_inv_sort1 … H) -H #H #_ destruct //
+qed.
+
+lemma cnu_ctop (h) (G): ∀i. ⦃G,⋆⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
+#h #G * [| #i ] #n #X #H
+[ elim (cpm_inv_zero1 … H) -H *
+ [ #H #_ destruct //
+ | #Y #X1 #X2 #_ #_ #H destruct
+ | #m #Y #X1 #X2 #_ #_ #H destruct
+ ]
+| elim (cpm_inv_lref1 … H) -H *
+ [ #H #_ destruct //
+ | #Z #Y #X0 #_ #_ #H destruct
+ ]
+]
+qed.
+
+lemma cnu_zero (h) (G) (L): ∀I. ⦃G,L.ⓤ{I}⦄ ⊢ ⥲[h] 𝐍⦃#0⦄.
+#h #G #L #I #n #X #H
+elim (cpm_inv_zero1 … H) -H *
+[ #H #_ destruct //
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
+lemma cnu_gref (h) (G) (L): ∀l. ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃§l⦄.
+#h #G #L #l1 #n #X #H
+elim (cpm_inv_gref1 … H) -H #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/rt_transition/cnr.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties with normal terms for r-transition *******************)
+
+lemma cnu_abst (h) (p) (G) (L):
+ ∀W. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃W⦄ → ∀T.⦃G,L.ⓛW⦄ ⊢ ⥲[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃ⓛ{p}W.T⦄.
+#h #p #G #L #W1 #HW1 #T1 #HT1 #n #X #H
+elim (cpm_inv_abst1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+<(HW1 … HW12) -W2 /3 width=2 by tueq_bind/
+qed.
+
+lemma cnu_abbr_neg (h) (G) (L):
+ ∀V. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ∀T.⦃G,L.ⓓV⦄ ⊢ ⥲[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃-ⓓV.T⦄.
+#h #G #L #V1 #HV1 #T1 #HT1 #n #X #H
+elim (cpm_inv_abbr1 … H) -H *
+[ #V2 #T2 #HV12 #HT12 #H destruct
+ <(HV1 … HV12) -V2 /3 width=2 by tueq_bind/
+| #X1 #_ #_ #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/rt_transition/cpm_simple.ma".
+include "basic_2/rt_transition/cnr.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties with simple terms and normal terms for r-transition **)
+
+lemma cnu_appl_simple (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃ⓐV.T⦄.
+#h #G #L #V1 #T1 #HV1 #HT1 #HS #n #X #H
+elim (cpm_inv_appl1_simple … H HS) -H -HS #V2 #T2 #HV12 #HT12 #H destruct
+lapply (HV1 … HV12) -HV1 -HV12 #H destruct
+/3 width=2 by tueq_appl/
+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 "static_2/syntax/tueq_tueq.ma".
+include "basic_2/rt_transition/cpm_tueq.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cnu_tueq_trans (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄ → ∀T2.T1 ≅ T2 → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄.
+#h #G #L #T1 #HT1 #T2 #HT12 #n #T0 #HT20
+@(tueq_canc_sn … HT12)
+elim (tueq_cpm_trans … HT12 … HT20) -T2 #T2 #HT13 #HT30
+/3 width=3 by tueq_trans/
+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 "static_2/relocation/lifts_tueq.ma".
+include "basic_2/rt_transition/cpm_drops.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cnu_atom_drops (h) (b) (G) (L):
+ ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
+#h #b #G #L #i #Hi #n #X #H
+elim (cpm_inv_lref1_drops … H) -H * [ // || #m ] #K #V1 #V2 #HLK
+lapply (drops_gen b … HLK) -HLK #HLK
+lapply (drops_mono … Hi … HLK) -L #H destruct
+qed.
+
+lemma cnu_unit_drops (h) (I) (G) (L):
+ ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
+#h #I #G #L #K #i #HLK #n #X #H
+elim (cpm_inv_lref1_drops … H) -H * [ // || #m ] #Y #V1 #V2 #HLY
+lapply (drops_mono … HLK … HLY) -L #H destruct
+qed.
+
+(* Properties with generic relocation ***************************************)
+
+lemma cnu_lifts (h) (G): d_liftable1 … (cnu h G).
+#h #G #K #T #HT #b #f #L #HLK #U #HTU #n #U0 #H
+elim (cpm_inv_lifts_sn … H … HLK … HTU) -b -L #T0 #HTU0 #HT0
+lapply (HT … HT0) -G -K /2 width=6 by tueq_lifts_bi/
+qed-.
+
+(* Inversion lemmas with generic relocation *********************************)
+
+lemma cnu_inv_lifts (h) (G): d_deliftable1 … (cnu h G).
+#h #G #L #U #HU #b #f #K #HLK #T #HTU #n #T0 #H
+elim (cpm_lifts_sn … H … HLK … HTU) -b -K #U0 #HTU0 #HU0
+lapply (HU … HU0) -G -L /2 width=6 by tueq_inv_lifts_bi/
+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 "static_2/relocation/lifts_tueq.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties with uniform relocation for terms ********************)
+
+lemma cnu_lref (h) (I) (G) (L):
+ ∀i. ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄ → ⦃G,L.ⓘ{I}⦄ ⊢ ⥲[h] 𝐍⦃#↑i⦄.
+#h #I #G #L #i #Hi #n #X #H
+elim (cpm_inv_lref1 … H) -H *
+[ #H #_ destruct //
+| #J #K #V #HV #HVX #H destruct
+ lapply (Hi … HV) -Hi -HV #HV
+ elim (tueq_lifts_dx … HV … HVX) -V #Xi #Hi #HX
+ lapply (lifts_inv_lref1_uni … Hi) -Hi #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/rt_transition/cnr_tdeq.ma".
+include "basic_2/rt_transition/cnu_drops.ma".
