arity/subst0 arity_subst0
asucc/fwd asucc_gen_sort
asucc/fwd asucc_gen_head
+
cnt/props cnt_lift
-C/props clt_wf__q_ind
-C/props clt_wf_ind
csuba/arity csuba_arity
csuba/arity csuba_arity_rev
lift1/fwd lift1_cons_tail
lift1/fwd lifts1_nil
lift1/fwd lifts1_cons
-lift/props thead_x_lift_y_y
lift/props lifts_tapp
lift/props lifts_inj
llt/props lweight_repl
--- /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/unwind/sstas.ma".
+include "basic_2/reducibility/ysc.ma".
+include "basic_2/computation/cprs.ma".
+
+(* "BIG TREE" ORDER FOR CLOSURES ********************************************)
+
+definition ygt: ∀h. sd h → bi_relation lenv term ≝
+ λh,g. bi_TC … (ysc h g).
+
+interpretation "'big tree' order (closure)"
+ 'BTGreaterThan h g L1 T1 L2 T2 = (ygt h g L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma ygt_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term.
+ (∀L2,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ → R L2 T2) →
+ (∀L,T,L2,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ → R L T → R L2 T2) →
+ ∀L2,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ → R L2 T2.
+#h #g #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
+@(bi_TC_ind … IH1 IH2 L2 T2 H)
+qed-. (**) (* /3 width=6 by bi_TC_ind/ fails *)
+
+lemma ygt_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term.
+ (∀L1,T1. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ → R L1 T1) →
+ (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → R L T → R L1 T1) →
+ ∀L1,T1. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ → R L1 T1.
+/3 width=6 by bi_TC_ind_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
+ h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+/2 width=4/ qed-.
+
+lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L, T⦄ →
+ h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+/2 width=4/ qed-.
+
+lemma ygt_cprs_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
+ ∀T2. L ⊢ T ➡* T2 → h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T2⦄.
+#h #g #L1 #L #T1 #T #HLT1 #T2 #H @(cprs_ind … H) -T2 //
+#T0 #T2 #_ #HT02 #IHT0 -HLT1
+elim (term_eq_dec T0 T2) #HT02 destruct //
+@(ygt_strap1 … IHT0) /3 width=1/
+qed-.
+
+lemma ygt_sstas_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
+ ∀T2. ⦃h, L⦄ ⊢ T •*[g] T2 → h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T2⦄.
+#h #g #L1 #L #T1 #T #HLT1 #T2 #H @(sstas_ind … H) -T2 //
+#T0 #T2 #l #_ #HT02 #IHT0 -HLT1
+@(ygt_strap1 … IHT0) -IHT0 /2 width=2/
+qed-.
+
+lemma cprs_ygt_trans: ∀h,g,L,T1,T. L ⊢ T1 ➡* T →
+ ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L #T1 #T #H @(cprs_ind … H) -T //
+#T0 #T #_ #HT0 #IHT10 #L2 #T2 #HLT2
+elim (term_eq_dec T0 T) #HT0 destruct /2 width=1/
+@IHT10 -IHT10 @(ygt_strap2 … HLT2) /3 width=1/
+qed-.
+
+lemma sstas_ygt_trans: ∀h,g,L,T1,T. ⦃h, L⦄ ⊢ T1 •*[g] T →
+ ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L #T1 #T #H @(sstas_ind … H) -T //
+#T0 #T #l #_ #HT0 #IHT10 #L2 #T2 #HLT2
+@IHT10 -IHT10 @(ygt_strap2 … HLT2) /2 width=2/
+qed-.
+
+lemma fw_ygt: ∀h,g,L1,L2,T1,T2. ♯{L2, T2} < ♯{L1, T1} → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+/3 width=1/ qed.
+
+lemma cprs_ygt: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
+#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2
+[ #H elim H -H //
+| #T #T2 #_ #HT2 #IHT1 #H
+ elim (term_eq_dec T1 T) #HT1 destruct
+ [ -IHT1 /4 width=1 by ysc_cpr, bi_inj/ (**) (* auto too slow without trace *)
+ | -H /4 width=3 by inj, ygt_cprs_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/computation/ygt.ma".
+
+(* "BIG TREE" ORDER FOR CLOSURES ********************************************)
+
+theorem ygt_trans: ∀h,g. bi_transitive … (ygt h g).
+/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/dynamic/snv_ltpss_sn.ma".
+include "basic_2/dynamic/snv_cpr_ssta.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for closures ***************)
+
+axiom snv_ltpr_tpr_aux: ∀h,g,n. (
+ ∀L1,T1. ♯{L1, T1} < n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
+ ) → (
+ ∀L1,T1. ♯{L1, T1} < n → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
+ ) →
+ ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
+(*
+#h #g #n #IH2 #IH1 #L1 * * [||||*]
+[ #k #Hn #H1 #L2 #_ #X #H2 destruct -IH2 -IH1 -L1
+ >(tpr_inv_atom1 … H2) -X //
+| #i #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
+ elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
+ >(tpr_inv_atom1 … H2) -X
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
+ lapply (IH1 … HV1 … HK12 V2 ?) -IH1 -HV1 -HK12 //
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ ] -HV12 /2 width=5/
+| #p #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
+ elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
+ elim (snv_inv_bind … H1) -H1 #HV1 #HT1
+ elim (tpr_inv_bind1 … H2) -H2 *
+ [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
+ lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
+ lapply (cpr_intro (L2.ⓑ{I}V2) … T2 0 1 HT10 ?) -HT10 /2 width=1/ -HT02 #HT12
+ lapply (IH1 … HV1 … HL12 V2 ?) -HV1 //
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ ] #HV2
+ lapply (IH1 … HT1 (L2.ⓑ{I}V2) … T2 ?) -IH1 -HT1 /3 width=1/
+ | #T2 #HT12 #HXT2 #H1 #H2 destruct
+ lapply (IH1 … HT1 (L2.ⓓV1) … T2 ?) -IH1 -HT1 // /2 width=2/
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ ] -HT12 -HL12 #HT2
+ lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
+ ]
+| #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct
+ elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
+ elim (tpr_inv_appl1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ lapply (IH1 … HV1 … HL12 V2 ?)
