--- /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/substitution/cpye_lift.ma".
+include "basic_2/reduction/lpx_cpye.ma".
+include "basic_2/computation/cpxs_cpxs.ma".
+include "basic_2/computation/lpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+
+(* Forward lemmas on evaluation for extended substitution *******************)
+
+lemma cpx_cpye_fwd_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+ ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃U1⦄ →
+ ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
+ ⦃G, L1⦄ ⊢ U1 ➡*[h, g] U2.
+#h #g #G #L1 #L2 #H @(lpxs_ind_dx … H) -L1
+[ /3 width=9 by cpx_cpxs, cpx_cpye_fwd_lpx/
+| #L1 #L #HL1 #_ #IHL2 #T1 #T2 #HT12 #U1 #d #e #HTU1 #U2 #HTU2
+ elim (cpye_total G L T2 d e) #X2 #HTX2
+ lapply (cpx_cpye_fwd_lpx … HT12 … HL1 … HTU1 … HTX2) -T1
+ /4 width=9 by lpx_cpxs_trans, cpxs_strap2/ (**) (* full auto too long: 41s *)
+]
+qed-.
(* Advanced properties ******************************************************)
-fact le_repl_sn_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
-// qed-.
-
-axiom cpxs_cpys_conf_lpxs: ∀h,g,G,d,e.
- ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h, g] T1 →
- ∀T2. ⦃G, L0⦄ ⊢ T0 ▶*[d, e] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h, g] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] T & ⦃G, L0⦄ ⊢ T2 ➡*[h, g] T.
-
-axiom cpxs_conf_lpxs_lpys: ∀h,g,G,d,e.
- ∀I,L0,V0,T0,T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡*[h, g] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h, g] L1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ▶*[d, e] V2 →
- ∃∃T. ⦃G, L1.ⓑ{I}V0⦄ ⊢ T1 ▶*[⫯d, e] T & ⦃G, L0.ⓑ{I}V2⦄ ⊢ T0 ➡*[h, g] T.
-
-
-include "basic_2/reduction/cpx_cpys.ma".
-
-fact pippo_aux: ∀h,g,G,L1,T,T1,d,e. ⦃G, L1⦄ ⊢ T ▶*[d, e] T1 → e = ∞ →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- ∃∃T2. ⦃G, L2⦄ ⊢ T ▶*[d, e] T2 & ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 &
- L1 ⋕[T, d] L2 ↔ T1 = T2.
-#h #g #G #L1 #T #T1 #d #e #H @(cpys_ind_alt … H) -G -L1 -T -T1 -d -e [ * ]
-[ /5 width=5 by lpxs_fwd_length, lleq_sort, ex3_intro, conj/
-| #i #G #L1 elim (lt_or_ge i (|L1|)) [2: /6 width=6 by lpxs_fwd_length, lleq_free, le_repl_sn_aux, ex3_intro, conj/ ]
- #Hi #d elim (ylt_split i d) [ /5 width=5 by lpxs_fwd_length, lleq_skip, ex3_intro, conj/ ]
- #Hdi #e #He #L2 elim (lleq_dec (#i) L1 L2 d) [ /4 width=5 by lpxs_fwd_length, ex3_intro, conj/ ]
- #HnL12 #HL12 elim (ldrop_O1_lt L1 i) // -Hi #I #K1 #V1 #HLK1
- elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpxs_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- elim (lift_total V2 0 (i+1)) #W2 #HVW2
- @(ex3_intro … W2) /2 width=7 by cpxs_delta, cpys_subst/ -I -K1 -V1 -Hdi
- @conj #H [ elim HnL12 // | destruct elim (lift_inv_lref2_be … HVW2) // ]
-| /5 width=5 by lpxs_fwd_length, lleq_gref, ex3_intro, conj/
-| #I #G #L1 #K1 #V #V1 #T1 #i #d #e #Hdi #Hide #HLK1 #HV1 #HVT1 #_ #He #L2 #HL12 destruct
- elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpxs_inv_pair1 … H) -H #K2 #W #HK12 #HVW #H destruct
- elim (cpxs_cpys_conf_lpxs … HVW … HV1 … HK12) -HVW -HV1 -HK12 #W1 #HW1 #VW1
- elim (lift_total W1 0 (i+1)) #U1 #HWU1
- lapply (ldrop_fwd_drop2 … HLK1) -HLK1 #HLK1
- @(ex3_intro … U1) /2 width=10 by cpxs_lift, cpys_subst/
-| #a #I #G #L #V #V1 #T #T1 #d #e #HV1 #_ #IHV1 #IHT1 #He #L2 #HL12
- elim (IHV1 … HL12) // -IHV1 #V2 #HV2 #HV12 * #H1V #H2V
- elim (IHT1 … (L2.ⓑ{I}V2)) /4 width=3 by lpxs_cpx_trans, lpxs_pair, cpys_cpx/ -IHT1 -He #T2 #HT2 #HT12 * #H1T #H2T
- elim (cpxs_conf_lpxs_lpys … HT12 … HL12 … HV1) -HT12 -HL12 -HV1 #T0 #HT20 #HT10
- @(ex3_intro … (ⓑ{a,I}V2.T0))
- [ @cpys_bind // @(cpys_trans_eq … T2) /3 width=5 by lsuby_cpys_trans, lsuby_succ/
- | /2 width=1 by cpxs_bind/
- | @conj #H destruct
- [ elim (lleq_inv_bind … H) -H #HV #HT >H1V -H1V //
- | @lleq_bind /2 width=1/
-
-
- /3 width=5 by lsuby_cpys_trans, lsuby_succ/
-| #I #G #L #V #V1 #T #T1 #d #e #HV1 #HT1 #IHV1 #IHT1 #He #L2 #HL12
- elim (IHV1 … HL12) // -IHV1 #V2 #HV2 #HV12 * #H1V #H2V
- elim (IHT1 … HL12) // -IHT1 #T2 #HT2 #HT12 * #H1T #H2T -He -HL12
- @(ex3_intro … (ⓕ{I}V2.T2)) /2 width=1 by cpxs_flat, cpys_flat/
- @conj #H destruct [2: /3 width=1 by lleq_flat/ ]
- elim (lleq_inv_flat … H) -H /3 width=1 by eq_f2/
-]
-
-
-
- [
- | @cpxs_bind //
- @(lpx_cpxs_trans … HT12)
-|
-]
-
axiom lleq_lpxs_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ∀K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ⋕[T, d] K2.