+include "basic_2/rt_transition/cnu_cnr.ma".
+include "basic_2/rt_transition/cnu_cnr_simple.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Properties with context-free sort-irrelevant equivalence for terms *******)
+
+lemma cnu_dec_tdeq (h) (G) (L):
+ ∀T1. ∨∨ ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄
+ | ∃∃n,T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥).
+#h #G #L #T1
+@(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
+[ #s #HG #HL #HT destruct -IH
+ /3 width=5 by cnu_sort, or_introl/
+| #i #HG #HL #HT destruct -IH
+ elim (drops_F_uni L i)
+ [ /3 width=7 by cnu_atom_drops, or_introl/
+ | * * [ #I | * #V ] #K #HLK
+ [ /3 width=8 by cnu_unit_drops, or_introl/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_delta_drops/ ] #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_ell_drops/ ] #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ ]
+ ]
+| #l #HG #HL #HT destruct -IH
+ /3 width=5 by cnu_gref, or_introl/
+| #p * [ cases p ] #V1 #T1 #HG #HL #HT destruct
+ [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2 -IH
+ elim (tdeq_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
+ [ elim (tdeq_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
+ @or_intror @(ex2_2_intro … X1) [1,2: /2 width=4 by cpm_zeta/ ] #H
+ /2 width=7 by tdeq_lifts_inv_pair_sn/
+ | @or_intror @(ex2_2_intro … (+ⓓV1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | elim (cnr_dec_tdeq h G L V1) [ elim (IH G (L.ⓓV1) T1) [| * | // ] | * ] -IH
+ [ #HT1 #HV1 /3 width=7 by cnu_abbr_neg, or_introl/
+ | #n #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_2_intro … (-ⓓV1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_2_intro … (-ⓓV2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | elim (cnr_dec_tdeq h G L V1) [ elim (IH G (L.ⓛV1) T1) [| * | // ] | * ] -IH
+ [ #HT1 #HV1 /3 width=7 by cnu_abst, or_introl/
+ | #n #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_2_intro … (ⓛ{p}V1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_2_intro … (ⓛ{p}V2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ ]
+| * #V1 #T1 #HG #HL #HT destruct [| -IH ]
+ [ elim (cnr_dec_tdeq h G L V1) [ elim (IH G L T1) [| * | // ] | * ] -IH
+ [ #HT1 #HV1
+ elim (simple_dec_ex T1) [| * #p * #W1 #U1 #H destruct ]
+ [ /3 width=7 by cnu_appl_simple, or_introl/
+ | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
+ @or_intror @(ex2_2_intro … (ⓓ{p}W1.ⓐX1.U1)) [1,2: /2 width=3 by cpm_theta/ ] #H
+ elim (tdeq_inv_pair … H) -H #H destruct
+ | @or_intror @(ex2_2_intro … (ⓓ{p}ⓝW1.V1.U1)) [1,2: /2 width=2 by cpm_beta/ ] #H
+ elim (tdeq_inv_pair … H) -H #H destruct
+ ]
+ | #n #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_2_intro … (ⓐV1.T2)) [1,2: /2 width=2 by cpm_appl/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_2_intro … (ⓐV2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | @or_intror @(ex2_2_intro … T1) [1,2: /2 width=2 by cpm_eps/ ] #H
+ /2 width=4 by tdeq_inv_pair_xy_y/
+ ]
+]
+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 "static_2/relocation/lifts_tueq.ma".
+include "basic_2/rt_transition/cpm.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Properties with tail sort-irrelevant equivalence on terms ****************)
+
+lemma cpm_tueq_conf (h) (n) (G) (L) (T0):
+ ∀T1. ⦃G,L⦄ ⊢ T0 ➡[n,h] T1 → ∀T2. T0 ≅ T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T2 ➡[n,h] T & T1 ≅ T.
+#h #n #G #L #T0 #T1 #H @(cpm_ind … H) -G -L -T0 -T1 -n
+[ /2 width=3 by ex2_intro/
+| #G #L #s0 #X2 #H2
+ elim (tueq_inv_sort1 … H2) -H2 #s2 #H destruct
+ /3 width=3 by tueq_sort, ex2_intro/
+| #n #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #X2 #H2
+ >(tueq_inv_lref1 … H2) -X2
+ elim (IH V0) [| // ] -IH #V #HV0 #HV1
+ elim (tueq_lifts_sn … HV1 … HVW1) -V1
+ /3 width=3 by cpm_delta, ex2_intro/
+| #n #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #X2 #H2
+ >(tueq_inv_lref1 … H2) -X2
+ elim (IH V0) [| // ] -IH #V #HV0 #HV1
+ elim (tueq_lifts_sn … HV1 … HVW1) -V1
+ /3 width=3 by cpm_ell, ex2_intro/
+| #n #I #G #K0 #V1 #W1 #i #_ #IH #HVW1 #X2 #H2
+ >(tueq_inv_lref1 … H2) -X2