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ | //
+ ] #HV2
+ lapply (IH1 … HT1 … HL12 T2 ?)
+ [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
+ | //
+ ] #HT2
+ lapply (IH1 … HT1 … HTU1) -IH1 // #H
+ elim (snv_inv_bind … H) -H #HW1 #HU1
+ elim (IH2 … HVW1 … HL12 … HV12 HV1) -IH2 -HVW1 -HV12 -HV1 // #W2 #HVW2 #HW12
+ lapply (fpcs_canc_sn L1 L1 … W10 W1 … HW12) -HW12 /3 width=1/ -W10 #HW12
+ @(snv_appl … HV2 HT2 HVW2)
+*)
+
+fact snv_cpr_aux: ∀h,g,n. (
+ ∀L1,T1. ♯{L1, T1} < n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
+ ) → (
+ ∀L1,T1. ♯{L1, T1} < n → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
+ ) →
+ ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀T2. L1 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T2 :[g].
+#h #g #n #IH2 #IH1 #L1 #T1 #Hn #HT1 #T2 * #T0 #HT10 #HT02
+lapply (snv_ltpr_tpr_aux … Hn HT1 … HT10) -Hn -HT1 -HT10 [ // | skip | /3 width=6/ | /3 width=5/ ] -n -T1 #HT0
+lapply (snv_tpss_conf … HT0 … HT02) -T0 //
+qed-.
+
+fact snv_lfpr_aux: ∀h,g,n. (
+ ∀L1,T1. ♯{L1, T1} < n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
+ ) → (
+ ∀L1,T1. ♯{L1, T1} < n → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
+ ) →
+ ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃h, L2⦄ ⊩ T1 :[g].
+#h #g #n #IH2 #IH1 #L1 #T1 #Hn #HT1 #L2 * #L #HL1 #HL2
+lapply (snv_ltpr_tpr_aux … Hn HT1 … HL1 ??) -Hn -HT1 -HL1 [ // | skip | /3 width=6/ | /3 width=5/ ] -n -L1 #HT1
+lapply (snv_ltpss_sn_conf … HL2 … HT1) -L //
+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_ltpss_sn.ma".
+include "basic_2/computation/dxprs_lift.ma".
+include "basic_2/equivalence/lsubse_ssta.ma".
+include "basic_2/equivalence/fpcs_cpcs.ma".
+include "basic_2/equivalence/lfpcs_fpcs.ma".
+include "basic_2/dynamic/snv_ssta.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-sensitive parallel reduction for terms *************)
+
+fact ssta_ltpr_tpr_aux: ∀h,g,n. (
+ ∀L,T2. ♯{L,T2} < n →
+ ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
+ ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
+ ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
+ L ⊢ U1 ⬌* U2 ∧ l1 = l2
+ ) → (
+ ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
+ ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
+ ) → (
+ ∀L1,T1. ♯{L1,T1} < n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
+ ) →
+ ∀L1,T1. ♯{L1,T1} = n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
+#h #g #n #IH3 #IH2 #IH1 #L1 * * [|||| *]
+[ #k #_ #Y #l #H1 #L2 #HL12 #X #H2 #_ -IH3 -IH2 -IH1
+ elim (ssta_inv_sort1 … H1) -H1 #Hkl #H destruct
+ >(tpr_inv_atom1 … H2) -X /4 width=6/
+| #i #Hn #U1 #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
+ elim (ssta_inv_lref1 … H1) -H1 * #K1
+ >(tpr_inv_atom1 … H2) -X
+ elim (snv_inv_lref … H3) -H3 #I0 #K0 #V0 #H #HV1
+ [ #V1 #W1 #HLK1 #HVW1 #HWU1
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+ elim (ltpr_ldrop_conf … HLK1 … HL12) #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IH1 … HVW1 K2 … HV12) -IH1 -HVW1 -HV12 // -HV1 -HKV1 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #H1
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H2
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (fpcs_lift … HW12 … H1 H2 … HWU1 … HWU2) -H1 -H2 -W1 [ /3 width=1/ ] /3 width=6/
+ | #V1 #W1 #l0 #HLK1 #HVW1 #HVU1 #H destruct
+ lapply (ldrop_mono … H … HLK1) -H #H destruct
+ lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 #X #H #HLK2
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IH1 … HVW1 K2 … HV12) -IH1 -HVW1 // -HV1 -HK12 -HKV1 #W2 #HVW2 #_ -W1
+ elim (lift_total V2 0 (i+1)) #U2 #HVU2
+ lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 /4 width=6/