(*
(* *)
(**************************************************************************)
-include "basic_2/computation/lpxs_lleq.ma".
+include "basic_2/reduction/cpx_cpys.ma".
+include "basic_2/computation/lpxs_cpye.ma".
include "basic_2/computation/csx_alt.ma".
include "basic_2/computation/lsx_lpxs.ma".
(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-(* Advanced forward lemmas **************************************************)
-
-include "basic_2/computation/cpxs_lleq.ma".
+(* Advanced properties ******************************************************)
+axiom lpxs_cpye_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀T. ⦃G, L2⦄ ⊢ T ▶*[0, ∞] 𝐍⦃U⦄ →
+ G ⊢ ⋕⬊*[h, g, T] L2.
(*
-lemma lsx_cpxs_conf: ∀h,g,G,L1,T1,T2. G ⊢ ⋕⬊*[h, g, T1] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 →
- G ⊢ ⋕⬊*[h, g, T2] L2.
-#h #g #G #L1 #T1 #T2 #H @(lsx_ind … H) -L1
-#L1 #HL1 #IHL1 #L2 #HL12 #HT12 @lsx_intro
-#L3 #HL23 #HnL23 elim (lleq_dec T2 L1 L2 0) [| -HnL23 ]
-[ #HT2 @(IHL1 L3) /2 width=3 by lpxs_trans/
-
- @(lsx_lpxs_trans … HL23)
-| #HnT2 @(lsx_lpxs_trans … HL23) @(IHL1 … L2) //
- #HT1 @HnT2 @(lleq_cpxs_conf_dx … HT12) //
+#h #g #G #L1 #U #H @(csx_ind_alt … H) -U
+#U0 #_ #IHU0 #L0 #HL10 #T #H0 @lsx_intro
+#L2 #HL02 #HnT elim (cpye_total G L2 T 0 (∞))
+#U2 #H2 elim (eq_term_dec U0 U2) #H destruct
+[ -IHU0
+| -HnT /4 width=9 by lpxs_trans, lpxs_cpxs_trans, cpx_cpye_fwd_lpxs/
]
*)
-
-lemma lsx_fwd_bind_dx_lpxs: ∀h,g,a,I,G,L1,V1. ⦃G, L1⦄ ⊢ ⬊*[h, g] V1 →
- ∀L2,T. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V1.T] L2 →
- ∀V2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L2⦄ ⊢ V1 ➡*[h, g] V2 →
- G ⊢ ⋕⬊*[h, g, T] L2.ⓑ{I}V2.
-#h #g #a #I #G #L1 #V1 #H @(csx_ind_alt … H) -V1
-#V1 #_ #IHV1 #L2 #T #H @(lsx_ind … H) -L2
-#L2 #HL2 #IHL2 #V2 #HL12 #HV12 @lsx_intro
-#Y #H #HnT elim (lpxs_inv_pair1 … H) -H
-#L3 #V3 #HL23 #HV23 #H destruct
-lapply (lpxs_trans … HL12 … HL23) #HL13
-elim (eq_term_dec V2 V3)
-[ (* -HV13 -HL2 -IHV1 -HL12 *) #H destruct
- @IHL2 -IHL2 // -HL23 -HL13 /3 width=2 by lleq_fwd_bind_O/
-| -IHL2 -HnT #HnV23 elim (eq_term_dec V1 V2)
- [ #H -HV12 destruct
- (* @(lsx_lpxs_trans … (L2.ⓑ{I}V2)) /2 width=1 by lpxs_pair/ *)
- @(IHV1 … HnV23) -IHV1 -HnV23 /2 width=3 by lpxs_cpxs_trans/ -HL12 -HL13
- @(lsx_lpxs_trans … HL23)
-
-
-lemma lsx_fwd_bind_dx_lpxs: ∀h,g,a,I,G,L1,V. ⦃G, L1⦄ ⊢ ⬊*[h, g] V →
- ∀L2,T. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- G ⊢ ⋕⬊*[h, g, T] L2.ⓑ{I}V.