+ elim (IH (#i)) [| // ] -IH #V #HV0 #HV1
+ elim (tueq_lifts_sn … HV1 … HVW1) -V1
+ /3 width=3 by cpm_lref, ex2_intro/
+| #n #p #I #G #L #V0 #V1 #T0 #T1 #HV01 #_ #_ #IHT #X2 #H2
+ elim (tueq_inv_bind1 … H2) -H2 #T2 #HT02 #H destruct
+ elim (IHT … HT02) -T0 #T #HT2 #HT1
+ /3 width=3 by cpm_bind, tueq_bind, ex2_intro/
+| #n #G #L #V0 #V1 #T0 #T1 #HV10 #_ #_ #IHT #X2 #H2
+ elim (tueq_inv_appl1 … H2) -H2 #T2 #HT02 #H destruct
+ elim (IHT … HT02) -T0 #T #HT2 #HT1
+ /3 width=3 by cpm_appl, tueq_appl, ex2_intro/
+| #n #G #L #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X2 #H2
+ elim (tueq_inv_cast1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (IHV … HV02) -V0 #V #HV2 #HV1
+ elim (IHT … HT02) -T0 #T #HT2 #HT1
+ /3 width=5 by cpm_cast, tueq_cast, ex2_intro/
+| #n #G #L #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X2 #H2
+ elim (tueq_inv_bind1 … H2) -H2 #U2 #HU02 #H destruct
+ elim (tueq_inv_lifts_sn … HU02 … HTU0) -U0 #T2 #HTU2 #HT02
+ elim (IH … HT02) -T0 #T #HT2 #HT1
+ /3 width=3 by cpm_zeta, ex2_intro/
+| #n #G #L #V0 #T0 #T1 #_ #IH #X2 #H2
+ elim (tueq_inv_cast1 … H2) -H2 #V2 #T2 #_ #HT02 #H destruct
+ elim (IH … HT02) -V0 -T0
+ /3 width=3 by cpm_eps, ex2_intro/
+| #n #G #L #V0 #T0 #T1 #_ #IH #X2 #H2
+ elim (tueq_inv_cast1 … H2) -H2 #V2 #T2 #HV02 #_ #H destruct
+ elim (IH … HV02) -V0 -T1
+ /3 width=3 by cpm_ee, ex2_intro/
+| #n #p #G #L #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HW01 #_ #_ #_ #IHT #X2 #H2
+ elim (tueq_inv_appl1 … H2) -H2 #X #H2 #H destruct
+ elim (tueq_inv_bind1 … H2) -H2 #T2 #HT02 #H destruct
+ elim (IHT … HT02) -T0
+ /4 width=3 by cpm_beta, tueq_cast, tueq_bind, ex2_intro/
+| #n #p #G #L #V0 #V1 #U1 #W0 #W1 #T0 #T1 #HV01 #HW01 #_ #_ #_ #IHT #HVU1 #X2 #H2
+ elim (tueq_inv_appl1 … H2) -H2 #X #H2 #H destruct
+ elim (tueq_inv_bind1 … H2) -H2 #T2 #HT02 #H destruct
+ elim (IHT … HT02) -T0 #T #HT2 #HT1
+ /4 width=3 by cpm_theta, tueq_appl, tueq_bind, ex2_intro/
+]
+qed-.
+
+lemma tueq_cpm_trans (h) (n) (G) (L) (T0):
+ ∀T1. T1 ≅ T0 → ∀T2. ⦃G,L⦄ ⊢ T0 ➡[n,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n,h] T & T ≅ T2.
+#h #n #G #L #T0 #T1 #HT10 #T2 #HT02
+elim (cpm_tueq_conf … HT02 T1)
+/3 width=3 by tueq_sym, ex2_intro/
+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/notation/relations/predtywhead_4.ma".
-include "static_2/syntax/item_sh.ma".
-include "static_2/syntax/lenv.ma".
-include "static_2/syntax/genv.ma".
-
-(* WHD NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ****)
-
-inductive cwhx (h:sh) (G:genv): relation2 lenv term ≝
-| cwhx_sort: ∀L,s. cwhx h G L (⋆s)
-| cwhx_abst: ∀p,L,W,T. cwhx h G L (ⓛ{p}W.T)
-| cwhx_ldef: ∀L,V,T. cwhx h G (L.ⓓV) T → cwhx h G L (-ⓓV.T)
-.
-
-interpretation
- "whd normality for unbound context-sensitive parallel rt-transition (term)"
- 'PRedTyWHead h G L T = (cwhx h G L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cwhx_inv_lref_aux (h) (G):
- ∀Y,X. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃X⦄ →
- ∀i. X = #i → ⊥.
-#h #G #Y #X * - X -Y
-[ #L #s #i #H destruct
-| #p #L #W #T #i #H destruct
-| #L #V #T #_ #i #H destruct
-]
-qed-.
-
-lemma cwhx_inv_lref (h) (G):
- ∀L,i. ⦃G,L⦄ ⊢ ⬈[h] 𝐖𝐇⦃#i⦄ → ⊥.
-/2 width=7 by cwhx_inv_lref_aux/ qed-.
-
-fact cwhx_inv_gref_aux (h) (G):
- ∀Y,X. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃X⦄ →
- ∀l. X = §l → ⊥.
-#h #G #Y #X * - X -Y
-[ #L #s #l #H destruct
-| #p #L #W #T #l #H destruct
-| #L #V #T #_ #l #H destruct
-]
-qed-.
-
-lemma cwhx_inv_gref (h) (G):
- ∀L,l. ⦃G,L⦄ ⊢ ⬈[h] 𝐖𝐇⦃§l⦄ → ⊥.
-/2 width=7 by cwhx_inv_gref_aux/ qed-.
-
-fact cwhx_inv_abbr_aux (h) (G):
- ∀Y,X. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃X⦄ →
- ∀V,T. X = +ⓓV.T → ⊥.
-#h #G #Y #X * - X -Y
-[ #L #s #X1 #X2 #H destruct
-| #p #L #W #T #X1 #X2 #H destruct
-| #L #V #T #_ #X1 #X2 #H destruct
-]
-qed-.
-
-lemma cwhx_inv_abbr (h) (G):
- ∀Y,V,T. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃+ⓓV.T⦄ → ⊥.
-/2 width=8 by cwhx_inv_abbr_aux/ qed-.
-
-fact cwhx_inv_ldef_aux (h) (G):
- ∀Y,X. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃X⦄ →
- ∀V,T. X = -ⓓV.T → ⦃G,Y.ⓓV⦄ ⊢ ⬈[h] 𝐖𝐇⦃T⦄.