+ ]
+| #p #Hn #U1 #l #H1 -IH3 -IH1
+ elim (ssta_inv_gref1 … H1)
+| #a #I #V1 #T1 #Hn #Y #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
+ elim (ssta_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
+ elim (snv_inv_bind … H3) -H3 #_ #HT1
+ elim (tpr_inv_bind1 … H2) -H2 *
+ [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
+ elim (IH1 … HTU1 (L2.ⓑ{I}V2) … HT10) -IH1 -HTU1 -HT10 // -T1 /3 width=1/ -HL12 #U0 #HTU0 #HU10
+ lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 [ /2 width=1/ ] #HT02
+ elim (ssta_tps_conf … HTU0 … HT02) -T0 #U2 #HTU2 #HU02
+ lapply (cpr_intro … U0 … HU02) -HU02 // #HU02
+ lapply (fpcs_fpr_strap1 … HU10 (L2.ⓑ{I}V2) U2 ?) [ /2 width=1/ ] -U0 #HU12
+ lapply (fpcs_fwd_shift … HU12 a) -HU12 /3 width=3/
+ | #T2 #HT12 #HT2 #H1 #H2 destruct
+ elim (IH1 … HTU1 (L2.ⓓV1) … HT12) -IH1 -HTU1 -HT12 // -T1 [2: /3 width=1/ ] -HL12 #U2 #HTU2 #HU12
+ lapply (fpcs_fwd_shift … HU12 true) -HU12 #HU12
+ elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
+ lapply (fpcs_fpr_strap1 … HU12 L2 U ?) -HU12 [ /3 width=3/ ] -U2 /2 width=3/
+ ]
+| #V1 #T1 #Hn #Y #l #H1 #L2 #HL12 #X #H2 #H3 destruct
+ elim (ssta_inv_appl1 … H1) -H1 #U1 #HTU1 #H destruct
+ elim (snv_inv_appl … H3) -H3 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
+ elim (tpr_inv_appl1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
+ elim (IH1 … HTU1 … HL12 … HT12 HT1) -IH1 -HTU1 -HL12 -HT12 -HT1 // /3 width=5/
+ | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
+ elim (snv_inv_bind … HT1) -HT1 #HW #HT2
+ elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
+ elim (dxprs_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #_ #H destruct
+ lapply (cprs_div … HW10 … HW0) -W0 #HW1
+ elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
+ elim (snv_ssta … HW) #V #l1 #HWV
+ elim (IH3 … HW1 … HW … HWX1 … HWV) -IH3 -HWX1 // [2: /2 width=5/ ] -IH2 #_ #H destruct -X1
+ elim (IH1 … HVW1 … HL12 … HV12) -HVW1 // -HV1 #W2 #HVW2 #HW12
+ elim (IH1 … HWV … HL12 W) -HWV // -HW #V0 #HWV0 #_
+ elim (IH1 … HTU2 (L2.ⓛW) … HT20 HT2) -IH1 -HTU2 -HT20 -HT2 // [2: /2 width=1/ ] #U20 #HTU20 #HU20
+ lapply (lfpr_inv_fpr L1 L2 … W) [ /2 width=1/ ] -HL12 #HL12
+ elim (lsubse_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
+ [ #U #HTU20 #HU20 -HWV0 -HL12 -W1 -W2
+ @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
+ @(fpcs_fprs_strap2 ? L1 … (ⓓ{b}V2.U2)) [ /4 width=1/ ] -V1
+ /4 width=4 by fpcs_fwd_shift, fpcs_canc_dx, cpcs_fpcs, fpcs_fwd_abst13/
+ | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20
+ /6 width=6 by lsubse_abbr, fpcs_inv_cpcs, fpcs_canc_sn, fpcs_fprs_strap1, cpcs_fpcs, bi_inj/
+ ]
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
+ elim (ssta_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
+ elim (snv_inv_bind … HT1) -HT1 #_ #HT0
+ elim (IH1 … HTU0 (L2.ⓓW2) … HT02 HT0) -IH1 -HTU0 -HT02 -HT0 // -T0 [2: /2 width=1/ ] -HL12 -HW02 #U2 #HTU2 #HU02
+ lapply (fpcs_fwd_shift … HU02 b) -HU02 #HU02
+ lapply (fpcs_flat_dx_tpr … HU02 … HV10 Appl) -HV10 -HU02 #HU02
+ lapply (fpcs_fpr_strap1 … HU02 L2 (ⓓ{b}W2.ⓐV2.U2) ?) -HU02 [ @ltpr_tpr_fpr // /2 width=3/ ] -V0 /4 width=3/
+ ]
+| #U0 #T1 #Hn #U1 #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
+ lapply (ssta_inv_cast1 … H1) -H1 #HTU1
+ elim (snv_inv_cast … H3) -H3 #T0 #l0 #_ #HT1 #HT10 #_
+ elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
+ elim (tpr_inv_cast1 … H2) -H2
+ [ * #U2 #T2 #_ #HT12 #H destruct
+ elim (IH1 … HTU1 … HL12 … HT12 HT1) -IH1 -HTU1 -HL12 -HT12 -HT1 // -T1 -U0 /3 width=3/
+ | #HT1X
+ elim (IH1 … HTU1 … HL12 … HT1X HT1) -IH1 -HTU1 -HL12 -HT1X -HT1 // -U0 -T1 /2 width=3/
+ ]
+]
+qed-.