-#h #g #a #I #G #L1 #V #H @(csx_ind_alt … H) -V
-#V1 #_ #IHV1 #L2 #T #H @(lsx_ind … H) -L2
-#L2 #HL2 #IHL2 #HL12 @lsx_intro
-#Y #H #HnT elim (lpxs_inv_pair1 … H) -H
-#L3 #V3 #HL23 #HV13 #H destruct
-lapply (lpxs_trans … HL12 … HL23) #HL13
-elim (eq_term_dec V1 V3)
-[ -HV13 -HL2 -IHV1 -HL12 #H destruct
- @IHL2 -IHL2 // -HL23 -HL13 /3 width=2 by lleq_fwd_bind_O/
-| -IHL2 -HnT #HnV13
- @(IHV1 … HnV13) -IHV1 -HnV13 /2 width=3 by lpxs_cpxs_trans/ -HL12 -HL13
- @(lsx_lpxs_trans … HL23)
-
-
-(* Advanced inversion lemmas ************************************************)
-
-
-
(* Main properties **********************************************************)
-axiom csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → G ⊢ ⋕⬊*[h, g, T] L.
-(*
-#h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
-#G1 #L1 #T1 #IH #G2 #L2 * *
-[ #k #HG #HL #HT destruct //
-| #i #HG #HL #HT destruct
- #H
-| #p #HG #HL #HT destruct //
-| #a #I #V2 #T2 #HG #HL #HT #H destruct
- elim (csx_fwd_bind … H) -H
- #HV2 #HT2
-| #I #V2 #T2 #HG #HL #HT #H destruct
- elim (csx_fwd_flat … H) -H /3 width=1 by lsx_flat/
-*)
+lemma csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → G ⊢ ⋕⬊*[h, g, T] L.
+#h #g #G #L #T #HT elim (cpye_total G L T 0 (∞))
+#U #HTU elim HTU
+/4 width=5 by lpxs_cpye_csx_lsx, csx_cpx_trans, cpys_cpx/
+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/substitution/cpys_alt.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on context-sensitive extended multiple substitution for terms *)
+
+lemma cpys_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt … H) -G -L -T1 -T2 -d -e
+/2 width=7 by cpx_delta, cpx_bind, cpx_flat/
+qed.
+
+lemma cpy_cpx: ∀h,g,G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+/3 width=3 by cpy_cpys, cpys_cpx/ qed.
+
+lemma cpx_cpy_trans: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
+ ∀T2,d,e. ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T #H elim H -G -L -T1 -T
+[ #I #G #L #X #d #e #H elim (cpy_inv_atom1 … H) //
+ * /2 width=3 by cpy_cpx/
+| #G #L #k #l #Hkl #X #d #e #H >(cpy_inv_sort1 … H) -X /2 width=2 by cpx_sort/
+| #I #G #L #K #V1 #V #W #i #HLK #_ #HVW #IHV1 #X #d #e #H
+ lapply (ldrop_fwd_drop2 … HLK) #H0
+ lapply (cpy_weak … H 0 (∞) ? ?) -H // #H
+ elim (cpy_inv_lift1_be … H … H0 … HVW) -H -H0 -HVW
+ /3 width=7 by cpx_delta/
+| #a #I #G #L #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #X #d #e #H elim (cpy_inv_bind1 … H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct
+ /5 width=7 by cpx_bind, lsuby_cpy_trans, lsuby_succ/
+| #I #G #L #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #X #d #e #H elim (cpy_inv_flat1 … H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct /3 width=3 by cpx_flat/
+| #G #L #V1 #U1 #U #T #_ #HTU #IHU1 #T2 #d #e #HT2
+ lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
+ elim (lift_total T2 0 1) #U2 #HTU2
+ lapply (cpy_lift_be … HT2 (L.ⓓV1) … (Ⓕ) … HTU … HTU2 ? ?) -T
+ /3 width=3 by cpx_zeta, ldrop_drop/
+| /3 width=3 by cpx_tau/
+| /3 width=3 by cpx_ti/
+| #a #G #L #V1 #V #W1 #W #T1 #T #_ #_ #_ #IHV1 #IHW1 #IHT1 #X #d #e #HX
+ elim (cpy_inv_bind1 … HX) -HX #Y #T2 #HY #HT2 #H destruct
+ elim (cpy_inv_flat1 … HY) -HY #W2 #V2 #HW2 #HV2 #H destruct
+ /5 width=7 by cpx_beta, lsuby_cpy_trans, lsuby_succ/
+| #a #G #L #V1 #V #U #W1 #W #T1 #T #_ #HVU #_ #_ #IHV1 #IHW1 #IHT1 #X #d #e #HX
+ elim (cpy_inv_bind1 … HX) -HX #W2 #Y #HW2 #HY #H destruct
+ elim (cpy_inv_flat1 … HY) -HY #U2 #T2 #HU2 #HT2 #H destruct
+ lapply (cpy_weak … HU2 0 (∞) ? ?) -HU2 // #HU2
+ elim (cpy_inv_lift1_be … HU2 L … HVU) -U
+ /5 width=7 by cpx_theta, lsuby_cpy_trans, lsuby_succ, ldrop_drop/
+]
+qed-.
+
+lemma cpx_cpys_trans: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
+ ∀T2,d,e. ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L #T1 #T #HT1 #T2 #d #e #H @(cpys_ind … H) -T2
+/2 width=5 by cpx_cpy_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/substitution/cpye_cpye.ma".
+include "basic_2/reduction/lpx_cpys.ma".
+
+axiom cpx_cpys_conf_lpx: ∀h,g,G,d,e.
+ ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡[h, g] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h, g] L1 →
+ ∀T2. ⦃G, L0⦄ ⊢ T0 ▶*[d, e] T2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] T & ⦃G, L0⦄ ⊢ T2 ➡[h, g] T.
+
+(* SN EXTENDED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *********************)
+
+(* Forward lemmas on evaluation for extended substitution *******************)
+
+lemma cpx_cpys_cpye_fwd_lpx: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] U1 →
+ ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
+ ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2.