-#h #G #Y #X * - X -Y
-[ #L #s #X1 #X2 #H destruct
-| #p #L #W #T #X1 #X2 #H destruct
-| #L #V #T #HT #X1 #X2 #H destruct //
-]
-qed-.
-
-lemma cwhx_inv_ldef (h) (G):
- ∀Y,V,T. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃-ⓓV.T⦄ → ⦃G,Y.ⓓV⦄ ⊢ ⬈[h] 𝐖𝐇⦃T⦄.
-/2 width=3 by cwhx_inv_ldef_aux/ qed-.
-
-fact cwhx_inv_appl_aux (h) (G):
- ∀Y,X. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃X⦄ →
- ∀V,T. X = ⓐV.T → ⊥.
-#h #G #Y #X * - X -Y
-[ #L #s #X1 #X2 #H destruct
-| #p #L #W #T #X1 #X2 #H destruct
-| #L #V #T #_ #X1 #X2 #H destruct
-]
-qed-.
-
-lemma cwhx_inv_appl (h) (G):
- ∀Y,V,T. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃ⓐV.T⦄ → ⊥.
-/2 width=8 by cwhx_inv_appl_aux/ qed-.
-
-fact cwhx_inv_cast_aux (h) (G):
- ∀Y,X. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃X⦄ →
- ∀U,T. X = ⓝU.T → ⊥.
-#h #G #Y #X * - X -Y
-[ #L #s #X1 #X2 #H destruct
-| #p #L #W #T #X1 #X2 #H destruct
-| #L #V #T #_ #X1 #X2 #H destruct
-]
-qed-.
-
-lemma cwhx_inv_cast (h) (G):
- ∀Y,U,T. ⦃G,Y⦄ ⊢ ⬈[h] 𝐖𝐇⦃ⓝU.T⦄ → ⊥.
-/2 width=8 by cwhx_inv_cast_aux/ 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 "static_2/relocation/drops.ma".
-include "basic_2/rt_transition/cwhx.ma".
-
-(* WHD NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ****)
-
-(* Properties with generic slicing ******************************************)
-
-lemma cwhx_lifts (h) (G): d_liftable1 … (cwhx h G).
-#h #G #K #T #H elim H -K -T
-[ #K #s #b #f #L #HLK #X #H
- lapply (lifts_inv_sort1 … H) -H #H destruct //
-| #p #K #V #T #b #f #L #HLK #X #H
- elim (lifts_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct //
-| #K #V #T #_ #IH #b #f #L #HLK #X #H
- elim (lifts_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct
- /5 width=4 by cwhx_ldef, drops_skip, ext2_pair/
-]
-qed-.
-
-(* Inversion lemmas with generic slicing ************************************)
-
-lemma cwhx_inv_lifts (h) (G): d_deliftable1 … (cwhx h G).
-#h #G #L #U #H elim H -L -U
-[ #L #s #b #f #K #HLK #X #H
- lapply (lifts_inv_sort2 … H) -H #H destruct //
-| #p #L #W #U #b #f #K #HLK #X #H
- elim (lifts_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct //
-| #L #W #U #_ #IH #b #f #K #HLK #X #H
- elim (lifts_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct
- /5 width=4 by cwhx_ldef, drops_skip, ext2_pair/
-]
-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 "static_2/static/rdeq_fqup.ma".
-include "basic_2/rt_transition/cwhx.ma".
-
-(* WHD NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ****)
-
-(* Properties with sort-irrelevant equivalence ******************************)
-
-lemma rdeq_tdeq_cwhx_trans (h) (G):
- ∀L2,T2. ⦃G,L2⦄ ⊢ ⬈[h] 𝐖𝐇⦃T2⦄ →
- ∀T1. T1 ≛ T2 →
- ∀L1. L1 ≛[T1] L2 → ⦃G,L1⦄ ⊢ ⬈[h] 𝐖𝐇⦃T1⦄.
-#h #G #L2 #T2 #H elim H -L2 -T2
-[ #L2 #s2 #X1 #HX #L1 #HL
- elim (tdeq_inv_sort2 … HX) -HX #s1 #H destruct -s2 //
-| #p #L2 #W2 #T2 #X1 #HX #L1 #HL
- elim (tdeq_inv_pair2 … HX) -HX #W1 #T1 #_ #_ #H destruct -W2 -T2 //
-| #L2 #V2 #T2 #_ #IH #X1 #HX #L1 #HL
- elim (tdeq_inv_pair2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
- elim (rdeq_inv_bind … HL) -HL #HV1 #HT1
- /5 width=2 by cwhx_ldef, rdeq_bind_repl_dx, ext2_pair/
-]
-qed-.
-
-lemma tdeq_cwhx_trans (h) (G) (L):
- ∀T2. ⦃G,L⦄ ⊢ ⬈[h] 𝐖𝐇⦃T2⦄ →
- ∀T1. T1 ≛ T2 → ⦃G,L⦄ ⊢ ⬈[h] 𝐖𝐇⦃T1⦄.
-/3 width=5 by rdeq_tdeq_cwhx_trans/ qed-.
<table name="basic_2_sum"/>
<subsection name="B">Stage "B"</subsection>
+ <news class="beta" date="2019 April 16.">
+ Extended (λδ-2) and restricted (λδ-1) validity is decidable
+ (anniversary milestone).
