--- /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/reducibility/fpr.ma".
+
+(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************)
+
+definition fpc: bi_relation lenv term ≝
+ λL1,T1,L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ ∨ ⦃L2, T2⦄ ➡ ⦃L1, T1⦄.
+
+interpretation
+ "context-free parallel conversion (closure)"
+ 'FocalizedPConv L1 T1 L2 T2 = (fpc L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpc_refl: bi_reflexive … fpc.
+/2 width=1/ qed.
+
+lemma fpc_sym: bi_symmetric … fpc.
+#L1 #L2 #T1 #T2 * /2 width=1/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fpc_fwd_fpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄.
+#L1 #L2 #T1 #T2 * /2 width=4/
+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/conversion/fpc.ma".
+
+(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************)
+
+(* Main properties **********************************************************)
+
+theorem fpc_conf: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ →
+ ∀L2,T2. ⦃L0, T0⦄ ⬌ ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄.
+/3 width=4/ qed.
⦃h, L⦄ ⊢ V •[g, l + 1] W → L ⊢ W ➡* W0 →
⦃h, L⦄ ⊢ T •➡*[g] ⓛ{a}W0.U → snv h g L (ⓐV.T)
| snv_cast: ∀L,W,T,U,l. snv h g L W → snv h g L T →
- ⦃h, L⦄ ⊢ T •[g, l + 1] U → L ⊢ W ⬌* U → snv h g L (ⓝW.T)
+ ⦃h, L⦄ ⊢ T •[g, l + 1] U → L ⊢ U ⬌* W → snv h g L (ⓝW.T)
.
interpretation "stratified native validity (term)"
fact snv_inv_cast_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀W,T. X = ⓝW.T →
∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] &
- L ⊢ W ⬌* U & ⦃h, L⦄ ⊢ T •[g, l + 1] U.
+ ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W.
#h #g #L #X * -L -X
[ #L #k #W #T #H destruct
| #I #L #K #V #i #_ #_ #W #T #H destruct
| #a #I #L #V #T0 #_ #_ #W #T #H destruct
| #a #L #V #W0 #W00 #T0 #U #l #_ #_ #_ #_ #_ #W #T #H destruct
-| #L #W0 #T0 #U0 #l #HW0 #HT0 #HTU0 #HWU0 #W #T #H destruct /2 width=4/
+| #L #W0 #T0 #U0 #l #HW0 #HT0 #HTU0 #HUW0 #W #T #H destruct /2 width=4/
]
qed.
lemma snv_inv_cast: ∀h,g,L,W,T. ⦃h, L⦄ ⊩ ⓝW.T :[g] →
- ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] &
- L ⊢ W ⬌* U & ⦃h, L⦄ ⊢ T •[g, l + 1] U.
+ ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] &
+ ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W.
/2 width=3/ qed-.
lapply (xprs_aaa … HT … HTU) -HTU #H
elim (aaa_inv_abst … H) -H #B0 #A #H1 #HU #H2 destruct
lapply (aaa_mono … H1 … HW0) -W0 #H destruct /3 width=4/
-| #L #W #T #U #l #_ #_ #HTU #HWU * #B #HW * #A #HT
- lapply (aaa_cpcs_mono … HWU … HW A ?) -HWU /2 width=7/ -HTU #H destruct /3 width=3/
+| #L #W #T #U #l #_ #_ #HTU #HUW * #B #HW * #A #HT
+ lapply (aaa_cpcs_mono … HUW A … HW) -HUW /2 width=7/ -HTU #H destruct /3 width=3/
]
qed-.
lemma snv_csn: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → L ⊢ ⬊* T.
#h #g #L #T #H elim (snv_aaa … H) -H /2 width=2/
-qed.