+#h #g #G #L1 #T1 #T2 #HT12 #L2 #HL12 #U1 #d #e #HTU1
+elim (cpx_cpys_conf_lpx … HT12 … HL12 … HTU1) -T1
+/3 width=9 by cpx_cpys_trans_lpx, cpye_cpys_conf/
+qed-.
+
+lemma cpx_cpye_fwd_lpx: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃U1⦄ →
+ ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
+ ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2.
+#h #g #G #L1 #T1 #T2 #HT12 #L2 #HL12 #U1 #d #e *
+/2 width=9 by cpx_cpys_cpye_fwd_lpx/
+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/relocation/cpy_lift.ma".
+include "basic_2/substitution/cpys.ma".
+include "basic_2/reduction/lpx_ldrop.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on context-sensitive extended substitution for terms **********)
+
+lemma cpx_cpy_trans_lpx: ∀h,g,G,L1,T1,T. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀T2,d,e. ⦃G, L2⦄ ⊢ T ▶[d, e] T2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L1 #T1 #T #H elim H -G -L1 -T1 -T
+[ #J #G #L1 #L2 #HL12 #T2 #d #e #H elim (cpy_inv_atom1 … H) -H //
+ * #I #K2 #V2 #i #_ #_ #HLK2 #HVT2 #H destruct
+ elim (lpx_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpx_inv_pair2 … H) -H #K1 #V1 #_ #HV12 #H destruct
+ /2 width=7 by cpx_delta/
+| #G #L1 #k #l #Hkl #L2 #_ #X #d #e #H >(cpy_inv_sort1 … H) -X /2 width=2 by cpx_sort/
+| #I #G #L1 #K1 #V1 #V #T #i #HLK1 #_ #HVT #IHV1 #L2 #HL12 #T2 #d #e #HT2
+ elim (lpx_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (lpx_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_drop2 … HLK2) -V0 #HLK2
+ lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
+ elim (cpy_inv_lift1_be … HT2 … HLK2 … HVT) -HT2 -HLK2 -HVT
+ /3 width=7 by cpx_delta/
+| #a #I #G #L1 #V1 #V #T1 #T #HV1 #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_bind1 … H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct /4 width=5 by lpx_pair, cpx_bind/
+| #I #G #L1 #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_flat1 … H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5 by cpx_flat/
+| #G #L1 #V1 #U1 #U #T #_ #HTU #IHU1 #L2 #HL12 #T2 #d #e #HT2
+ lapply (cpy_weak … HT2 0 (∞) ? ?) -HT2 // #HT2
+ elim (lift_total T2 0 1) #U2 #HTU2
+ lapply (cpy_lift_be … HT2 (L2.ⓓV1) … (Ⓕ) … HTU … HTU2 ? ?) -T
+ /4 width=5 by cpx_zeta, lpx_pair, ldrop_drop/
+| /3 width=5 by cpx_tau/
+| /3 width=5 by cpx_ti/
+| #a #G #L1 #V1 #V #W1 #W #T1 #T #HV1 #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
+ elim (cpy_inv_bind1 … HX) -HX #Y #T2 #HY #HT2 #H destruct
+ elim (cpy_inv_flat1 … HY) -HY #W2 #V2 #HW2 #HV2 #H destruct
+ /5 width=11 by lpx_pair, cpx_beta, lsuby_cpy_trans, lsuby_succ/
+| #a #G #L1 #V1 #V #U #W1 #W #T1 #T #_ #HVU #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
+ elim (cpy_inv_bind1 … HX) -HX #W2 #Y #HW2 #HY #H destruct
+ elim (cpy_inv_flat1 … HY) -HY #U2 #T2 #HU2 #HT2 #H destruct
+ lapply (cpy_weak … HU2 0 (∞) ? ?) -HU2 // #HU2
+ elim (cpy_inv_lift1_be … HU2 L2 … HVU) -U
+ /4 width=7 by lpx_pair, cpx_theta, ldrop_drop/
+]
+qed-.
+
+lemma cpx_cpys_trans_lpx: ∀h,g,G,L1,T1,T. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+ ∀T2,d,e. ⦃G, L2⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L1 #T1 #T #HT1 #L2 #HL12 #T2 #d #e #H @(cpys_ind … H) -T2
+/2 width=7 by cpx_cpy_trans_lpx/
+qed-.
(* Inversion lemmas on relocation *******************************************)
-lemma cny_lift_inv_le: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
+lemma cny_inv_lift_le: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
⇧[d, e] T ≡ U → dt + et ≤ d → ⦃G, K⦄ ⊢ ▶[dt, et] 𝐍⦃T⦄.
#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdetd #T2 #HT12
elim (lift_total T2 d e) #U2 #HTU2
lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
qed-.
-lemma cny_lift_inv_be: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
+lemma cny_inv_lift_be: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
⇧[d, e] T ≡ U → dt ≤ d → yinj d + e ≤ dt + et → ⦃G, K⦄ ⊢ ▶[dt, et-e] 𝐍⦃T⦄.
#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdtd #Hdedet #T2 #HT12
lapply (yle_fwd_plus_ge_inj … Hdedet) // #Heet
lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
qed-.
-lemma cny_lift_inv_ge: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
+lemma cny_inv_lift_ge: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
⇧[d, e] T ≡ U → yinj d + e ≤ dt → ⦃G, K⦄ ⊢ ▶[dt-e, et] 𝐍⦃T⦄.