+ </news>
<news class="beta" date="2019 March 25.">
Preservation of validity for rt-computation
does not need the sort degree parameter
]
[ { "context-sensitive native validity" * } {
[ [ "restricted refinement for lenvs" ] "lsubv ( ? ⊢ ? ⫃![?,?] ? )" "lsubv_drops" + "lsubv_lsubr" + "lsubv_lsuba" + "lsubv_cpms" + "lsubv_cpcs" + "lsubv_cnv" + "lsubv_lsubv" * ]
- [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" + "( ⦃?,?⦄ ⊢ ? ![?] )" + "( ⦃?,?⦄ ⊢ ? !*[?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_cpme" + "cnv_cpes" + "cnv_cpes_dec" + "cnv_cpcs" + "cnv_preserve_sub" + "cnv_preserve" * ]
+ [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" + "( ⦃?,?⦄ ⊢ ? ![?] )" + "( ⦃?,?⦄ ⊢ ? !*[?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_cpme" + "cnv_cpue" + "cnv_eval" + "cnv_cpes" + "cnv_cpcs" + "cnv_preserve_sub" + "cnv_preserve" + "cnv_preserve_cpes" * ]
}
]
}
class "prune"
[ { "rt-equivalence" * } {
[ { "context-sensitive parallel r-equivalence" * } {
- [ [ "for terms" ] "cpcs ( ⦃?,?⦄ ⊢ ? ⬌*[?] ? )" "cpcs_drops" + "cpcs_lsubr" + "cpcs_aaa" + "cpcs_cprs" + "cpcs_lprs" + "cpcs_cpc" + "cpcs_cpcs" * ]
+ [ [ "for terms" ] "cpcs ( ⦃?,?⦄ ⊢ ? ⬌*[?] ? )" "cpcs_drops" + "cpcs_lsubr" + "cpcs_aaa" + "cpcs_csx" + "cpcs_cprs" + "cpcs_lprs" + "cpcs_cpc" + "cpcs_cpcs" * ]
}
]
[ { "t-bound context-sensitive parallel rt-equivalence" * } {
- [ [ "for terms" ] "cpes ( ⦃?,?⦄ ⊢ ? ⬌*[?,?,?] ? )" "cpes_aaa" * ]
+ [ [ "for terms" ] "cpes ( ⦃?,?⦄ ⊢ ? ⬌*[?,?,?] ? )" "cpes_aaa" + "cpes_cprs" * ]
}
]
}
]
class "sky"
[ { "rt-computation" * } {
+ [ { "t-unbound context-sensitive parallel rt-computation" * } {
+ [ [ "evaluation for terms" ] "cpue ( ⦃?,?⦄ ⊢ ? ⥲*[?] 𝐍⦃?⦄ )" "cpue_csx" * ]
+ }
+ ]
[ { "context-sensitive parallel r-computation" * } {
[ [ "evaluation for terms" ] "cpre ( ⦃?,?⦄ ⊢ ? ➡*[?] 𝐍⦃?⦄ )" "cpre_csx" + "cpre_cpms" + "cpre_cpre" * ]
[ [ "for lenvs on all entries" ] "lprs ( ⦃?,?⦄ ⊢ ➡*[?] ? )" "lprs_tc" + "lprs_ctc" + "lprs_length" + "lprs_drops" + "lprs_aaa" + "lprs_lpr" + "lprs_lpxs" + "lprs_cpms" + "lprs_cprs" + "lprs_lprs" * ]
]
[ { "t-bound context-sensitive parallel rt-computation" * } {
[ [ "evaluation for terms" ] "cpme ( ⦃?,?⦄ ⊢ ? ➡*[?,?] 𝐍⦃?⦄ )" "cpme_aaa" * ]
- [ [ "for terms" ] "cpms" + "( ⦃?,?⦄ ⊢ ? ➡*[?,?] ? )" "cpms_drops" + "cpms_lsubr" + "cpms_rdeq" + "cpms_aaa" + "cpms_cwhx" + "cpms_lpr" + "cpms_cpxs" + "cpms_fpbs" + "cpms_fpbg" + "cpms_cpms" * ]
+ [ [ "for terms" ] "cpms" + "( ⦃?,?⦄ ⊢ ? ➡*[?,?] ? )" "cpms_drops" + "cpms_lsubr" + "cpms_rdeq" + "cpms_aaa" + "cpms_lpr" + "cpms_cnu" + "cpms_cpxs" + "cpms_fpbs" + "cpms_fpbg" + "cpms_cpms" * ]
}
]
[ { "unbound context-sensitive parallel rst-computation" * } {
]
class "cyan"
[ { "rt-transition" * } {
- [ { "unbound parallel rst-transition" * } {
- [ [ "for closures" ] "fpbq" + "( ⦃?,?,?⦄ ≽[?] ⦃?,?,?⦄ )" "fpbq_aaa" + "fpbq_fpb" * ]
- [ [ "proper for closures" ] "fpb" + "( ⦃?,?,?⦄ ≻[?] ⦃?,?,?⦄ )" "fpb_rdeq" + "fpb_fdeq" * ]
+ [ { "t-unbound context-sensitive parallel rt-transition" * } {
+ [ [ "normal form for terms" ] "cnu ( ⦃?,?⦄ ⊢ ⥲[?] 𝐍⦃?⦄ )" "cnu_tdeq" + "cnu_lifts" + "cnu_drops" + "cnu_cnr" + "cnu_cnr_simple" + "cnu_cnu" * ]
}
]
[ { "context-sensitive parallel r-transition" * } {
}
]
[ { "t-bound context-sensitive parallel rt-transition" * } {
- [ [ "for terms" ] "cpm" + "( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpm_simple" + "cpm_tdeq" + "cpm_drops" + "cpm_lsubr" + "cpm_fsle" + "cpm_aaa" + "cpm_cpx" * ]
+ [ [ "for terms" ] "cpm" + "( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpm_simple" + "cpm_tdeq" + "cpm_tueq" + "cpm_drops" + "cpm_lsubr" + "cpm_fsle" + "cpm_aaa" + "cpm_cpx" * ]
+ }
+ ]
+ [ { "unbound parallel rst-transition" * } {
+ [ [ "for closures" ] "fpbq" + "( ⦃?,?,?⦄ ≽[?] ⦃?,?,?⦄ )" "fpbq_aaa" + "fpbq_fpb" * ]
+ [ [ "proper for closures" ] "fpb" + "( ⦃?,?,?⦄ ≻[?] ⦃?,?,?⦄ )" "fpb_rdeq" + "fpb_fdeq" * ]
}
]
[ { "unbound context-sensitive parallel rt-transition" * } {
- [ [ "whd normal form for terms" ] "cwhx" + "( ⦃?,?⦄ ⊢ ⬈[?] 𝐖𝐇⦃?⦄ )" "cwhx_drops" + "cwhx_rdeq" * ]
[ [ "normal form for terms" ] "cnx" + "( ⦃?,?⦄ ⊢ ⬈[?] 𝐍⦃?⦄ )" "cnx_simple" + "cnx_drops" + "cnx_basic" + "cnx_cnx" * ]
[ [ "for lenvs on referred entries" ] "rpx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "rpx_length" + "rpx_drops" + "rpx_fqup" + "rpx_fsle" + "rpx_rdeq" + "rpx_lpx" + "rpx_rpx" * ]
[ [ "for lenvs on all entries" ] "lpx" + "( ⦃?,?⦄ ⊢ ⬈[?] ? )" "lpx_length" + "lpx_drops" + "lpx_fquq" + "lpx_fsle" + "lpx_rdeq" + "lpx_aaa" * ]
-../../matitac.opt `cat partial.txt`
+../../matitac.opt ground_2
+../../matitac.opt static_2
+../../matitac.opt basic_2
+../../matitac.opt apps_2
+++ /dev/null
-ground_2
-static_2
-basic_2
-apps_2
--- /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 FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( T1 ≅ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'ApproxEq $T1 $T2 }.
λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
∀T2. ⬆*[f] T2 ≘ U2 → R T1 T2.
+definition liftable2_dx: predicate (relation term) ≝
+ λR. ∀T1,T2. R T1 T2 → ∀f,U2. ⬆*[f] T2 ≘ U2 →
+ ∃∃U1. ⬆*[f] T1 ≘ U1 & R U1 U2.
+
definition deliftable2_dx: predicate (relation term) ≝
λR. ∀U1,U2. R U1 U2 → ∀f,T2. ⬆*[f] T2 ≘ U2 →
∃∃T1. ⬆*[f] T1 ≘ U1 & R T1 T2.
(* Basic properties *********************************************************)
+lemma liftable2_sn_dx (R): symmetric … R → liftable2_sn R → liftable2_dx R.
+#R #H2R #H1R #T1 #T2 #HT12 #f #U2 #HTU2
+elim (H1R … T1 … HTU2) -H1R /3 width=3 by ex2_intro/
+qed-.
+
lemma deliftable2_sn_dx (R): symmetric … R → deliftable2_sn R → deliftable2_dx R.
#R #H2R #H1R #U1 #U2 #HU12 #f #T2 #HTU2
elim (H1R … U1 … HTU2) -H1R /3 width=3 by ex2_intro/
]
qed-.
-lemma lifts_push_zero (f): ⬆*[⫯f]#O ≘ #0.
+lemma lifts_push_zero (f): ⬆*[⫯f]#0 ≘ #0.
/2 width=1 by lifts_lref/ qed.
lemma lifts_push_lref (f) (i1) (i2): ⬆*[f]#i1 ≘ #i2 → ⬆*[⫯f]#(↑i1) ≘ #(↑i2).
]
qed-.
+lemma tdeq_lifts_dx: liftable2_dx tdeq.
+/3 width=3 by tdeq_lifts_sn, liftable2_sn_dx, tdeq_sym/ qed-.
+
lemma tdeq_lifts_bi: liftable2_bi tdeq.
/3 width=6 by tdeq_lifts_sn, liftable2_sn_bi/ 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 "static_2/syntax/tueq.ma".
+include "static_2/relocation/lifts_lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Properties with tail sort-irrelevant equivalence for terms ***************)
+
+lemma tueq_lifts_sn: liftable2_sn tueq.
+#T1 #T2 #H elim H -T1 -T2
+[ #s1 #s2 #f #X #H >(lifts_inv_sort1 … H) -H
+ /3 width=3 by lifts_sort, tueq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref1 … H) -H
+ /3 width=3 by lifts_lref, tueq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref1 … H) -H
+ /2 width=3 by lifts_gref, tueq_gref, ex2_intro/
+| #p #I #V #T1 #T2 #_ #IHT #f #X #H elim (lifts_inv_bind1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHT … HTU1) -T1
+ /3 width=3 by lifts_bind, tueq_bind, ex2_intro/
+| #V #T1 #T2 #_ #IHT #f #X #H elim (lifts_inv_flat1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHT … HTU1) -T1
+ /3 width=3 by lifts_flat, tueq_appl, ex2_intro/
+| #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H elim (lifts_inv_flat1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HVW1) -V1 elim (IHT … HTU1) -T1
+ /3 width=5 by lifts_flat, tueq_cast, ex2_intro/
+]
+qed-.
+
+lemma tueq_lifts_dx: liftable2_dx tueq.
+/3 width=3 by tueq_lifts_sn, liftable2_sn_dx, tueq_sym/ qed-.
+
+lemma tueq_lifts_bi: liftable2_bi tueq.
+/3 width=6 by tueq_lifts_sn, liftable2_sn_bi/ qed-.
+
+(* Inversion lemmas with tail sort-irrelevant equivalence for terms *********)
+
+lemma tueq_inv_lifts_sn: deliftable2_sn tueq.