+
+fact ssta_cpr_aux: ∀h,g,n. (
+ ∀L,T2. ♯{L,T2} < n →
+ ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
+ ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
+ ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
+ L ⊢ U1 ⬌* U2 ∧ l1 = l2
+ ) → (
+ ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
+ ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
+ ) → (
+ ∀L1,T1. ♯{L1,T1} < n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
+ ) →
+ ∀L1,T1. ♯{L1,T1} = n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀T2. L1 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L1⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
+#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #T2 * #T0 #HT10 #HT02 #HT1
+elim (ssta_ltpr_tpr_aux … IH3 IH2 … Hn … HTU1 … HT10 HT1)
+-T1 -IH3 -IH2 [3: // |2: skip |4: /3 width=5/ ] -n #U0 #HTU0 #HU10
+lapply (fpcs_inv_cpcs … HU10) -HU10 #HU10
+elim (ssta_tpss_conf … HTU0 … HT02) -T0 #U2 #HTU2 #HU02
+lapply (cpcs_cpr_strap1 … HU10 U2 ?) -HU10 /2 width=3/
+qed-.
+
+fact ssta_lfpr_aux: ∀h,g,n. (
+ ∀L,T2. ♯{L,T2} < n →
+ ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
+ ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
+ ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
+ L ⊢ U1 ⬌* U2 ∧ l1 = l2
+ ) → (
+ ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
+ ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
+ ) → (
+ ∀L1,T1. ♯{L1,T1} < n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
+ ) →
+ ∀L1,T1. ♯{L1,T1} = n →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T1 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
+#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #L2 * #L #HL1 #HL2 #HT1
+elim (ssta_ltpr_tpr_aux … IH3 IH2 … Hn … HTU1 … HL1 … HT1) -HTU1 -HL1 -HT1
+-IH3 -IH2 [3: // |2: skip |4: /3 width=5/ ] -n #U0 #HTU0 #HU10
+elim (ssta_ltpss_sn_conf … HTU0 … HL2) -HTU0 #U2 #HTU2 #HU02
+lapply (fpcs_fpr_strap1 … HU10 L U2 ?) -HU10 /3 width=3/ -HU02 #HU12
+lapply (fpcs_fpr_strap1 … HU12 L2 U2 ?) /2 width=3/ -h -L1 -T1 -U1 -U0 -l /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/ygt.ma".
+include "basic_2/dynamic/snv_cpr.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel computation for closures *************)
+
+fact ssta_cprs_aux: ∀h,g,L0,T0. (
+ ∀L,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T2⦄ →
+ ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
+ ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
+ ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
+ L ⊢ U1 ⬌* U2 ∧ l1 = l2
+ ) → (
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
+ ) → (
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡* T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
+ ) →
+ ∀L1,T1,T2. L1 ⊢ T1 ➡* T2 → L0 = L1 → T0 = T1 →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L1⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #T2 #H
+@(cprs_ind_dx … H) -T1 [ /2 width=3/ ]
+#T1 #T #HT1T #HTT2 #IH #H1 #H2 #U1 #l #HTU1 #HT1 destruct
+elim (term_eq_dec T1 T) #H destruct [ /2 width=1/ ] -IH
+elim (ssta_cpr_aux … HTU1 … HT1T HT1) -HTU1
+[2: // |3: skip |4,5,6: /3 width=9 by inj, dxprs_strap2, fw_ygt/ ] -IH3 #U #HTU #HU1
+lapply (snv_cpr_aux … HT1 … HT1T) -HT1
+[ // | skip |3,4: /3 width=6 by inj, fw_ygt/ ] -IH2 #HT
+elim (IH1 … HTU HT … HTT2) -IH1 -HTU -HT -HTT2 // [2: /3 width=1/ ] -T #U2 #HTU2 #HU2
+lapply (fpcs_inv_cpcs … HU2) -HU2 #HU2
+lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
+qed-.
+
+fact snv_cprs_aux: ∀h,g,L0,T0. (
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ →
+ ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡* T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
+ ) → (
+ ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
+ ) →
+ ∀L1,T1. L0 = L1 → T0 = T1 → ⦃h, L1⦄ ⊩ T1 :[g] →
+ ∀T2. L1 ⊢ T1 ➡* T2 → ⦃h, L1⦄ ⊩ T2 :[g].
+#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H1 #H2 #HT1 #T2 #H
+@(cprs_ind … H) -T2 // -HT1
+#T #T2 #HT1T #HTT2 #HT destruct
+lapply (snv_cpr_aux … HT … HTT2) -HT -HTT2 [1,5: // |2: skip ]
+/4 width=6 by cprs_ygt_trans, inj, fw_ygt/
+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/snv_lift.ma".