#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hdedt #T2 #HT12
elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #Hddte #Hedt
lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
qed-.
-(* Advanced properties ******************************************************)
+(* Advanced inversion lemmas on relocation **********************************)
-fact cny_subst_aux: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
- ⇩[i+1] L ≡ K → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄ →
- ⇧[O, i+1] V ≡ W → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄.
-#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HV #HVW
-lapply (cny_lift_be … HV … HLK … HVW ? ?) // -HV -HLK -HVW
-#HW @(cny_narrow … HW) -HW //
+lemma cny_inv_lift_ge_up: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄ → ⇩[s, d, e] L ≡ K →
+ ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+ ⦃G, K⦄ ⊢ ▶[d, dt + et - (yinj d + e)] 𝐍⦃T⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HU1 #HLK #HTU1 #Hddt #Hdtde #Hdedet
+lapply (cny_narrow … HU1 (d+e) (dt+et-(d+e)) ? ?) -HU1 [ >ymax_pre_sn_comm ] // #HU1
+lapply (cny_inv_lift_ge … HU1 … HLK … HTU1 ?) // -L -U1
+>yplus_minus_inj //
qed-.
-lemma cny_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
- ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄ →
- ⇧[O, i+1] V ≡ W → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄.
-/3 width=13 by cny_subst_aux, ldrop_fwd_drop2/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact cny_inv_subst_aux: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
- ⇩[i+1] L ≡ K → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
- ⇧[O, i+1] V ≡ W → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
+lemma cny_inv_lift_subst: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i+1] L ≡ K → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
+ ⇧[O, i+1] V ≡ W → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HW #HVW
-lapply (cny_narrow … HW (i+1) (⫰(d+e-i)) ? ?) -HW
-[ >yplus_SO2 <yplus_succ_swap >ylt_inv_O1
- [ >ymax_pre_sn_comm /2 width=2 by ylt_fwd_le/
- | lapply (monotonic_ylt_minus_dx … Hide i ?) //
- ]
+lapply (cny_inv_lift_ge_up … HW … HLK … HVW ? ? ?) //
+>yplus_O1 <yplus_inj >yplus_SO2
+[ /2 width=1 by ylt_fwd_le_succ1/
| /2 width=3 by yle_trans/
-| #HW lapply (cny_lift_inv_ge … HW … HLK … HVW ?) // -L -W
- >yplus_inj >yminus_refl //
+| >yminus_succ2 //
]
qed-.
-lemma cny_inv_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
- ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
- ⇧[O, i+1] V ≡ W → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
-/3 width=13 by cny_inv_subst_aux, ldrop_fwd_drop2/ qed-.
+(* Advanced properties ******************************************************)
+
+(* Note: this should be applicable in a forward manner *)
+lemma cny_lift_ge_up: ∀G,L,K,T,U,s,d,dt,e,et. ⦃G, K⦄ ⊢ ▶[yinj d, dt + et - (yinj d + yinj e)] 𝐍⦃T⦄ →
+ ⇩[s, d, e] L ≡ K → ⇧[d, e] T ≡ U →
+ yinj d ≤ dt → dt ≤ yinj d + yinj e → yinj d + yinj e ≤ dt + et →
+ ⦃G, L⦄ ⊢ ▶[dt, et] 𝐍⦃U⦄.
+#G #L #K #T1 #U1 #s #d #dt #e #et #HT1 #HLK #HTU1 #Hddt #Hdtde #Hdedet
+lapply (cny_lift_be … HT1 … HLK … HTU1 ? ?) // -K -T1
+#HU1 @(cny_narrow … HU1) -HU1 // (**) (* auto fails *)
+qed-.
+
+lemma cny_lift_subst: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i+1] L ≡ K → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄ →
+ ⇧[O, i+1] V ≡ W → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄.
+#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HV #HVW
+@(cny_lift_ge_up … HLK … HVW) // >yplus_O1 <yplus_inj >yplus_SO2
+[ >yminus_succ2 //
+| /2 width=3 by yle_trans/
+| /2 width=1 by ylt_fwd_le_succ1/
+]
+qed-.
include "ground_2/ynat/ynat_max.ma".
include "basic_2/notation/relations/psubst_6.ma".
include "basic_2/grammar/genv.ma".
-include "basic_2/grammar/cl_shift.ma".
-include "basic_2/relocation/ldrop_append.ma".
include "basic_2/relocation/lsuby.ma".
(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
/2 width=2 by cpy_weak_top/
qed-.
-lemma cpy_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
- ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → d + e ≤ dt + et →
- ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #T1 #d #e #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ]
- [ elim (ylt_yle_false … Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/
- | elim (le_inv_plus_l … Hid) #Hdie #Hei
- elim (lift_split … HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie
- #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hei
- @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HVW1) -V1 -IHW12 //
- elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
- <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
- /3 width=2 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
- /3 width=2 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ∀i. i ≤ d + e →
∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d, i-d] T & ⦃G, L⦄ ⊢ T ▶[i, d+e-i] T2.
#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
]
qed-.
-lemma cpy_append: ∀G,d,e. l_appendable_sn … (cpy d e G).