+#U1 #U2 #H elim H -U1 -U2
+[ #s1 #s2 #f #X #H >(lifts_inv_sort2 … H) -H
+ /3 width=3 by lifts_sort, tueq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref2 … H) -H
+ /3 width=3 by lifts_lref, tueq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref2 … H) -H
+ /2 width=3 by lifts_gref, tueq_gref, ex2_intro/
+| #p #I #W #U1 #U2 #_ #IHU #f #X #H elim (lifts_inv_bind2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHU … HTU1) -U1
+ /3 width=3 by lifts_bind, tueq_bind, ex2_intro/
+| #W #U1 #U2 #_ #IHU #f #X #H elim (lifts_inv_flat2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHU … HTU1) -U1
+ /3 width=3 by lifts_flat, tueq_appl, ex2_intro/
+| #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X #H elim (lifts_inv_flat2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW … HVW1) -W1 elim (IHU … HTU1) -U1
+ /3 width=5 by lifts_flat, tueq_cast, ex2_intro/
+]
+qed-.
+
+lemma tueq_inv_lifts_dx: deliftable2_dx tueq.
+/3 width=3 by tueq_inv_lifts_sn, deliftable2_sn_dx, tueq_sym/ qed-.
+
+lemma tueq_inv_lifts_bi: deliftable2_bi tueq.
+/3 width=6 by tueq_inv_lifts_sn, deliftable2_sn_bi/ 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 "static_2/notation/relations/approxeq_2.ma".
+include "static_2/syntax/term.ma".
+
+(* TAIL SORT-IRRELEVANT EQUIVALENCE ON TERMS ********************************)
+
+inductive tueq: relation term ≝
+| tueq_sort: ∀s1,s2. tueq (⋆s1) (⋆s2)
+| tueq_lref: ∀i. tueq (#i) (#i)
+| tueq_gref: ∀l. tueq (§l) (§l)
+| tueq_bind: ∀p,I,V,T1,T2. tueq T1 T2 → tueq (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
+| tueq_appl: ∀V,T1,T2. tueq T1 T2 → tueq (ⓐV.T1) (ⓐV.T2)
+| tueq_cast: ∀V1,V2,T1,T2. tueq V1 V2 → tueq T1 T2 → tueq (ⓝV1.T1) (ⓝV2.T2)
+.
+
+interpretation
+ "context-free tail sort-irrelevant equivalence (term)"
+ 'ApproxEq T1 T2 = (tueq T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tueq_refl: reflexive … tueq.
+#T elim T -T * [|||| * ]
+/2 width=1 by tueq_sort, tueq_lref, tueq_gref, tueq_bind, tueq_appl, tueq_cast/
+qed.
+
+lemma tueq_sym: symmetric … tueq.
+#T1 #T2 #H elim H -T1 -T2
+/2 width=3 by tueq_sort, tueq_bind, tueq_appl, tueq_cast/
+qed-.
+
+(* Left basic inversion lemmas **********************************************)
+
+fact tueq_inv_sort1_aux: ∀X,Y. X ≅ Y → ∀s1. X = ⋆s1 →
+ ∃s2. Y = ⋆s2.
+#X #Y * -X -Y
+[ #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
+| #i #s #H destruct
+| #l #s #H destruct
+| #p #I #V #T1 #T2 #_ #s #H destruct
+| #V #T1 #T2 #_ #s #H destruct
+| #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
+]
+qed-.
+
+lemma tueq_inv_sort1: ∀Y,s1. ⋆s1 ≅ Y →
+ ∃s2. Y = ⋆s2.
+/2 width=4 by tueq_inv_sort1_aux/ qed-.
+
+fact tueq_inv_lref1_aux: ∀X,Y. X ≅ Y → ∀i. X = #i → Y = #i.
+#X #Y * -X -Y //
+[ #s1 #s2 #j #H destruct
+| #p #I #V #T1 #T2 #_ #j #H destruct
+| #V #T1 #T2 #_ #j #H destruct
+| #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
+]
+qed-.
+
+lemma tueq_inv_lref1: ∀Y,i. #i ≅ Y → Y = #i.
+/2 width=5 by tueq_inv_lref1_aux/ qed-.
+
+fact tueq_inv_gref1_aux: ∀X,Y. X ≅ Y → ∀l. X = §l → Y = §l.
+#X #Y * -X -Y //
+[ #s1 #s2 #k #H destruct
+| #p #I #V #T1 #T2 #_ #k #H destruct
+| #V #T1 #T2 #_ #k #H destruct
+| #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
+]
+qed-.
+
+lemma tueq_inv_gref1: ∀Y,l. §l ≅ Y → Y = §l.
+/2 width=5 by tueq_inv_gref1_aux/ qed-.
+
+fact tueq_inv_bind1_aux: ∀X,Y. X ≅ Y → ∀p,I,V,T1. X = ⓑ{p,I}V.T1 →
+ ∃∃T2. T1 ≅ T2 & Y = ⓑ{p,I}V.T2.
+#X #Y * -X -Y
+[ #s1 #s2 #q #J #W #U1 #H destruct
+| #i #q #J #W #U1 #H destruct
+| #l #q #J #W #U1 #H destruct
+| #p #I #V #T1 #T2 #HT #q #J #W #U1 #H destruct /2 width=3 by ex2_intro/
+| #V #T1 #T2 #_ #q #J #W #U1 #H destruct
+| #V1 #V2 #T1 #T2 #_ #_ #q #J #W #U1 #H destruct
+]
+qed-.
+
+lemma tueq_inv_bind1: ∀p,I,V,T1,Y. ⓑ{p,I}V.T1 ≅ Y →
+ ∃∃T2. T1 ≅ T2 & Y = ⓑ{p,I}V.T2.
+/2 width=3 by tueq_inv_bind1_aux/ qed-.
+
+fact tueq_inv_appl1_aux: ∀X,Y. X ≅ Y → ∀V,T1. X = ⓐV.T1 →
+ ∃∃T2. T1 ≅ T2 & Y = ⓐV.T2.
+#X #Y * -X -Y
+[ #s1 #s2 #W #U1 #H destruct
+| #i #W #U1 #H destruct
+| #l #W #U1 #H destruct
+| #p #I #V #T1 #T2 #_ #W #U1 #H destruct
+| #V #T1 #T2 #HT #W #U1 #H destruct /2 width=3 by ex2_intro/
+| #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct
+]
+qed-.