-include "basic_2/dynamic/snv_fpr_ssta.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for closures ***************)
-
-fact snv_ltpr_tpr_aux: ∀h,g,n. (
- ∀L1,T1. ♯{L1, T1} < n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
- ) → (
- ∀L1,T1. ♯{L1, T1} < n → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
- ) →
- ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
-#h #g #n #IH2 #IH1 #L1 * * [||||*]
-[ #k #Hn #H1 #L2 #_ #X #H2 destruct -IH2 -IH1 -L1
- >(tpr_inv_atom1 … H2) -X //
-| #i #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
- elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
- >(tpr_inv_atom1 … H2) -X
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
- lapply (IH1 … HV1 … HK12 V2 ?) -IH1 -HV1 -HK12 //
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- ] -HV12 /2 width=5/
-| #p #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
- elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
- elim (snv_inv_bind … H1) -H1 #HV1 #HT1
- elim (tpr_inv_bind1 … H2) -H2 *
- [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
- lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
- lapply (cpr_intro (L2.ⓑ{I}V2) … T2 0 1 HT10 ?) -HT10 /2 width=1/ -HT02 #HT12
- lapply (IH1 … HV1 … HL12 V2 ?) -HV1 //
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- ] #HV2
- lapply (IH1 … HT1 (L2.ⓑ{I}V2) … T2 ?) -IH1 -HT1 /3 width=1/
- | #T2 #HT12 #HXT2 #H1 #H2 destruct
- lapply (IH1 … HT1 (L2.ⓓV1) … T2 ?) -IH1 -HT1 // /2 width=2/
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- ] -HT12 -HL12 #HT2
- lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
- ]
-| #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct
- elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
- elim (tpr_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH1 … HV1 … HL12 V2 ?)
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- | //
- ] #HV2
- lapply (IH1 … HT1 … HL12 T2 ?)
- [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
- | //
- ] #HT2
- lapply (IH1 … HT1 … HTU1) -IH1 // #H
- elim (snv_inv_bind … H) -H #HW1 #HU1
- elim (IH2 … HVW1 … HL12 … HV12 HV1) -IH2 -HVW1 -HV12 -HV1 // #W2 #HVW2 #HW12
- lapply (fpcs_canc_sn L1 L1 … W10 W1 … HW12) -HW12 /3 width=1/ -W10 #HW12
- @(snv_appl … HV2 HT2 HVW2)
+++ /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_ltpss_sn.ma".
-include "basic_2/computation/dxprs_lift.ma".
-include "basic_2/equivalence/lsubse_ssta.ma".
-include "basic_2/equivalence/fpcs_cpcs.ma".
-include "basic_2/equivalence/lfpcs_fpcs.ma".
-include "basic_2/dynamic/snv_ssta.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for closures ***************)
-
-fact ssta_ltpr_tpr_aux: ∀h,g,n. (
- ∀L,T2. ♯{L,T2} < n →
- ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
- ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
- ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
- L ⊢ U1 ⬌* U2 ∧ l1 = l2
- ) → (
- ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
- ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
- ) → (
- ∀L1,T1. ♯{L1,T1} < n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
- ) →
- ∀L1,T1. ♯{L1,T1} = n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
-#h #g #n #IH3 #IH2 #IH1 #L1 * * [|||| *]
-[ #k #_ #Y #l #H1 #L2 #HL12 #X #H2 #_ -IH3 -IH2 -IH1
- elim (ssta_inv_sort1 … H1) -H1 #Hkl #H destruct
- >(tpr_inv_atom1 … H2) -X /4 width=6/
-| #i #Hn #U1 #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
- elim (ssta_inv_lref1 … H1) -H1 * #K1
- >(tpr_inv_atom1 … H2) -X
- elim (snv_inv_lref … H3) -H3 #I0 #K0 #V0 #H #HV1
- [ #V1 #W1 #HLK1 #HVW1 #HWU1
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- elim (IH1 … HVW1 K2 … HV12) -IH1 -HVW1 -HV12 // -HV1 -HKV1 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #H1
- lapply (ldrop_fwd_ldrop2 … HLK2) #H2
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (fpcs_lift … HW12 … H1 H2 … HWU1 … HWU2) -H1 -H2 -W1 [ /3 width=1/ ] /3 width=6/
- | #V1 #W1 #l0 #HLK1 #HVW1 #HVU1 #H destruct
- lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 #X #H #HLK2
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- elim (IH1 … HVW1 K2 … HV12) -IH1 -HVW1 // -HV1 -HK12 -HKV1 #W2 #HVW2 #_ -W1
- elim (lift_total V2 0 (i+1)) #U2 #HVU2
- lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 /4 width=6/
- ]
-| #p #Hn #U1 #l #H1 -IH3 -IH1
- elim (ssta_inv_gref1 … H1)
-| #a #I #V1 #T1 #Hn #Y #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