+qed-.
| #L #W0 #U #W #l #_ #_ #HUW #HW0 #IHW0 #IHU #K #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #V0 #T #HVW0 #HTU #H destruct
elim (ssta_inv_lift1 … HUW … HLK … HTU) -HUW #V #HTV #HVW
- lapply (cpcs_inv_lift … HLK … HVW0 … HVW ?) // -W /3 width=4/
+ lapply (cpcs_inv_lift … HLK … HVW … HVW0 ?) // -W /3 width=4/
]
qed-.
fact snv_ssta_conf_aux: ∀h,g,L,T. (
∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] →
- ∀U0. ⦃h, L0⦄ ⊢ T0 •➡*[g] U0 →
- #{L0, T0} < #{L ,T} → ⦃h, L0⦄ ⊩ U0 :[g]
+ ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 →
+ #{L0, T0} < #{L, T} → ⦃h, L0⦄ ⊩ U0 :[g]
) →
∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] →
∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 →
|
| #a #L0 #V #W #W0 #T0 #V0 #l0 #HV #HT0 #HVW #HW0 #HTV0 #X #l #H #H1 #H2 destruct
elim (ssta_inv_appl1 … H) -H #U0 #HTU0 #H destruct
- lapply (IH1 … HT0 U0 ? ?) // [ /3 width=2/ ] -HTU0 #HU0
+ lapply (IH1 … HT0 … HTU0 ?) // #HU0
@(snv_appl … HV HU0 HVW HW0) -HV -HU0 -HVW -HW0
| #L0 #W #T0 #W0 #l0 #_ #HT0 #_ #_ #U0 #l #H #H1 #H2 destruct -W0
- lapply (ssta_inv_cast1 … H) -H #HTU0
- lapply (IH1 … HT0 U0 ? ?) -IH1 -HT0 // -W /3 width=2/
+ lapply (ssta_inv_cast1 … H) -H /2 width=5/
/3 width=5 by cprs_div, cprs_lsubs_trans/ (**) (* /3 width=5/ is a bit slow *)
qed.
-
(* Basic_1: was: pc3_lift *)
lemma cpcs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
--- /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/conversion/fpc.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+definition fpcs: bi_relation lenv term ≝ bi_TC … fpc.
+
+interpretation "context-free parallel equivalence (closure)"
+ 'FocalizedPConvStar L1 T1 L2 T2 = (fpcs L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fpcs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
+ (∀L,L2,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L2 T2.
+/3 width=7 by bi_TC_star_ind/ qed-.
+
+lemma fpcs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
+ (∀L1,L,T1,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L1 T1.
+/3 width=7 by bi_TC_star_ind_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fpcs_refl: bi_reflexive … fpcs.
+/2 width=1/ qed.
+
+lemma fpcs_sym: bi_symmetric … fpcs.
+/3 width=1/ qed.
+
+lemma fpcs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma fpcs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/2 width=4/ qed.
+
+lemma fpcs_fpr_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=1/ qed.
+
+lemma fpcs_fpr_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡ ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=1/ qed.
+
+lemma fpcs_fpr_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ →
+ ∀L2,T2. ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed.
+
+lemma fpcs_fpr_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ →
+ ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed.
+
+lemma fpcs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ →
+ ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed.
+
+lemma fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ qed-.
+
+lemma fpcs_fpr_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡ ⦃L1, T1⦄ →
+ ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4/ 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/computation/fprs_aaa.ma".
+include "basic_2/equivalence/fpcs_fpcs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Main properties about atomic arity assignment on terms *******************)
+
+theorem aaa_fpcs_mono: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ →
+ ∀A1. L1 ⊢ T1 ⁝ A1 → ∀A2. L2 ⊢ T2 ⁝ A2 →
+ A1 = A2.
+#L1 #L2 #T1 #T2 #H12 #A1 #HT1 #A2 #HT2
+elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
+lapply (aaa_fprs_conf … HT1 … H1) -L1 -T1 #HT1
+lapply (aaa_fprs_conf … HT2 … H2) -L2 -T2 #HT2
+lapply (aaa_mono … HT1 … HT2) -L -T //
+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/computation/fprs_cprs.ma".
+include "basic_2/equivalence/cpcs_cpcs.ma".
+include "basic_2/equivalence/fpcs_fprs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Properties on context-sensitive parallel equivalence for terms ***********)
+
+lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄.
+#L #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *)
+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/computation/fprs_fprs.ma".
+include "basic_2/conversion/fpc_fpc.ma".