+
+lemma tueq_inv_appl1: ∀V,T1,Y. ⓐV.T1 ≅ Y →
+ ∃∃T2. T1 ≅ T2 & Y = ⓐV.T2.
+/2 width=3 by tueq_inv_appl1_aux/ qed-.
+
+fact tueq_inv_cast1_aux: ∀X,Y. X ≅ Y → ∀V1,T1. X = ⓝV1.T1 →
+ ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & Y = ⓝV2.T2.
+#X #Y * -X -Y
+[ #s1 #s2 #W1 #U1 #H destruct
+| #i #W1 #U1 #H destruct
+| #l #W1 #U1 #H destruct
+| #p #I #V #T1 #T2 #_ #W1 #U1 #H destruct
+| #V #T1 #T2 #_ #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #HV #HT #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma tueq_inv_cast1: ∀V1,T1,Y. ⓝV1.T1 ≅ Y →
+ ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & Y = ⓝV2.T2.
+/2 width=3 by tueq_inv_cast1_aux/ qed-.
+
+(* Right basic inversion lemmas *********************************************)
+
+lemma tueq_inv_bind2: ∀p,I,V,T2,X1. X1 ≅ ⓑ{p,I}V.T2 →
+ ∃∃T1. T1 ≅ T2 & X1 = ⓑ{p,I}V.T1.
+#p #I #V #T2 #X1 #H
+elim (tueq_inv_bind1 p I V T2 X1)
+[ #T1 #HT #H destruct ]
+/3 width=3 by tueq_sym, ex2_intro/
+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 "static_2/syntax/tueq.ma".
+
+(* TAIL SORT-IRRELEVANT EQUIVALENCE ON TERMS ********************************)
+
+(* Main properties **********************************************************)
+
+theorem tueq_trans: Transitive … tueq.
+#T1 #T #H elim H -T1 -T
+[ #s1 #s #X #H
+ elim (tueq_inv_sort1 … H) -s /2 width=1 by tueq_sort/
+| #i1 #i #H //
+| #l1 #l #H //
+| #p #I #V #T1 #T #_ #IHT #X #H
+ elim (tueq_inv_bind1 … H) -H /3 width=1 by tueq_bind/
+| #V #T1 #T #_ #IHT #X #H
+ elim (tueq_inv_appl1 … H) -H /3 width=1 by tueq_appl/
+| #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
+ elim (tueq_inv_cast1 … H) -H /3 width=1 by tueq_cast/
+]
+qed-.
+
+theorem tueq_canc_sn: left_cancellable … tueq.
+/3 width=3 by tueq_trans, tueq_sym/ qed-.
+
+theorem tueq_canc_dx: right_cancellable … tueq.
+/3 width=3 by tueq_trans, tueq_sym/ qed-.
+
+theorem tueq_repl: ∀T1,T2. T1 ≅ T2 →
+ ∀U1. T1 ≅ U1 → ∀U2. T2 ≅ U2 → U1 ≅ U2.
+/3 width=3 by tueq_canc_sn, tueq_trans/ qed-.
[ { "generic and uniform relocation" * } {
[ [ "for binders" ] "lifts_bind" + "( ⬆*[?] ? ≘ ? )" "lifts_weight_bind" + "lifts_lifts_bind" * ]
[ [ "for term vectors" ] "lifts_vector" + "( ⬆*[?] ? ≘ ? )" "lifts_lifts_vector" * ]
- [ [ "for terms" ] "lifts" + "( ⬆*[?] ? ≘ ? )" "lifts_simple" + "lifts_weight" + "lifts_tdeq" + "lifts_lifts" * ]
+ [ [ "for terms" ] "lifts" + "( ⬆*[?] ? ≘ ? )" "lifts_simple" + "lifts_weight" + "lifts_tdeq" + "lifts_tueq" + "lifts_lifts" * ]
}
]
[ { "syntactic equivalence" * } {
[ [ "for terms" ] "theq" + "( ? ⩳ ? )" "theq_simple" + "theq_tdeq" + "theq_theq" + "theq_simple_vector" * ]
}
]
- [ { "degree-based equivalence" * } {
+ [ { "tail sort-irrelevant equivalence" * } {
+ [ [ "" ] "tueq" + "( ? ≅ ? )" "tueq_tueq" * ]
+ }
+ ]
+ [ { "sort-irrelevant equivalence" * } {
[ [ "" ] "tdeq_ext" + "( ? ≛ ? )" + "( ? ⊢ ? ≛ ? )" * ]
[ [ "" ] "tdeq" + "( ? ≛ ? )" "tdeq_tdeq" * ]
}
["+"; "⨭"; "⨮"; "⨁"; "⊕"; "⊞"; ];
["-"; "÷"; "⊢"; "⊩"; "⧟"; "⊟"; ];
["="; "≝"; "≡"; "≘"; "≗"; "≐"; "≑"; "≛"; "≚"; "≙"; "⌆"; "⧦"; "⊜"; "≋"; "⩳"; "≅"; "⩬"; "≂"; "≃"; "≈"; ];
- ["→"; "↦"; "⇝"; "⤞"; "⇾"; "⤍"; "⤏"; "⤳"; ] ;
+ ["â\86\92"; "⥲"; "â\86¦"; "â\87\9d"; "â¤\9e"; "â\87¾"; "â¤\8d"; "â¤\8f"; "⤳"; ] ;
["⇒"; "⤇"; "➾"; "⇨"; "➡"; "⬈"; "➤"; "➸"; "⇉"; "⥰"; ] ;
["^"; "↑"; "⇡"; ] ;
["⇑"; "⇧"; "⬆"; ] ;
projects / helm.git / commitdiff