-#G #d #e #K #T1 #T2 #H elim H -G -K -T1 -T2 -d -e
-/2 width=1 by cpy_atom, cpy_bind, cpy_flat/
-#I #G #K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpy_subst I … (L@@K0) … HVW) // (**) (* /4/ does not work *)
-@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/
+(* Basic forward lemmas *****************************************************)
+
+lemma cpy_fwd_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
+ ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → d + e ≤ dt + et →
+ ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #T1 #d #e #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ]
+ [ elim (ylt_yle_false … Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ elim (lift_split … HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie
+ #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hei
+ @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HVW1) -V1 -IHW12 //
+ elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
+ <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
+ /3 width=2 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
+ /3 width=2 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e normalize
+/3 width=1 by monotonic_le_plus_l, le_plus/
qed-.
(* Basic inversion lemmas ***************************************************)
(* Basic_1: was: subst1_gen_lift_eq *)
lemma cpy_inv_lift1_eq: ∀G,T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 → U1 = U2.
-#G #T1 #U1 #d #e #HTU1 #L #U2 #HU12 elim (cpy_up … HU12 … HTU1) -HU12 -HTU1
+#G #T1 #U1 #d #e #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
/2 width=4 by cpy_inv_refl_O2/
qed-.
-(* Basic forward lemmas *****************************************************)
-
-lemma cpy_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e normalize
-/3 width=1 by monotonic_le_plus_l, le_plus/
-qed-.
-
-lemma cpy_fwd_shift1: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #d #e #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X #d #e
- >shift_append_assoc normalize #H
- elim (cpy_inv_bind1 … H) -H
- #V0 #T0 #_ #HT10 #H destruct
- elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
- >append_length >HL12 -HL12
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
- /2 width=3 by trans_eq/
-]
-qed-.
-
(* Basic_1: removed theorems 25:
subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
--- /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/substitution/cpys_cny.ma".
+include "basic_2/substitution/cpys_cpys.ma".
+include "basic_2/substitution/cpye.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+(* Advanced properties ******************************************************)
+
+lemma cpye_cpys_conf: ∀G,L,T,T2,d,e. ⦃G, L⦄ ⊢ T ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀T1. ⦃G, L⦄ ⊢ T ▶*[d, e] T1 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+#G #L #T #T2 #d #e * #H2 #HT2 #T1 #H1 elim (cpys_conf_eq … H1 … H2) -T
+#T0 #HT10 #HT20 >(cpys_inv_cny1 … HT2 … HT20) -T2 //
+qed-.
+
\ No newline at end of file
⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ →
⇧[O, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃W2⦄.
#I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK *
-/4 width=13 by cpys_subst, cny_subst_aux, ldrop_fwd_drop2, conj/
+/4 width=13 by cpys_subst, cny_lift_subst, ldrop_fwd_drop2, conj/
qed.
-lemma cpys_total: ∀G,L,T1,d,e. ∃T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
+lemma cpye_total: ∀G,L,T1,d,e. ∃T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
#G #L #T1 @(fqup_wf_ind_eq … G L T1) -G -L -T1
#Z #Y #X #IH #G #L * *
[ #k #HG #HL #HT #d #e destruct -IH /2 width=2 by ex_intro/
/3 width=1 by or4_intro0, or4_intro1, or4_intro2, conj/
| * #I #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
@or4_intro3 @(ex5_4_intro … HLK … HVT2) (**) (* explicit constructor *)
- /4 width=13 by cny_inv_subst_aux, ldrop_fwd_drop2, conj/
+ /4 width=13 by cny_inv_lift_subst, ldrop_fwd_drop2, conj/
]
qed-.
elim (ylt_yle_false … H) //
qed-.
+lemma cpye_inv_lref1_lget: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀I,K,V1. ⇩[i] L ≡ K.ⓑ{I}V1 →
+ ∨∨ d + e ≤ yinj i ∧ T2 = #i
+ | yinj i < d ∧ T2 = #i
+ | ∃∃V2. d ≤ yinj i & yinj i < d + e &
+ ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
+ ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #d #e #i #H #I #K #V1 #HLK elim (cpye_inv_lref1 … H) -H *
+[ #H elim (lt_refl_false i) -T2 -d
+ @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/
+| /3 width=1 by or3_intro0, conj/
+| /3 width=1 by or3_intro1, conj/
+| #Z #Y #X1 #X2 #Hdi #Hide #HLY #HX12 #HXT2
+ lapply (ldrop_mono … HLY … HLK) -HLY -HLK #H destruct
+ /3 width=3 by or3_intro2, ex4_intro/
+]
+qed-.
+
+lemma cpye_inv_lref1_subst_ex: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀I,K,V1. d ≤ yinj i → yinj i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 →
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
+ ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #d #e #i #H #I #K #V1 #Hdi #Hide #HLK
+elim (cpye_inv_lref1_lget … H … HLK) -H * /2 width=3 by ex2_intro/
+#H elim (ylt_yle_false … H) //
+qed-.
+
lemma cpye_inv_lref1_subst: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
∀I,K,V1,V2. d ≤ yinj i → yinj i < d + e →
⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ T2 →
⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄.
-#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2 elim (cpye_inv_lref1 … H) -H *
-[ #H elim (lt_refl_false i) -V2 -T2 -d
- @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/
-|2,3: #H elim (ylt_yle_false … H) //
-| #Z #Y #X1 #X2 #_ #_ #HLY #HX12 #HXT2
- lapply (ldrop_mono … HLY … HLK) -HLY -HLK #H destruct
- lapply (lift_inj … HXT2 … HVT2) -HXT2 -HVT2 #H destruct //
-]
+#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2
+elim (cpye_inv_lref1_subst_ex … H … HLK) -H -HLK //
+#X2 #H0 #HXT2 lapply (lift_inj … HXT2 … HVT2) -HXT2 -HVT2 #H destruct //
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma cpye_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdetd
+elim (cpys_inv_lift1_le … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_le … HU2 … HLK … HTU2 ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et - e] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet
+elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_be … HU2 … HLK … HTU2 ? ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ yinj d + e ≤ dt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt - e, et] 𝐍⦃T2⦄ & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hdedt
+elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_ge … HU2 … HLK … HTU2 ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] 𝐍⦃U2⦄ →
+ ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] 𝐍⦃T2⦄ &
+ ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et * #HU12 #HU2 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) -U1 // #T2 #HT12 #HTU2
+lapply (cny_inv_lift_ge_up … HU2 … HLK … HTU2 ? ? ?) -L
+/3 width=3 by ex2_intro, conj/
+qed-.