+include "basic_2/equivalence/fpcs_fprs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fpcs_inv_fprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ & ⦃L2, T2⦄ ➡* ⦃L, T⦄.
+#L1 #L2 #T1 #T2 #H @(fpcs_ind … H) -L2 -T2
+[ /3 width=4/
+| #L #L2 #T #T2 #_ #HT2 * #L0 #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
+ [ elim (fprs_strip … HT2 … HT0) -L -T #L #T #HT2 #HT0
+ lapply (fprs_strap1 … HT10 … HT0) -L0 -T0 /2 width=4/
+ | lapply (fprs_strap2 … HT2 … HT0) -L -T /2 width=4/
+ ]
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpr_fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
+elim (fprs_strip … HT2 … HT1) /2 width=4 by fpr_fprs_div/
+qed-.
+
+lemma fprs_fpr_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L2, T2⦄ ⬌* ⦃L1, T1⦄.
+#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
+elim (fprs_strip … HT2 … HT1) /2 width=4 by fprs_fpr_div/
+qed-.
+
+lemma fprs_conf: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L #L1 #L2 #T #T1 #T2 #HT1 #HT2
+elim (fprs_conf … HT1 … HT2) /2 width=4/
+qed-.
+
+lemma fpcs_strip: ∀L0,L1,T0,T1. ⦃L0, T0⦄ ⬌ ⦃L1, T1⦄ →
+ ∀L2,T2. ⦃L0, T0⦄ ⬌* ⦃L2, T2⦄ →
+ ∃∃L,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ & ⦃L2, T2⦄ ⬌ ⦃L, T⦄.
+/3 width=4/ qed.
+
+(* Main properties **********************************************************)
+
+theorem fpcs_trans: bi_transitive … fpcs.
+/2 width=4/ qed.
+
+theorem fpcs_canc_sn: ∀L,L1,L2,T,T1,T2. ⦃L, T⦄ ⬌* ⦃L1, T1⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *)
+
+theorem fpcs_canc_dx: ∀L1,L2,L,T1,T2,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L2, T2⦄ ⬌* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4 by fpcs_trans, fpcs_sym/ qed. (**) (* /3 width=3/ is too slow *)
--- /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/computation/fprs.ma".
+include "basic_2/equivalence/fpcs.ma".
+
+(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+
+(* Properties on context-free parallel computation for closures *************)
+
+(* Note: lemma 1000 *)
+lemma fpcs_fprs_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /3 width=4/
+qed.
+
+lemma fpcs_fprs_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡* ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /3 width=4/
+qed.
+
+lemma fpcs_fprs_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind … H) -L2 -T2 /width=1/ /2 width=4/
+qed.
+
+lemma fpcs_fprs_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #H #L2 #T2 #HT2 @(fprs_ind_dx … H) -L1 -T1 /width=1/ /2 width=4/
+qed.
+
+lemma fpcs_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #HT1 #L2 #T2 #H @(fprs_ind_dx … H) -L2 -T2 /width=1/ /2 width=4/
+qed.
+
+lemma fpcs_fprs_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡* ⦃L1, T1⦄ → ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #H #T2 #HT2 @(fprs_ind … H) -L1 -T1 /width=1/ /3 width=4 by fpcs_fpr_conf/ (**) (* /2 width=4/ does not work *)
+qed.
+
+lemma fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+#L1 #L #T1 #T #HT1 #T2 #L2 #H @(fprs_ind_dx … H) -L2 -T2 /2 width=1/ /2 width=4/
+qed.
+
+lemma fprs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡* ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=7 by bi_step, fprs_div/ qed-.
+
+lemma fpr_fprs_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡* ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/3 width=4 by bi_step, fprs_div/ 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/static/ssta.ma".
+include "basic_2/computation/cprs.ma".
+include "basic_2/equivalence/cpcs.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **)
+
+(* Note: this is not transitive *)
+inductive lsubse (h:sh) (g:sd h): relation lenv ≝
+| lsubse_atom: lsubse h g (⋆) (⋆)
+| lsubse_pair: ∀I,L1,L2,V. lsubse h g L1 L2 →
+ lsubse h g (L1. ⓑ{I} V) (L2. ⓑ{I} V)
+| lsubse_abbr: ∀L1,L2,V1,V2,W1,W2,l. L1 ⊢ W1 ⬌* W2 →
+ ⦃h, L1⦄ ⊢ V1 •[g, l + 1] W1 → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 →
+ lsubse h g L1 L2 → lsubse h g (L1. ⓓV1) (L2. ⓛW2)
+.