- elim (ssta_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
- elim (snv_inv_bind … H3) -H3 #_ #HT1
- elim (tpr_inv_bind1 … H2) -H2 *
- [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
- elim (IH1 … HTU1 (L2.ⓑ{I}V2) … HT10) -IH1 -HTU1 -HT10 // -T1 /3 width=1/ -HL12 #U0 #HTU0 #HU10
- lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 [ /2 width=1/ ] #HT02
- elim (ssta_tps_conf … HTU0 … HT02) -T0 #U2 #HTU2 #HU02
- lapply (cpr_intro … U0 … HU02) -HU02 // #HU02
- lapply (fpcs_fpr_strap1 … HU10 (L2.ⓑ{I}V2) U2 ?) [ /2 width=1/ ] -U0 #HU12
- lapply (fpcs_fwd_shift … HU12 a) -HU12 /3 width=3/
- | #T2 #HT12 #HT2 #H1 #H2 destruct
- elim (IH1 … HTU1 (L2.ⓓV1) … HT12) -IH1 -HTU1 -HT12 // -T1 [2: /3 width=1/ ] -HL12 #U2 #HTU2 #HU12
- lapply (fpcs_fwd_shift … HU12 true) -HU12 #HU12
- elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
- lapply (fpcs_fpr_strap1 … HU12 L2 U ?) -HU12 [ /3 width=3/ ] -U2 /2 width=3/
- ]
-| #V1 #T1 #Hn #Y #l #H1 #L2 #HL12 #X #H2 #H3 destruct
- elim (ssta_inv_appl1 … H1) -H1 #U1 #HTU1 #H destruct
- elim (snv_inv_appl … H3) -H3 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
- elim (tpr_inv_appl1 … H2) -H2 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
- elim (IH1 … HTU1 … HL12 … HT12 HT1) -IH1 -HTU1 -HL12 -HT12 -HT1 // /3 width=5/
- | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
- elim (snv_inv_bind … HT1) -HT1 #HW #HT2
- elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
- elim (dxprs_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #_ #H destruct
- lapply (cprs_div … HW10 … HW0) -W0 #HW1
- elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
- elim (snv_ssta … HW) #V #l1 #HWV
- elim (IH3 … HW1 … HW … HWX1 … HWV) -IH3 -HWX1 // [2: /2 width=5/ ] -IH2 #_ #H destruct -X1
- elim (IH1 … HVW1 … HL12 … HV12) -HVW1 // -HV1 #W2 #HVW2 #HW12
- elim (IH1 … HWV … HL12 W) -HWV // -HW #V0 #HWV0 #_
- elim (IH1 … HTU2 (L2.ⓛW) … HT20 HT2) -IH1 -HTU2 -HT20 -HT2 // [2: /2 width=1/ ] #U20 #HTU20 #HU20
- lapply (lfpr_inv_fpr L1 L2 … W) [ /2 width=1/ ] -HL12 #HL12
- elim (lsubse_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
- [ #U #HTU20 #HU20 -HWV0 -HL12 -W1 -W2
- @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
- @(fpcs_fprs_strap2 ? L1 … (ⓓ{b}V2.U2)) [ /4 width=1/ ] -V1
- /4 width=4 by fpcs_fwd_shift, fpcs_canc_dx, cpcs_fpcs, fpcs_fwd_abst13/
- | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20
- /6 width=6 by lsubse_abbr, fpcs_inv_cpcs, fpcs_canc_sn, fpcs_fprs_strap1, cpcs_fpcs, bi_inj/
- ]
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
- elim (ssta_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
- elim (snv_inv_bind … HT1) -HT1 #_ #HT0
- elim (IH1 … HTU0 (L2.ⓓW2) … HT02 HT0) -IH1 -HTU0 -HT02 -HT0 // -T0 [2: /2 width=1/ ] -HL12 -HW02 #U2 #HTU2 #HU02
- lapply (fpcs_fwd_shift … HU02 b) -HU02 #HU02
- lapply (fpcs_flat_dx_tpr … HU02 … HV10 Appl) -HV10 -HU02 #HU02
- lapply (fpcs_fpr_strap1 … HU02 L2 (ⓓ{b}W2.ⓐV2.U2) ?) -HU02 [ @ltpr_tpr_fpr // /2 width=3/ ] -V0 /4 width=3/
- ]
-| #U0 #T1 #Hn #U1 #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
- lapply (ssta_inv_cast1 … H1) -H1 #HTU1
- elim (snv_inv_cast … H3) -H3 #T0 #l0 #_ #HT1 #HT10 #_
- elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
- elim (tpr_inv_cast1 … H2) -H2
- [ * #U2 #T2 #_ #HT12 #H destruct
- elim (IH1 … HTU1 … HL12 … HT12 HT1) -IH1 -HTU1 -HL12 -HT12 -HT1 // -T1 -U0 /3 width=3/
- | #HT1X
- elim (IH1 … HTU1 … HL12 … HT1X HT1) -IH1 -HTU1 -HL12 -HT1X -HT1 // -U0 -T1 /2 width=3/
- ]
-]
-qed-.
-
-fact ssta_ltpr_cpr_aux: ∀h,g,n. (
- ∀L,T2. ♯{L,T2} < n →
- ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
- ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
- ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
- L ⊢ U1 ⬌* U2 ∧ l1 = l2
- ) → (
- ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
- ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
- ) → (
- ∀L1,T1. ♯{L1,T1} < n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
- ) →
- ∀L1,T1. ♯{L1,T1} = n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
-#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #L2 #HL12 #T2 * #T0 #HT10 #HT02 #HT1
-elim (ssta_ltpr_tpr_aux … IH3 IH2 … Hn … HTU1 … HL12 … HT10 HT1)
--T1 -IH3 -IH2 -HL12 [2: /3 width=5/ ] -n #U0 #HTU0 #HU10
-elim (ssta_tpss_conf … HTU0 … HT02) -T0 #U2 #HTU2 #HU02
-lapply (fpcs_fpr_strap1 … HU10 L2 U2 ?) -HU10 /2 width=3/ -HTU2 /3 width=3/
-qed-.
