+++ /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/notation/relations/btpredalt_8.ma".
-include "basic_2/reduction/fpb_fleq.ma".
-include "basic_2/reduction/fpbq.ma".
-
-(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
-
-(* Basic properties *********************************************************)
-
-lemma fleq_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by fpbq_lleq/
-qed.
-
-lemma fpb_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
-qed.
-
-(* Advanced eliminators *****************************************************)
-
-lemma fpbq_ind_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ∀R:Prop.
- (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → R) →
- (⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R) →
- ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ → R.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #R #HI1 #HI2 #H elim (fpbq_fpbqa … H) /2 width=1 by/
-qed-.
-
-(* aternative definition of fpb *********************************************)
-
-lemma fpb_fpbq_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ ∧ (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥).
-/3 width=10 by fpb_fpbq, fpb_inv_fleq, conj/ qed.
-
-lemma fpbq_inv_fpb_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 @(fpbq_ind_alt … H) -H //
-#H elim H0 -H0 //
-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/lsubr_length.ma".
+include "basic_2/rt_transition/lfpx_fsle.ma".
+
+lemma pippo: ∀L1,L2. L2 ⫃ L1 → ∀T. ⦃L1, T⦄ ⊆ ⦃L2, T⦄.
+#L1 #L2 #HL #T
+elim (frees_total L1 T) #f1 #Hf1
+elim (frees_total L2 T) #f2 #Hf2
+lapply (lsubr_lsubf … Hf1 … HL … Hf2) #H
+lapply (lsubf_fwd_sle … H) -H #H12
+lapply (lsubr_fwd_length … HL) -HL #HL
+/3 width=8 by lveq_length_eq, ex4_4_intro/
+qed.
+
+lemma R_fle_comp_LTC: ∀R. R_fle_compatible R → R_fle_compatible (LTC … R).
+#R #HR #L #T1 #T2 #H elim H -T2
+/3 width=3 by fle_trans_dx/
+qed-.
(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
-(* Basic_2A1: includes: fpbq_lleq *)
+(* Basic_2A1: includes: fleq_fpbq fpbq_lleq *)
inductive fpbq (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
| fpbq_fquq : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h o G1 L1 T1 G1 L1 T2
| fpbq_lfpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h, T1] L2 → fpbq h o G1 L1 T1 G1 L2 T1
-| ffpq_lfdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
+| fpbq_ffdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
.
interpretation
--- /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/rt_transition/fpb_ffdeq.ma".
+include "basic_2/rt_transition/fpbq.ma".
+
+(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
+
+(* Properties with proper parallel rst-transition for closures **************)
+
+lemma fpb_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lfpx, fqu_fquq/
+qed.
+
+(* Basic_2A1: fpb_fpbq_alt *)
+lemma fpb_fpbq_ffdneq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+ ∧∧ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ & (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥).
+/3 width=10 by fpb_fpbq, fpb_inv_ffdeq, conj/ qed-.
+
+(* Inversrion lemmas with proper parallel rst-transition for closures *******)
+
+(* Basic_2A1: uses: fpbq_ind_alt *)
+lemma fpbq_inv_fpb: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
+ ∨∨ ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄
+ | ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+[ #G2 #L2 #T2 * [2: * #H1 #H2 #H3 destruct ]
+ /3 width=1 by fpb_fqu, ffdeq_intro_sn, or_intror, or_introl/
+| #T2 #H elim (tdeq_dec h o T1 T2)
+ /4 width=1 by fpb_cpx, ffdeq_intro_sn, or_intror, or_introl/
+| #L2 elim (lfdeq_dec h o L1 L2 T1)
+ /4 width=1 by fpb_lfpx, ffdeq_intro_sn, or_intror, or_introl/
+| /2 width=1 by or_introl/
+]
+qed-.
+
+(* Basic_2A1: fpbq_inv_fpb_alt *)
+lemma fpbq_ffdneq_inv_fpb: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
+ (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #H0
+elim (fpbq_inv_fpb … H) -H // #H elim H0 -H0 //
+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/rt_transition/lfpx_fsle.ma".
-(*
-lemma R_fle_comp_LTC: ∀R. R_fle_compatible R → R_fle_compatible (LTC … R).
-#R #HR #L #T1 #T2 #H elim H -T2
-/3 width=3 by fle_trans_dx/
-qed-.
