helm/www/lambdadelta/xslt/chc_45.xsl
helm/www/lambdadelta/xslt/xhtbl.xsl
+matita/matita/contribs/lambdadelta/.depend
matita/matita/contribs/lambdadelta/nodes
matita/matita/contribs/lambdadelta/token
matita/matita/contribs/lambdadelta/2A1
/3 width=4 by lfxs_gref, tc_lfxs_step_dx, inj/
qed.
-lemma tc_lfxs_sym: ∀R. lexs_frees_confluent (cext2 R) cfull →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (tc_lfxs R T).
-#R #H1R #H2R #T #L1 #L2 #H elim H -L2
-/4 width=3 by lfxs_sym, tc_lfxs_step_sn, inj/
-qed-.
-
lemma tc_lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
∀L1,L2,T. L1 ⪤**[R1, T] L2 → L1 ⪤**[R2, T] L2.
#R1 #R2 #HR #L1 #L2 #T #H elim H -L2
(* Note: s_rs_transitive_lex_inv_isid could be invoked in the last auto but makes it too slow *)
lemma tc_lfxs_inv_lex_lfeq: ∀R. c_reflexive … R →
- lexs_frees_confluent (cext2 R) cfull →
+ lfxs_fle_compatible R →
s_rs_transitive … R (λ_.lex R) →
lfeq_transitive R →
∀L1,L2,T. L1 ⪤**[R, T] L2 →
elim (lexs_sdj_split … ceq_ext … HL2 f0 ?) -HL2
[ #L0 #HL0 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ]
lapply (lexs_sdj … HL0 f1 ?) /2 width=1 by sdj_isid_sn/ #H
- elim (H2R … Hf1 … H) -H #f2 #Hf2 #Hf21
+ elim (frees_lexs_conf … Hf1 … H) // -H2R -H #f2 #Hf2 #Hf21
lapply (sle_lexs_trans … HL02 … Hf21) -f1 // #HL02
lapply (lexs_co ?? cfull (LTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1
/8 width=11 by lexs_inv_tc_dx, lexs_tc_dx, lexs_co, ext2_tc, ext2_refl, step, ex2_intro/ (**) (* full auto too slow *)
(* *)
(**************************************************************************)
+include "basic_2/static/lfxs_lfxs.ma".
include "basic_2/i_static/tc_lfxs.ma".
(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+(* Advanced properties ******************************************************)
+
+lemma tc_lfxs_sym: ∀R. lfxs_fle_compatible R →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (tc_lfxs R T).
+#R #H1R #H2R #T #L1 #L2 #H elim H -L2
+/4 width=3 by lfxs_sym, tc_lfxs_step_sn, inj/
+qed-.
+
(* Main properties **********************************************************)
theorem tc_lfxs_trans: ∀R,T. Transitive … (tc_lfxs R T).
include "basic_2/static/lfxs_lex.ma".
-include "basic_2/static/lfxs_fle.ma".
include "basic_2/rt_transition/cpx_etc.ma".
include "basic_2/rt_computation/lfpxs_lpxs.ma".
lemma tc_lfxs_inv_lex_lfeq: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 →
∃∃L. ⦃G, L1⦄ ⊢ ⬈*[h] L & L ≡[T] L2.
-/3 width=5 by lfpx_frees_conf, lpx_cpxs_trans, lfeq_cpx_trans, tc_lfxs_inv_lex_lfeq/ qed-.
+/3 width=5 by lfpx_fle_comp, lpx_cpxs_trans, lfeq_cpx_trans, tc_lfxs_inv_lex_lfeq/ qed-.
) →
∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → R L.
#h #o #G #T #R #H0 #L1 #H elim H -L1
-/5 width=1 by lfdeq_sym, SN_intro/
+/5 width=1 by SN_intro/
qed-.
(* Basic properties *********************************************************)
lemma lfsx_intro: ∀h,o,G,L1,T.
(∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → (L1 ≛[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
-/5 width=1 by lfdeq_sym, SN_intro/ qed.
+/5 width=1 by SN_intro/ qed.
(* Basic_2A1: uses: lsx_sort *)
lemma lfsx_sort: ∀h,o,G,L,s. G ⊢ ⬈*[h, o, ⋆s] 𝐒⦃L⦄.
include "basic_2/rt_transition/cpx_lfxs.ma".
include "basic_2/rt_transition/cpm_cpx.ma".
-include "basic_2/rt_transition/cpr_ext.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
(* Properties with context-sensitive free variables *************************)
-lemma cpm_frees_conf: ∀n,h,G. R_frees_confluent (cpm n h G).
-/3 width=6 by cpm_fwd_cpx, cpx_frees_conf/ qed-.
+lemma cpm_fle_comp: ∀n,h,G. R_fle_compatible (cpm n h G).
+/3 width=6 by cpm_fwd_cpx, cpx_fle_comp/ qed-.
-lemma lfpr_frees_conf: ∀h,G. lexs_frees_confluent (cpr_ext h G) cfull.
-/5 width=9 by cpm_fwd_cpx, lfpx_frees_conf, lexs_co, cext2_co/ qed-.
+lemma lfpr_fle_comp: ∀h,G. lfxs_fle_compatible (cpm 0 h G).
+/4 width=5 by cpm_fwd_cpx, lfpx_fle_comp, lfxs_co/ qed-.