-
-fact ssta_fpr_aux: ∀h,g,n. (
- ∀L,T2. ♯{L,T2} < n →
- ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
- ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
- ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
- L ⊢ U1 ⬌* U2 ∧ l1 = l2
- ) → (
- ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
- ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
- ) → (
- ∀L1,T1. ♯{L1,T1} < n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
- ) →
- ∀L1,T1. ♯{L1,T1} = n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
-#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #L2 #T2 #H12 #HT1
-elim (fpr_inv_all … H12) -H12 #L #HL1 #HT12 #HL2
-elim (ssta_ltpr_cpr_aux … IH3 IH2 … Hn … HTU1 … HL1 … HT12 HT1)
--T1 -IH3 -IH2 -HL1 [2: /3 width=5/ ] -n #U #HTU #HU1
-elim (ssta_ltpss_sn_conf … HTU … HL2) -HTU #U2 #HTU2 #HU2
-lapply (fpcs_fpr_strap1 … HU1 L2 U2 ?) -HU1 /2 width=3/ -HTU2 /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/dynamic/snv_fpr.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel computation for closures *************)
-
-fact ssta_fprs_aux: ∀h,g,n. (
- ∀L,T2. ♯{L,T2} < n →
- ∀T1. L ⊢ T1 ⬌* T2 → ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
- ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
- ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
- L ⊢ U1 ⬌* U2 ∧ l1 = l2
- ) → (
- ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
- ∀U. ⦃h, L⦄ ⊢ T •*➡*[g] U → ⦃h, L⦄ ⊩ U :[g]
- ) → (
- ∀L1,T1. ♯{L1,T1} < n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
- ) →
- ∀L1,T1. ♯{L1,T1} = n →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
-#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #L2 #T2 #H12 #HT1
-@(fprs_ind … H12) -L2 -T2 [-IH1 /2 width=3/ ] (**) (* auto fails without -IH1 *)
-#L #L2 #T #T2 #HT1 #HT2 * #U #HTU #HU1
-(*
-lapply (IH2 …
-
-elim (ssta_fpr_aux … IH3 … Hn … HTU1 … HT2 HT1)
--T1 -IH3 -IH2 -HL1 [2: /3 width=5/ ] -n #U #HTU #HU1
-*)
\ No newline at end of file
(* Advanced inversion lemmas ************************************************)
lemma fpcs_inv_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
- ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄.
+ ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄.
#I #L1 #L2 #V1 #V2 #T1 #T2 #H12
elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
elim (fprs_inv_pair1 … H1) -H1 #K1 #U1 #_ #HTU1 #H destruct
+++ /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( h ⊢ break ⦃ L1, break T1 ⦄ • ⥸ break [ g ] break ⦃ L2 , break T2 ⦄ )"
- non associative with precedence 45
- for @{ 'YPRed $h $g $L1 $T1 $L2 $T2 }.
-
-include "basic_2/substitution/csup.ma".
-include "basic_2/reducibility/xpr.ma".
-
-(* HYPER PARALLEL REDUCTION ON CLOSURES *************************************)
-
-inductive ypr (h) (g) (L1) (T1): relation2 lenv term ≝
-| ypr_cpr : ∀T2. L1 ⊢ T1 ➡ T2 → ypr h g L1 T1 L1 T2
-| ypr_ssta: ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[g, l + 1] T2 → ypr h g L1 T1 L1 T2
-| ypr_csup: ∀L2,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ypr h g L1 T1 L2 T2
-.
-
-interpretation
- "hyper parallel reduction (closure)"
- 'YPRed h g L1 T1 L2 T2 = (ypr h g L1 T1 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma ypr_refl: ∀h,g. bi_reflexive … (ypr h g).
-/2 width=1/ qed.
-
-lemma xpr_ypr: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •➡[g] T2 → h ⊢ ⦃L, T1⦄ •⥸[g] ⦃L, T2⦄.
-#h #g #L #T1 #T2 * /2 width=1/ /2 width=2/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( h ⊢ break ⦃ L1, break T1 ⦄ • ⥸ * break [ g ] break ⦃ L2 , break T2 ⦄ )"
- non associative with precedence 45
- for @{ 'YPRedStar $h $g $L1 $T1 $L2 $T2 }.
-
-include "basic_2/reducibility/ypr.ma".
-
-(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************)
-
-definition yprs: ∀h. sd h → bi_relation lenv term ≝
- λh,g. bi_TC … (ypr h g).
-
-interpretation "hyper parallel computation (closure)"
- 'YPRedStar h g L1 T1 L2 T2 = (yprs h g L1 T1 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma yprs_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term. R L1 T1 →
- (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ •⥸[g] ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → R L2 T2.
-/3 width=7 by bi_TC_star_ind/ qed-.
-
-lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 →
- (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ •⥸[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ •⥸*[g] ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → R L1 T1.
-/3 width=7 by bi_TC_star_ind_dx/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma yprs_refl: ∀h,g. bi_reflexive … (yprs h g).
-/2 width=1/ qed.
-
-lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ •⥸[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄.
-/2 width=4/ qed.
-
-lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ •⥸[g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ •⥸*[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, 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/unfold/csups.ma".
-include "basic_2/computation/yprs.ma".