+
+lemma cpye_inv_lift1_subst: ∀G,L,W1,W2,d,e. ⦃G, L⦄ ⊢ W1 ▶*[d, e] 𝐍⦃W2⦄ →
+ ∀K,V1,i. ⇩[i+1] L ≡ K → ⇧[O, i+1] V1 ≡ W1 →
+ d ≤ yinj i → i < d + e →
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ & ⇧[O, i+1] V2 ≡ W2.
+#G #L #W1 #W2 #d #e * #HW12 #HW2 #K #V1 #i #HLK #HVW1 #Hdi #Hide
+elim (cpys_inv_lift1_subst … HW12 … HLK … HVW1) -W1 // #V2 #HV12 #HVW2
+lapply (cny_inv_lift_subst … HLK HW2 HVW2) -L
+/3 width=3 by ex2_intro, conj/
qed-.
/3 width=5 by cpys_strap1, cpy_weak_full/
qed-.
-lemma cpys_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
- ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
- d ≤ dt → d + e ≤ dt + et →
- ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
+(* Basic forward lemmas *****************************************************)
+
+lemma cpys_fwd_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+ ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → d + e ≤ dt + et →
+ ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
#G #L #U1 #U2 #dt #et #H #T1 #d #e #HTU1 #Hddt #Hdedet @(cpys_ind … H) -U2
[ /2 width=3 by ex2_intro/
| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
- elim (cpy_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
+ elim (cpy_fwd_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
]
qed-.
-lemma cpys_append: ∀G,d,e. l_appendable_sn … (cpys d e G).
-#G #d #e #K #T1 #T2 #H @(cpys_ind … H) -T2
-/3 width=3 by cpys_strap1, cpy_append/
+lemma cpys_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
+/2 width=3 by transitive_le/
qed-.
(* Basic inversion lemmas ***************************************************)
#G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2
/2 width=7 by cpy_inv_lift1_eq/
qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpys_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ♯{T1} ≤ ♯{T2}.
-#G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
-/2 width=3 by transitive_le/
-qed-.
-
-lemma cpys_fwd_shift1: ∀G,L,L1,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*[d, e] T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#G #L #L1 #T1 #T #d #e #H @(cpys_ind … H) -T
-[ /2 width=4 by ex2_2_intro/
-| #T #X #_ #HX * #L0 #T0 #HL10 #H destruct
- elim (cpy_fwd_shift1 … HX) -HX #L2 #T2 #HL02 #H destruct
- /2 width=4 by 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/relocation/cny.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Inversion lemmas on normality for extended substitution ******************)
+
+lemma cpys_inv_cny1: ∀G,L,T1,d,e. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T1⦄ →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → T1 = T2.
+#G #L #T1 #d #e #HT1 #T2 #H @(cpys_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct <(HT1 … HT2) -T //
+qed-.
elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
]
qed-.
+
+lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,d,e. ⦃G, L⦄ ⊢ W1 ▶*[d, e] W2 →
+ ∀K,V1,i. ⇩[i+1] L ≡ K → ⇧[O, i+1] V1 ≡ W1 →
+ d ≤ yinj i → i < d + e →
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] V2 & ⇧[O, i+1] V2 ≡ W2.
+#G #L #W1 #W2 #d #e #HW12 #K #V1 #i #HLK #HVW1 #Hdi #Hide
+elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) //
+>yplus_O1 <yplus_inj >yplus_SO2
+[ >yminus_succ2 /2 width=3 by ex2_intro/
+| /2 width=1 by ylt_fwd_le_succ1/
+| /2 width=3 by yle_trans/
+]
+qed-.
d ≤ dt → dt ≤ d + e → L1 ⋕[U, d] L2.
#L1 #L2 #U #dt * #HL12 #IH #T #d #e #HTU #Hddt #Hdtde @conj // -HL12
#U0 elim (IH U0) -IH #H12 #H21 @conj
-#HU0 elim (cpys_up … HU0 … HTU) // -HU0 /4 width=5 by cpys_weak/
+#HU0 elim (cpys_fwd_up … HU0 … HTU) // -HU0 /4 width=5 by cpys_weak/
qed-.
lemma lsuby_lleq_trans: ∀L2,L,T,d. L2 ⋕[T, d] L →
--- /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/substitution/cpye_lift.ma".
+include "basic_2/substitution/lleq_alt.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Forward lemmas on evaluation for extended substitution *******************)
+
+lemma lleq_fwd_cpye: ∀L1,L2,T,d. L1 ⋕[T, d] L2 → ∀G,T1. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] 𝐍⦃T1⦄ →
+ ∀T2. ⦃G, L2⦄ ⊢ T ▶*[d, ∞] 𝐍⦃T2⦄ → T1 = T2.