+
+interpretation
+ "local environment refinement (context-sensitive parallel equivalence)"
+ 'CrSubEqSE h g L1 L2 = (lsubse h g L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubse_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 = ⋆ → L2 = ⋆.
+#h #g #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubse_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊢•⊑[g] L2 → L2 = ⋆.
+/2 width=5 by lsubse_inv_atom1_aux/ qed-.
+
+fact lsubse_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ (∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
+ ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+ K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
+#h #g #L1 #L2 * -L1 -L2
+[ #J #K1 #U1 #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=10/
+]
+qed-.
+
+lemma lsubse_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊢•⊑[g] L2 →
+ (∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
+ ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+ K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
+/2 width=3 by lsubse_inv_pair1_aux/ qed-.
+
+fact lsubse_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
+#h #g #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubse_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊢•⊑[g] ⋆ → L1 = ⋆.
+/2 width=5 by lsubse_inv_atom2_aux/ qed-.
+
+fact lsubse_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 →
+ ∀I,K2,W2. L2 = K2. ⓑ{I} W2 →
+ (∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
+ ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+ K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
+#h #g #L1 #L2 * -L1 -L2
+[ #J #K2 #U2 #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=10/
+]
+qed-.
+
+lemma lsubse_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊢•⊑[g] K2. ⓑ{I} W2 →
+ (∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
+ ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+ K1 ⊢ W1 ⬌* W2 & h ⊢ K1 ⊢•⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
+/2 width=3 by lsubse_inv_pair2_aux/ qed-.
+
+(* Basic_forward lemmas *****************************************************)
+
+lemma lsubse_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 ≼[0, |L1|] L2.
+#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+lemma lsubse_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 → L1 ≼[0, |L2|] L2.
+#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsubse_refl: ∀h,g,L. h ⊢ L ⊢•⊑[g] L.
+#h #g #L elim L -L // /2 width=1/
+qed.
+
+lemma lsubse_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+/3 width=5 by lsubse_fwd_lsubs2, cprs_lsubs_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/equivalence/cpcs_cpcs.ma".
+include "basic_2/equivalence/lsubse.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **)
+
+(* Properties on context-sensitive parallel equivalence for terms ***********)
+
+lemma lsubse_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 →
+ ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
+/3 width=5 by lsubse_fwd_lsubs2, cpcs_lsubs_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/equivalence/lsubse.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **)
+
+(* Properties concerning basic local environment slicing ********************)
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubse_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 →
+ ∀K1,e. ⇩[0, e] L1 ≡ K1 →
+ ∃∃K2. h ⊢ K1 ⊢•⊑[g] K2 & ⇩[0, e] L2 ≡ K2.
+#h #g #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=6/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubse_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ⊢•⊑[g] L2 →
+ ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. h ⊢ K1 ⊢•⊑[g] K2 & ⇩[0, e] L1 ≡ K1.
+#h #g #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HW12 #HVW1 #HWV2 #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=6/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+]
+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/computation/xprs_lsubss.ma".
+*)
+include "basic_2/equivalence/lsubse.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR CONTEXT-SENSITIVE PARALLEL EQUIVALENCE **)
+
+(* Properties on stratified native type assignment **************************)
+
+axiom lsubse_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[g,l] U2 →
+ ∀L1. h ⊢ L1 ⊢•⊑[g] L2 →
+ ∃∃U1. ⦃h, L1⦄ ⊢ T •[g,l] U1 & L1 ⊢ U1 ⬌* U2.