-
-(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************)
-
-(* Properties on iterated supclosure ****************************************)
-
-lemma csups_yprs: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄.
-#h #g #L1 #L2 #T1 #T2 #H @(csups_ind … H) -L2 -T2 /3 width=1/ /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/xprs_cprs.ma".
-include "basic_2/computation/yprs.ma".
-
-(* HYPER PARALLEL COMPUTATION ON CLOSURES ***********************************)
-
-(* Properties on extended parallel computation for terms ********************)
-
-lemma xprs_yprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •➡*[g] T2 →
- h ⊢ ⦃L, T1⦄ •⥸*[g] ⦃L, T2⦄.
-#h #g #L #T1 #T2 #H @(xprs_ind … H) -T2 // /3 width=4/
-qed.
-
-lemma cprs_yprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ •⥸*[g] ⦃L, T2⦄.
-/3 width=1/ 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/yprs.ma".
-
-(* HYPER PARALLEL COMPUTATION ON TERMS **************************************)
-
-theorem yprs_trans: ∀h,g. bi_transitive … (yprs h g).
-/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 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( h ⊢ break ⦃ L1, break T1 ⦄ • ⭃ * break [ g ] break ⦃ L2 , break T2 ⦄ )"
- non associative with precedence 45
- for @{ 'YPRedStepStar $h $g $L1 $T1 $L2 $T2 }.
-
-include "basic_2/substitution/csup.ma".
-include "basic_2/computation/yprs.ma".
-
-(* ITERATED STEP OF HYPER PARALLEL COMPUTATION ON CLOSURES ******************)
-
-inductive ysteps (h) (g) (L1) (T1) (L2) (T2): Prop ≝
-| ysteps_intro: h ⊢ ⦃L1, T1⦄ •⥸*[g] ⦃L2, T2⦄ → (L1 = L2 → T1 = T2 → ⊥) →
- ysteps h g L1 T1 L2 T2
-.
-
-interpretation "iterated step of hyper parallel computation (closure)"
- 'YPRedStepStar h g L1 T1 L2 T2 = (ysteps h g L1 T1 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma ssta_ysteps: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U →
- h ⊢ ⦃L, T⦄ •⭃*[g] ⦃L, U⦄.
-#h #g #L #T #U #l #HTU
-@ysteps_intro /3 width=2/ #_ #H destruct
-elim (ssta_inv_refl … HTU)
-qed.
-
-lemma csup_ysteps: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ •⭃*[g] ⦃L2, T2⦄.
-#h #g #L1 #L2 #T1 #T2 #H
-lapply (csup_fwd_cw … H) #H1
-@ysteps_intro /3 width=1/ -H #H2 #H3 destruct
-elim (lt_refl_false … H1)
-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/yprs_csups.ma".
-include "basic_2/computation/ysteps.ma".
-
-(* ITERATED STEP OF HYPER PARALLEL COMPUTATION ON CLOSURES ******************)
-
-(* Properties on iterated supclosure ****************************************)
-
-lemma csups_ysteps: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ >* ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ •⭃*[g] ⦃L2, T2⦄.
-#h #g #L1 #L2 #T1 #T2 #H
-lapply (csups_fwd_cw … H) #H1
-@ysteps_intro /2 width=1/ -H #H2 #H3 destruct
-elim (lt_refl_false … H1)
-qed.
lemma lw_pair: ∀I,L,V. ♯{L} < ♯{L.ⓑ{I}V}.
/3 width=1/ qed.
-(* Basic_1: removed theorems 2: clt_cong clt_head clt_thead *)
+(* Basic_1: removed theorems 4: clt_cong clt_head clt_thead clt_wf_ind *)
+(* Basic_1: removed local theorems 1: clt_wf__q_ind *)
(* Basic_1: note: clt_thead should be renamed clt_ctail *)
non associative with precedence 45
for @{ 'FocalizedPRedAlt $L1 $L2 }.
+notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≻ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTSuccessor $h $g $L1 $T1 $L2 $T2 }.
+
(* Computation **************************************************************)
notation "hvbox( T1 ➡ * break term 46 T2 )"
non associative with precedence 45
for @{ 'DecomposedXSN $h $g $L $T }.
+notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ > break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTGreaterThan $h $g $L1 $T1 $L2 $T2 }.
+
(* Conversion ***************************************************************)
notation "hvbox( L ⊢ break term 46 T1 ⬌ break term 46 T2 )"
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ssta.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* "BIG TREE" SUCCESSOR ON CLOSURES *****************************************)
+
+inductive ysc (h) (g) (L1) (T1): relation2 lenv term ≝
+| ysc_fw : ∀L2,T2. ♯{L2, T2} < ♯{L1, T1} → ysc h g L1 T1 L2 T2
+| ysc_cpr : ∀T2. L1 ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → ysc h g L1 T1 L1 T2
+| ysc_ssta: ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[g, l + 1] T2 → ysc h g L1 T1 L1 T2
+.
+
+interpretation
+ "'big tree' successor (closure)"
+ 'BTSuccessor h g L1 T1 L2 T2 = (ysc h g L1 T1 L2 T2).
]
qed-.
+(* Basic_1: was: thead_x_lift_y_y *)
lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] ②{I} V. T ≡ T → ⊥.
#J #T elim T -T
[ * #i #V #d #e #H
projects / helm.git / commitdiff