+#L1 #L2 #T #d #H @(lleq_ind_alt … H) -L1 -L2 -T -d
+[ #L1 #L2 #d #k #_ #G #T1 #H1 #T2 #H2
+ >(cpye_inv_sort1 … H1) -H1 >(cpye_inv_sort1 … H2) -H2 //
+| #L1 #L2 #d #i #_ #Hid #G #T1 #H1 #T2 #H2
+ >(cpye_inv_lref1_free … H1) -H1 [ >(cpye_inv_lref1_free … H2) -H2 ]
+ /2 width=1 by or3_intro2/
+| #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #_ #IHV #G #T1 #H1 #T2 #H2
+ elim (cpye_inv_lref1_subst_ex … H1 … HLK1) -H1 -HLK1 //
+ elim (cpye_inv_lref1_subst_ex … H2 … HLK2) -H2 -HLK2 //
+ >yminus_Y_inj #V2 #HV2 #HVT2 #V1 #HV1 #HVT1
+ lapply (IHV … HV1 … HV2) -IHV -HV1 -HV2 #H destruct /2 width=5 by lift_mono/
+| #L1 #L2 #d #i #_ #HL1 #HL2 #G #T1 #H1 #T2 #H2
+ >(cpye_inv_lref1_free … H1) -H1 [ >(cpye_inv_lref1_free … H2) -H2 ]
+ /2 width=1 by or3_intro0/
+| #L1 #L2 #d #p #_ #G #T1 #H1 #T2 #H2
+ >(cpye_inv_gref1 … H1) -H1 >(cpye_inv_gref1 … H2) -H2 //
+| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #G #X1 #H1 #X2 #H2
+ elim (cpye_inv_bind1 … H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct
+ elim (cpye_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ /3 width=3 by eq_f2/
+| #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #G #X1 #H1 #X2 #H2
+ elim (cpye_inv_flat1 … H1) -H1 #V1 #T1 #HV1 #HT1 #H destruct
+ elim (cpye_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ /3 width=3 by eq_f2/
+]
+qed-.
}
]
[ { "context-sensitive extended computation" * } {
- [ "lpxs ( ⦃?,?⦄ ⊢ ➡*[?,?] ? )" "lpxs_alt ( ⦃?,?⦄ ⊢ ➡➡*[?,?] ? )" "lpxs_ldrop" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ]
+ [ "lpxs ( ⦃?,?⦄ ⊢ ➡*[?,?] ? )" "lpxs_alt ( ⦃?,?⦄ ⊢ ➡➡*[?,?] ? )" "lpxs_ldrop" + "lpxs_cpye" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ]
[ "cpxs ( ⦃?,?⦄ ⊢ ? ➡*[?,?] ? )" "cpxs_tstc" + "cpxs_tstc_vector" + "cpxs_lift" + "cpxs_lleq" + "cpxs_aaa" + "cpxs_cpxs" * ]
}
]
}
]
[ { "context-sensitive extended reduction" * } {
- [ "lpx ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lpx_ldrop" + "lpx_lleq" + "lpx_aaa" * ]
+ [ "lpx ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lpx_ldrop" + "lpx_cpys" + "lpx_cpye" + "lpx_lleq" + "lpx_aaa" * ]
[ "cpx ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpx_lift" + "cpx_cpys" + "cpx_lleq" + "cpx_cix" * ]
}
]
class "yellow"
[ { "substitution" * } {
[ { "lazy equivalence for local environments" * } {
- [ "lleq ( ? ⋕[?,?] ? )" "lleq_alt ( ? ⋕⋕[?,?] ? )" "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" + "lleq_ext" * ]
+ [ "lleq ( ? ⋕[?,?] ? )" "lleq_alt ( ? ⋕⋕[?,?] ? )" "lleq_ldrop" + "lleq_fqus" + "lleq_cpye" + "lleq_lleq" + "lleq_ext" * ]
}
]
[ { "evaluation for contxt-sensitive extended substitution" * } {
- [ "cpye ( ⦃?,?⦄ ⊢ ? ▶*[?,?] 𝐍⦃?⦄ )" "cpye_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] 𝐍⦃?⦄ )" "cpye_lift" * ]
+ [ "cpye ( ⦃?,?⦄ ⊢ ? ▶*[?,?] 𝐍⦃?⦄ )" "cpye_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] 𝐍⦃?⦄ )" "cpye_lift" + "cpye_cpye" * ]
}
]
[ { "contxt-sensitive extended multiple substitution" * } {
- [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
+ [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cny" + "cpys_cpys" * ]
}
]
[ { "iterated structural successor for closures" * } {
(* Properties ***************************************************************)
+fact le_repl_sn_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
+// qed-.
+
lemma monotonic_le_minus_l2: ∀x1,x2,y,z. x1 ≤ x2 → x1 - y - z ≤ x2 - y - z.
/3 width=1 by monotonic_le_minus_l/ qed.
(* inversion and forward lemmas on yle **************************************)
+lemma ylt_fwd_le_succ1: ∀m,n. m < n → ⫯m ≤ n.
+#m #n * -m -n /2 width=1 by yle_inj/
+qed-.
+
lemma ylt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
-#m #n * -m -n /3 width=1 by yle_pred_sn, yle_inj, yle_Y/
+#m #n * -m -n /3 width=1 by lt_to_le, yle_inj/
qed-.
lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
qed.
lemma ylt_succ: ∀m,n. m < n → ⫯m < ⫯n.
-#m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
+#m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
qed.
(* Properties on order ******************************************************)
projects / helm.git / commitdiff