+++ /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 "hvbox( T1 break ⊢ ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarSnAlt $T1 $d $e $T2 }.
-
-include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
-include "basic_2/unfold/ltpss_sn_ltpss_sn.ma".
-
-(* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
-
-(* alternative definition of ltpss_sn *)
-definition ltpssa: nat → nat → relation lenv ≝
- λd,e. TC … (ltpss_dx d e).
-
-interpretation "parallel unfold (local environment, sn variant) alternative"
- 'PSubstStarSnAlt L1 d e L2 = (ltpssa d e L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma ltpssa_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. L1 ⊢ ▶▶* [d, e] L → L ▶* [d, e] L2 → R L → R L2) →
- ∀L2. L1 ⊢ ▶▶* [d, e] L2 → R L2.
-#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma ltpssa_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. L1 ▶* [d, e] L → L ⊢ ▶▶* [d, e] L2 → R L → R L1) →
- ∀L1. L1 ⊢ ▶▶* [d, e] L2 → R L1.
-#d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma ltpssa_refl: ∀L,d,e. L ⊢ ▶▶* [d, e] L.
-/2 width=1/ qed.
-
-lemma ltpssa_tpss2: ∀I,L1,V1,V2,e. L1 ⊢ V1 ▶*[0, e] V2 →
- ∀L2. L1 ⊢ ▶▶* [0, e] L2 →
- L1.ⓑ{I}V1 ⊢ ▶▶* [O, e + 1] L2.ⓑ{I}V2.
-#I #L1 #V1 #V2 #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
-[ /3 width=1/ | /3 width=5/ ]
-qed.
-
-lemma ltpssa_tpss1: ∀I,L1,V1,V2,d,e. L1 ⊢ V1 ▶*[d, e] V2 →
- ∀L2. L1 ⊢ ▶▶* [d, e] L2 →
- L1.ⓑ{I}V1 ⊢ ▶▶* [d + 1, e] L2.ⓑ{I}V2.
-#I #L1 #V1 #V2 #d #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
-[ /3 width=1/ | /3 width=5/ ]
-qed.
-
-lemma ltpss_sn_ltpssa: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶▶* [d, e] L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // /2 width=1/
-qed.
-
-lemma ltpss_sn_dx_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
- ∀L2. L ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-#L1 #L #d #e #H elim H -L1 -L -d -e
-[ #d #e #X #H
- lapply (ltpss_dx_inv_atom1 … H) -H #H destruct //
-| #L #I #V #X #H
- lapply (ltpss_dx_inv_refl_O2 … H) -H #H destruct //
-| #L1 #L #I #V1 #V #e #_ #HV1 #IHL1 #X #H
- elim (ltpss_dx_inv_tpss21 … H ?) -H // <minus_plus_m_m
- #L2 #V2 #HL2 #HV2 #H destruct
- lapply (IHL1 … HL2) -L #HL12
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
- lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
-| #L1 #L #I #V1 #V #d #e #_ #HV1 #IHL1 #X #H
- elim (ltpss_dx_inv_tpss11 … H ?) -H // <minus_plus_m_m
- #L2 #V2 #HL2 #HV2 #H destruct
- lapply (IHL1 … HL2) -L #HL12
- lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
- lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
-]
-qed.
-
-lemma ltpss_dx_sn_trans_eq: ∀L1,L,d,e. L1 ▶* [d, e] L →
- ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-/3 width=3/ qed.
-
-lemma ltpssa_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
- ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
-/3 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma ltpssa_ltpss_sn: ∀L1,L2,d,e. L1 ⊢ ▶▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
-#L1 #L2 #d #e #H @(ltpssa_ind … H) -L2 // /2 width=3/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_sn_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
- ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
-#L0 #L1 #d1 #e1 #H #L2 #d2 #e2 #HL02
-lapply (ltpss_sn_ltpssa … H) -H #HL01
-elim (ltpssa_strip … HL01 … HL02) -L0
-/3 width=3 by ltpssa_ltpss_sn, ex2_intro/
-qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-lemma ltpss_sn_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
- L0 ⊢ U2 ▶* [d1, e1] T.
-#L0 #T2 #U2 #d2 #e2 #HTU2 #L1 #d1 #e1 #H
-lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L1 /2 width=3/ -HTU2
-#L #L1 #H #HL1 * #T #HT2 #HU2T
-lapply (ltpssa_ltpss_sn … H) -H #HL0
-lapply (ltpss_sn_dx_trans_eq … HL0 … HL1) -HL0 #HL01
-elim (ltpss_dx_tpss_conf … HT2 … HL1) -HT2 -HL1 #T0 #HT20 #HT0
-lapply (ltpss_sn_tpss_trans_eq … HT0 … HL01) -HT0 -HL01 #HT0
-lapply (tpss_trans_eq … HU2T HT0) -T /2 width=3/
-qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-lemma ltpss_sn_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
- L1 ⊢ ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
- ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L1 ⊢ T ▶* [d1, e1] U2.
-#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #H #HTU2
-lapply (ltpss_sn_ltpssa … H) -H #HL10
-@(ltpssa_ind_dx … HL10) -L1 /2 width=3/ -HTU2
-#L1 #L #HL1 #_ * #T #HT2 #HTU2
-elim (ltpss_dx_tpss_trans_down … HL1 HT2) -HT2 // #T0 #HT20 #HT0 -Hde2d1
-lapply (tpss_trans_eq … HT0 HTU2) -T #HT0U2
-lapply (ltpss_dx_tpss_trans_eq … HT0U2 … HL1) -HT0U2 -HL1 /2 width=3/
-qed.
-
-(* Main properties **********************************************************)
-
-theorem ltpssa_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ⊢ ▶▶* [d2, e2] L2 →
- ∃∃L. L1 ⊢ ▶▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
-/3 width=3/ qed.
-
-(* Note: this should go in ltpss_sn_ltpss_sn.ma *)
-theorem ltpss_sn_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
- ∀L2,d2,e2. L0 ⊢ ▶* [d2, e2] L2 →
- ∃∃L. L1 ⊢ ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
-#L0 #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2
-lapply (ltpss_sn_ltpssa … H1) -H1 #HL01
-lapply (ltpss_sn_ltpssa … H2) -H2 #HL02
-elim (ltpssa_conf … HL01 … HL02) -L0
-/3 width=3 by ltpssa_ltpss_sn, ex2_intro/
-qed.
projects / helm.git / blobdiff