-*)
-
-(* Note: "⦃L2, T1⦄ ⊆ ⦃L0, T1⦄" may not hold *)
-axiom lfpx_cpx_conf_fsle4: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
-(*
-#h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G0 L0 T0) -G0 -L0 -T0
-#G #L #T #IH #G0 #L0 * *
-[ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
- lapply (lfxs_fwd_length … HY) -HY #H0
- elim (cpx_inv_sort1 … HX) -HX #H destruct
- /3 width=1 by fsle_sort_length, and3_intro/
-| * [| #i ] #HG #HL #HT #X #HX #Y #HY destruct
- [ elim (cpx_inv_zero1 … HX) -HX
- [ #H destruct
- elim (lfxs_inv_zero … HY) -HY *
- [ #H1 #H2 destruct -IH /2 width=1 by and3_intro/
- | #I #K0 #K2 #V0 #V2 #HK02 #HV02 #H1 #H2 destruct
- lapply (lfxs_fwd_length … HK02) #HK
- elim H2R -H2R #H2R
- [ <(H2R G0) in HV02; -H2R #HV02
- elim (IH … HV02 … HK02) /2 width=2 by fqu_fqup, fqu_lref_O/ -IH -HV02 -HK02 #H1V #H2V #_
- /4 width=1 by fsle_trans_tc, fsle_zero_bi, and3_intro/
- | lapply (H2R … HV02 … HK02) -H2R -HV02 -HK02 -IH #HKV20
- /3 width=1 by fsle_zero_bi, and3_intro/
- ]
- | #f #I #K0 #K2 #Hf #HK02 #H1 #H2 destruct
- ]
- | * #I0 #K0 #V0 #V1 #HV01 #HV1X #H destruct
- elim (lfxs_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
- ]
- | elim (cpx_inv_lref1 … HX) -HX
- [ #H destruct
- elim (lfxs_inv_lref … HY) -HY *
- [ #H0 #H1 destruct /2 width=1 by and3_intro/
- | #I0 #I2 #K0 #K2 #HK02 #H1 #H2 destruct
- lapply (lfxs_fwd_length … HK02) #HK
- elim (IH … HK02) [|*: /2 width=4 by fqu_fqup/ ] -IH -HK02
- /3 width=5 by and3_intro, fsle_lifts_SO/
- ]
- | * #I0 #K0 #V1 #HV1 #HV1X #H0 destruct
- elim (lfxs_inv_lref_bind_sn … HY) -HY #I2 #K2 #HK02 #H destruct
- lapply (lfxs_fwd_length … HK02) #HK
- elim (IH … HK02) [|*: /2 width=4 by fqu_fqup/ ] -IH -HV1 -HK02
- /3 width=5 by fsle_lifts_SO, and3_intro/
- ]
- ]
-| #l #HG #HL #HT #X #HX #Y #HY destruct -IH
- lapply (lfxs_fwd_length … HY) -HY #H0
- >(cpx_inv_gref1 … HX) -X
- /3 width=1 by fsle_gref_length, and3_intro/
-| #p #I #V0 #T0 #HG #HL #HT #X #HX #Y #HY destruct
- lapply (lfxs_fwd_length … HY) #H0
- elim (lfxs_inv_bind … V0 ? HY) -HY // #HV0 #HT0
- elim (cpx_inv_bind1 … HX) -HX *
- [ #V1 #T1 #HV01 #HT01 #H destruct
- elim (IH … HV01 … HV0) -HV01 -HV0 // #H1V #H2V #H3V
- elim (IH … HT01 … HT0) -HT01 -HT0 -IH // #H1T #H2T #H3T
- /4 width=3 by fsle_bind_eq, fsle_fwd_pair_sn, and3_intro/
- | #T #HT #HXT #H1 #H2 destruct
- elim (IH G0 … V0… V0 … HV0) -HV0 // #H1V #H2V #H3V
- elim (IH … HT … HT0) -HT -HT0 -IH // #H1T #H2T #H3T
- /3 width=5 by fsle_bind, fsle_inv_lifts_sn, and3_intro/
- ]
-| #I #V0 #X0 #HG #HL #HT #X #HX #Y #HY destruct
- elim (lfxs_inv_flat … HY) -HY #HV0 #HX0
- elim (cpx_inv_flat1 … HX) -HX *
- [ #V1 #T1 #HV01 #HT01 #H destruct
- elim (IH … HV01 … HV0) -HV01 -HV0 // #H1V #H2V #H3V
- elim (IH … HT01 … HX0) -HT01 -HX0 -IH // #H1T #H2T #H3T
- /3 width=3 by fsle_flat, and3_intro/
- | #HX #H destruct
- elim (IH G0 … V0… V0 … HV0) -HV0 // #H1V #H2V #H3V
- elim (IH … HX … HX0) -HX -HX0 -IH // #H1T #H2T #H3T
- /4 width=3 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, and3_intro/
- | #HX #H destruct
- elim (IH … HX … HV0) -HX -HV0 // #H1V #H2V #H3V
- elim (IH G0 … X0… X0 … HX0) -HX0 -IH // #H1T #H2T #H3T
