+++ /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 "ground_2/xoa/ex_5_1.ma".
-include "ground_2/xoa/ex_9_3.ma".
-include "basic_2/rt_transition/cpm_tdeq.ma".
-include "basic_2/rt_computation/fpbg_fqup.ma".
-include "basic_2/dynamic/cnv_fsb.ma".
-
-(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-
-(* Inversion lemmas with restricted rt-transition for terms *****************)
-
-lemma cnv_cpr_tdeq_fwd_refl (h) (a) (G) (L):
- ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 → ⦃G,L⦄ ⊢ T1 ![h,a] → T1 = T2.
-#h #a #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 … 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 … 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_sn (h) (a) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![h,a] →
- ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
-#h #a #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 #HT
- lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV
- /2 width=4 by ex5_intro/
-| #X1 #HXT1 #HX1 #H1 #H destruct
- elim (cnv_fpbg_refl_false … 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_sn (h) (a) (n) (G) (L):
- ∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![h,a] →
- ∀X. ⦃G,L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛ X →
- ∃∃m,q,W,U1,T2. ad a m & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ V ➡*[1,h] W & ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
- & ⦃G,L⦄⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
-#h #a #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 #U1 #Hm #HV #HT #HVW #HTU1
- lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV
- /3 width=7 by ex8_5_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_sn (h) (a) (n) (G) (L):
- ∀U1,T1. ⦃G,L⦄ ⊢ ⓝU1.T1 ![h,a] →
- ∀X. ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
- ∃∃U0,U2,T2. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 & ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0
- & ⦃G,L⦄ ⊢ U1 ![h,a] & ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
- & ⦃G,L⦄ ⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
-#h #a #n #G #L #U1 #T1 #H0 #X #H1 #H2
-elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
-[ #U2 #T2 #HU12 #HT12 #H destruct
- elim (tdeq_inv_pair … H2) -H2 #_ #H2U12 #H2T12
- elim (cnv_inv_cast … H0) -H0 #U0 #HU1 #HT1 #HU10 #HT1U0
- /2 width=7 by ex9_3_intro/
-| #HT1X
- elim (cnv_fpbg_refl_false … 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 … H0) -a
- @(fpbg_tdeq_div … H2) -H2
- /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_ee, tdeq_inv_pair_xy_x/
-]
-qed-.
-
-lemma cpm_tdeq_inv_bind_dx (h) (a) (n) (p) (I) (G) (L):
- ∀X. ⦃G,L⦄ ⊢ X ![h,a] →
- ∀V,T2. ⦃G,L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 →
- ∃∃T1. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
-#h #a #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
-elim (tdeq_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
-elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
-/2 width=5 by ex5_intro/
-qed-.
-
-(* Eliminators with restricted rt-transition for terms **********************)
-
-lemma cpm_tdeq_ind (h) (a) (n) (G) (Q:relation3 …):
- (∀I,L. n = 0 → Q L (⓪{I}) (⓪{I})) →
- (∀L,s. n = 1 → Q L (⋆s) (⋆(⫯[h]s))) →
- (∀p,I,L,V,T1. ⦃G,L⦄⊢ V![h,a] → ⦃G,L.ⓑ{I}V⦄⊢T1![h,a] →
- ∀T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
- Q (L.ⓑ{I}V) T1 T2 → Q L (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
- ) →
- (∀m. ad a m →
- ∀L,V. ⦃G,L⦄ ⊢ V ![h,a] → ∀W. ⦃G,L⦄ ⊢ V ➡*[1,h] W →
- ∀p,T1,U1. ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![h,a] →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
- Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
- ) →
- (∀L,U0,U1,T1. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 → ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 →
- ∀U2. ⦃G,L⦄ ⊢ U1 ![h,a] → ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
- ∀T2. ⦃G,L⦄ ⊢ T1 ![h,a] → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
- Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
- ) →
- ∀L,T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → Q L T1 T2.
-#h #a #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
-@(insert_eq_0 … G) #F
-@(fqup_wf_ind_eq (Ⓣ) … F L T1) -L -T1 -F
-#G0 #L0 #T0 #IH #F #L * [| * [| * ]]
-[ #I #_ #_ #_ #_ #HF #X #H1X #H2X destruct -G0 -L0 -T0
- elim (cpm_tdeq_inv_atom_sn … H1X H2X) -H1X -H2X *
- [ #H1 #H2 destruct /2 width=1 by/
- | #s #H1 #H2 #H3 destruct /2 width=1 by/
- ]
-| #p #I #V #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
- elim (cpm_tdeq_inv_bind_sn … H0 … H1X H2X) -H0 -H1X -H2X #T2 #HV #HT1 #H1T12 #H2T12 #H destruct
- /3 width=5 by/
-| #V #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
- elim (cpm_tdeq_inv_appl_sn … H0 … H1X H2X) -H0 -H1X -H2X #m #q #W #U1 #T2 #Hm #HV #HVW #HTU1 #HT1 #H1T12 #H2T12 #H destruct
- /3 width=7 by/
-| #U1 #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
- elim (cpm_tdeq_inv_cast_sn … H0 … H1X H2X) -H0 -H1X -H2X #U0 #U2 #T2 #HU10 #HT1U0 #HU1 #H1U12 #H2U12 #HT1 #H1T12 #H2T12 #H destruct
- /3 width=5 by/
-]
-qed-.
-
-(* Advanced properties with restricted rt-transition for terms **************)
-
-lemma cpm_tdeq_free (h) (a) (n) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
- ∀F,K. ⦃F,K⦄ ⊢ T1 ➡[n,h] T2.
-#h #a #n #G #L #T1 #H0 #T2 #H1 #H2
-@(cpm_tdeq_ind … H0 … H1 H2) -L -T1 -T2
-[ #I #L #H #F #K destruct //
-| #L #s #H #F #K destruct //
-| #p #I #L #V #T1 #_ #_ #T2 #_ #_ #IH #F #K
- /2 width=1 by cpm_bind/
-| #m #_ #L #V #_ #W #_ #q #T1 #U1 #_ #_ #T2 #_ #_ #IH #F #K
- /2 width=1 by cpm_appl/
-| #L #U0 #U1 #T1 #_ #_ #U2 #_ #_ #_ #T2 #_ #_ #_ #IHU #IHT #F #K
- /2 width=1 by cpm_cast/
-]
-qed-.
-
-(* Advanced inversion lemmas with restricted rt-transition for terms ********)
-
-lemma cpm_tdeq_inv_bind_sn_void (h) (a) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![h,a] →
- ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
-#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
-elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
-/3 width=5 by ex5_intro, cpm_tdeq_free/
-qed-.
projects / helm.git / blobdiff