--- /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/dynamic/snv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Note: this is not transitive *)
+inductive lsubsv (h:sh) (g:sd h): relation lenv ≝
+| lsubsv_atom: lsubsv h g (⋆) (⋆)
+| lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 →
+ lsubsv h g (L1. ⓑ{I} V) (L2. ⓑ{I} V)
+| lsubsv_abbr: ∀L1,L2,V1,V2,W1,W2,l. ⦃h, L1⦄ ⊩ V1 :[g] → L1 ⊢ W2 ⬌* W1 →
+ ⦃h, L1⦄ ⊢ V1 •[g, l + 1] W1 → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 →
+ lsubsv h g L1 L2 → lsubsv h g (L1. ⓓV1) (L2. ⓛW2)
+.
+
+interpretation
+ "local environment refinement (stratified native validity)"
+ 'CrSubEqV h g L1 L2 = (lsubsv h g L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 = ⋆ → L2 = ⋆.
+#h #g #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊩:⊑[g] L2 → L2 = ⋆.
+/2 width=5 by lsubsv_inv_atom1_aux/ qed-.
+
+fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
+ ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+ K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
+#h #g #L1 #L2 * -L1 -L2
+[ #J #K1 #U1 #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HW21 #HVW1 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=10/
+]
+qed-.
+
+lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊩:⊑[g] L2 →
+ (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
+ ∃∃K2,W1,W2,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+ K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
+/2 width=3 by lsubsv_inv_pair1_aux/ qed-.
+
+fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
+#h #g #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊩:⊑[g] ⋆ → L1 = ⋆.
+/2 width=5 by lsubsv_inv_atom2_aux/ qed-.
+
+fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
+ ∀I,K2,W2. L2 = K2. ⓑ{I} W2 →
+ (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
+ ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+ K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
+#h #g #L1 #L2 * -L1 -L2
+[ #J #K2 #U2 #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HW21 #HVW1 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=11/
+]
+qed-.
+
+lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊩:⊑[g] K2. ⓑ{I} W2 →
+ (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
+ ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+ K1 ⊢ W2 ⬌* W1 & h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
+/2 width=3 by lsubsv_inv_pair2_aux/ qed-.
+
+(* Basic_forward lemmas *****************************************************)
+
+lemma lsubsv_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L1|] L2.
+#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+lemma lsubsv_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L2|] L2.
+#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsubsv_refl: ∀h,g,L. h ⊢ L ⊩:⊑[g] L.
+#h #g #L elim L -L // /2 width=1/
+qed.
+
+lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+/3 width=5 by lsubsv_fwd_lsubs2, cprs_lsubs_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/equivalence/cpcs_cpcs.ma".
+include "basic_2/dynamic/lsubsv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties on context-sensitive parallel equivalence for terms ***********)
+
+lemma lsubsv_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
+ ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
+/3 width=5 by lsubsv_fwd_lsubs2, cpcs_lsubs_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/dynamic/lsubsv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties concerning basic local environment slicing ********************)
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
+ ∀K1,e. ⇩[0, e] L1 ≡ K1 →
+ ∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & ⇩[0, e] L2 ≡ K2.
+#h #g #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HW21 #HVW1 #HWV2 #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=6/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+]
+qed.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
+ ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & ⇩[0, e] L1 ≡ K1.
+#h #g #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HW21 #HVW1 #HWV2 #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=6/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+]
+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/dynamic/lsubsv_ldrop.ma".
+include "basic_2/dynamic/lsubsv_ssta.ma".
+include "basic_2/dynamic/lsubsv_cpcs.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties concerning stratified native validity *************************)
+
+axiom lsubsv_xprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
+ ∀T1,T2. ⦃h, L2⦄ ⊢ T1 •➡*[g] T2 → ⦃h, L1⦄ ⊢ T1 •➡*[g] T2.
+(*
+/3 width=3 by lsubsv_fwd_lsubss, lsubss_xprs_trans/ qed-.
+*)
+axiom lsubsv_snv_trans: ∀h,g,L2,T. ⦃h, L2⦄ ⊩ T :[g] →
+ ∀L1. h ⊢ L1 ⊩:⊑[g] L2 → ⦃h, L1⦄ ⊩ T :[g].