- /4 width=3 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, and3_intro/
- | #p #V1 #W0 #W1 #T0 #T1 #HV01 #HW01 #HT01 #H1 #H2 #H3 destruct
- elim (lfxs_inv_bind … W0 ? HX0) -HX0 // #HW0 #HT0
- elim (IH … HV01 … HV0) -HV01 -HV0 // #H1V #H2V #H3V
- elim (IH … HW01 … HW0) -HW01 -HW0 // #H1W #H2W #H3W
- elim (IH … HT01 … HT0) -HT01 -HT0 -IH // #H1T #H2T #H3T
- lapply (fsle_fwd_pair_sn … H2T) -H2T #H2T
- lapply (fsle_fwd_pair_sn … H3T) -H3T #H3T
- @and3_intro [ /3 width=5 by fsle_flat, fsle_bind/ ] (**) (* full auto too slow *)
- @fsle_bind_sn_ge /4 width=1 by fsle_shift, fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, fsle_bind_dx_sn/
- | #p #V1 #X1 #W0 #W1 #T0 #T1 #HV01 #HVX1 #HW01 #HT01 #H1 #H2 #H3 destruct
- elim (lfxs_inv_bind … W0 ? HX0) -HX0 // #HW0 #HT0
- elim (IH … HV01 … HV0) -HV01 -HV0 // #H1V #H2V #H3V
- elim (IH … HW01 … HW0) -HW01 -HW0 // #H1W #H2W #H3W
- elim (IH … HT01 … HT0) -HT01 -HT0 -IH // #H1T #H2T #H3T
- lapply (fsle_fwd_pair_sn … H2T) -H2T #H2T
- lapply (fsle_fwd_pair_sn … H3T) -H3T #H3T
- @and3_intro[ /3 width=5 by fsle_flat, fsle_bind/ ] (**) (* full auto too slow *)
- @fsle_bind_sn_ge //
- [1,3: /3 width=1 by fsle_flat_dx_dx, fsle_bind_dx_sn/
- |2,4: /4 width=3 by fsle_flat_sn, fsle_flat_dx_sn, fsle_flat_dx_dx, fsle_shift, fsle_lifts_sn/
- ]
- ]
-]
-*)
(* Note: "⦃L2, T1⦄ ⊆ ⦃L2, T0⦄" does not hold *)
(* Note: Take L0 = K0.ⓓ(ⓝW.V), L2 = K0.ⓓW, T0 = #0, T1 = ⬆*[1]V *)
(* Note: This invalidates lfpxs_cpx_conf: "∀h,G. s_r_confluent1 … (cpx h G) (lfpxs h G)" *)
-(* Note: "⦃L2, T1⦄ ⊆ ⦃L0, T1⦄" may not hold *)
(* Basic_2A1: uses: lpx_cpx_frees_trans *)
lemma lfpx_cpx_conf_fsge: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
lemma lfpx_cpx_conf: ∀h,G. s_r_confluent1 … (cpx h G) (lfpx h G).
/2 width=5 by cpx_lfxs_conf/ qed-.
+
+lemma lfpx_cpx_conf_fsge_dx: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
+/3 width=5 by lfpx_cpx_conf, lfpx_fsge_comp/ qed-.
/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
qed-.
+theorem fsle_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1, T1⦄ ⊆ ⦃L, T⦄ →
+ ∀L2,T2. |L| = |L2| → ⦃L, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+#L1 #L #T1 #T #HL1
+* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
+#L2 #T2 #HL2
+* #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
+lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
+elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
+elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
+lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
+/3 width=10 by lveq_length_eq, sle_trans, ex4_4_intro/
+qed-.
+
theorem fsle_bind_sn_ge: ∀L1,L2. |L2| ≤ |L1| →
∀V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
class "cyan"
[ { "rt-transition" * } {
[ { "uncounted parallel rst-transition" * } {
- [ [ "for closures" ] "fpbq" + "( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpbq_aaa" * ]
+ [ [ "for closures" ] "fpbq" + "( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpbq_aaa" + "fpbq_fpb" * ]
[ [ "proper for closures" ] "fpb" + "( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpb_lfdeq" + "fpb_ffdeq" * ]
}
]
projects / helm.git / commitdiff