+(*
+#h #g #L2 #T #H elim H -L2 -T //
+[ #I2 #L2 #K2 #V2 #i #HLK2 #_ #IHV2 #L1 #HL12
+ elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsubsv_inv_pair2 … H) -H * #K1 [ | -IHV2 ]
+ [ #HK12 #H destruct /3 width=5/
+ | #V1 #l #HV1 #_ #_ #_ #H destruct /2 width=5/
+ ]
+| #a #I #L2 #V #T #_ #_ #IHV #IHT #L1 #HL12 /4 width=1/
+| #a #L2 #V #W #W0 #T #U #l #_ #_ #HVW #HW0 #HTU #IHV #IHT #L1 #HL12
+ lapply (IHV … HL12) -IHV #HV
+ lapply (IHT … HL12) -IHT #HT
+ lapply (lsubsv_ssta_trans … HVW … HL12) -HVW #HVW
+ lapply (lsubsv_cprs_trans … HL12 … HW0) -HW0 #HW0
+ lapply (lsubsv_xprs_trans … HL12 … HTU) -HL12 -HTU /2 width=8/
+| #L2 #W #T #U #l #_ #_ #HTU #HWU #IHW #IHT #L1 #HL12
+ lapply (IHW … HL12) -IHW #HW
+ lapply (IHT … HL12) -IHT #HT
+ lapply (lsubsv_ssta_trans … HTU … HL12) -HTU #HTU
+ lapply (lsubsv_cpcs_trans … HL12 … HWU) -HL12 -HWU /2 width=4/
+]
+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/computation/xprs_lsubss.ma".
+include "basic_2/dynamic/lsubsv.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
+
+(* Properties on stratified native type assignment **************************)
+
+axiom lsubsv_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[g,l] U2 →
+ ∀L1. h ⊢ L1 ⊩:⊑[g] L2 →
+ ∃∃U1. L1 ⊢ U2 ⬌* U1 & ⦃h, L1⦄ ⊢ T •[g,l] U1.
non associative with precedence 45
for @{ 'PConvStar $L $T1 $T2 }.
+notation "hvbox( h ⊢ break term 46 L1 ⊢ • ⊑ [ g ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'CrSubEqSE $h $g $L1 $L2 }.
+
notation "hvbox( ⦃ L1 ⦄ ⬌ * break ⦃ L2 ⦄ )"
non associative with precedence 45
for @{ 'FocalizedPConvStar $L1 $L2 }.
elim (aaa_inv_abst … HT1) -HT1 #B0 #A0 #HB0 #HA0 #H destruct
lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB
lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
- lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/
+ lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 [2,4: // |3,5: skip ] /2 width=1/ -T0 -L1 -V1 /4 width=7/
| #a #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
elim (aaa_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0
lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2
lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0
- lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 /width=5/ -V1 -T0 /2 width=1/ -L1 -W0 #HT2
+ lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 [2,4: // |3,5: skip ] /2 width=1/ -V1 -T0 -L1 -W0 #HT2
@(aaa_abbr … HW2) -HW2
@(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
]
<key name="ex">5 4</key>
<key name="ex">5 5</key>
<key name="ex">6 4</key>
+ <key name="ex">6 5</key>
<key name="ex">6 6</key>
<key name="ex">6 7</key>
- <key name="ex">7 4</key>
<key name="ex">7 7</key>
<key name="or">3</key>
<key name="or">4</key>
interpretation "multiple existental quantifier (6, 4)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_4 ? ? ? ? P0 P1 P2 P3 P4 P5).
+(* multiple existental quantifier (6, 5) *)
+
+inductive ex6_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→Prop) : Prop ≝
+ | ex6_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → ex6_5 ? ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (6, 5)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5).
+
(* multiple existental quantifier (6, 6) *)
inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
interpretation "multiple existental quantifier (6, 7)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
-(* multiple existental quantifier (7, 4) *)
-
-inductive ex7_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→Prop) : Prop ≝
- | ex7_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → ex7_4 ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (7, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
-
(* multiple existental quantifier (7, 7) *)
inductive ex7_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) }.
+(* multiple existental quantifier (6, 5) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P5) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P5) }.
+
(* multiple existental quantifier (6, 6) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) }.
-(* multiple existental quantifier (7, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P6) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) }.
-
(* multiple existental quantifier (7, 7) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
projects / helm.git / commitdiff