(* Properties with generic extension on referred entries ********************)
include "basic_2/static/fle_drops.ma".
include "basic_2/static/fle_fqup.ma".
-include "basic_2/static/fle_lsubf.ma".
include "basic_2/static/fle_fle.ma".
include "basic_2/static/lfxs_length.ma".
include "basic_2/rt_transition/cpx.ma".
(* *)
(**************************************************************************)
-include "basic_2/static/lfdeq.ma".
+include "basic_2/static/lfdeq_lfdeq.ma".
include "basic_2/rt_transition/cpx_lfxs.ma".
(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
(* *)
(**************************************************************************)
+include "basic_2/syntax/lveq_length.ma".
+include "basic_2/relocation/lexs_length.ma".
include "basic_2/relocation/drops_lexs.ma".
include "basic_2/static/frees_drops.ma".
include "basic_2/static/lsubf_frees.ma".
qed-.
(* Basic_2A1: uses: cpx_frees_trans *)
-lemma cpx_frees_conf: ∀h,G. R_frees_confluent (cpx h G).
-/4 width=7 by cpx_frees_conf_lexs, lexs_refl, ext2_refl/ qed-.
+lemma cpx_fle_comp: ∀h,G. R_fle_compatible (cpx h G).
+#h #G #L #T1 #T2 #HT12
+elim (frees_total L T1) #f1 #Hf1
+elim (cpx_frees_conf_lexs … Hf1 L … HT12) -HT12
+/3 width=8 by lexs_refl, ext2_refl, ex4_4_intro/
+qed-.
(* Basic_2A1: uses: lpx_frees_trans *)
-lemma lfpx_frees_conf: ∀h,G. lexs_frees_confluent (cpx_ext h G) cfull.
-/2 width=7 by cpx_frees_conf_lexs/ qed-.
+lemma lfpx_fle_comp: ∀h,G. lfxs_fle_compatible (cpx h G).
+#h #G #L1 #L2 #T * #f1 #Hf1 #HL12
+elim (cpx_frees_conf_lexs h … Hf1 … HL12 T) // #f2 #Hf2
+lapply (lexs_fwd_length … HL12)
+/3 width=8 by lveq_length_eq, ex4_4_intro/ (**) (* full auto fails *)
+qed-.
(* Properties with generic extension on referred entries ********************)
(* Note: lemma 1000 *)
(* Basic_2A1: uses: cpx_llpx_sn_conf *)
lemma cpx_lfxs_conf: ∀R,h,G. s_r_confluent1 … (cpx h G) (lfxs R).
-#R #h #G #L1 #T1 #T2 #H #L2 * #f1 #Hf1 elim (cpx_frees_conf … Hf1 … H) -T1
-/3 width=5 by sle_lexs_trans, ex2_intro/
+#R #h #G #L1 #T1 #T2 #H #L2 * #f1 #Hf1 elim (cpx_frees_conf_lexs … Hf1 L1 … H) -T1
+/3 width=5 by lexs_refl, ext2_refl, sle_lexs_trans, ex2_intro/
qed-.
(* Main properties **********************************************************)
theorem lfpr_conf: ∀h,G,T. confluent … (lfpr h G T).
-/3 width=6 by cpr_conf_lfpr, lfpr_frees_conf, lfxs_conf/ qed-.
+/3 width=6 by cpr_conf_lfpr, lfpr_fle_comp, lfxs_conf/ qed-.
theorem lfpr_bind: ∀h,G,L1,L2,V1. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 →
∀I,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, T] L2.ⓑ{I}V2 →
include "basic_2/relocation/lifts_tdeq.ma".
include "basic_2/static/lfxs_lfxs.ma".
include "basic_2/static/lfdeq_fqup.ma".
+include "basic_2/static/lfdeq_lfdeq.ma".
include "basic_2/rt_transition/cpx_lfxs.ma".
include "basic_2/rt_transition/lfpx.ma".
lemma lfpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀o,I,T.
∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[h, o, V] L2.
-/3 width=5 by lfpx_frees_conf, lfxs_pair_sn_split/ qed-.
+/3 width=5 by lfpx_fle_comp, lfxs_pair_sn_split/ qed-.
lemma lfpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀o,I,V.
∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[h, o, T] L2.
-/3 width=5 by lfpx_frees_conf, lfxs_flat_dx_split/ qed-.
+/3 width=5 by lfpx_fle_comp, lfxs_flat_dx_split/ qed-.
lemma lfpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀o,p.
∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
-/3 width=5 by lfpx_frees_conf, lfxs_bind_dx_split/ qed-.
+/3 width=5 by lfpx_fle_comp, lfxs_bind_dx_split/ qed-.
lemma lfpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀o,p,I,V.
∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[h, o, T] L2.
-/3 width=5 by lfpx_frees_conf, lfxs_bind_dx_split_void/ qed-.
+/3 width=5 by lfpx_fle_comp, lfxs_bind_dx_split_void/ qed-.
lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
#h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
qed-.
lemma lfpx_lfdeq_conf: ∀h,o,G,T. confluent2 … (lfpx h G T) (lfdeq h o T).
-/3 width=6 by lfpx_frees_conf, cpx_tdeq_conf_lexs, frees_lfdeq_conf_lexs, lfxs_conf/ qed-.
+/3 width=6 by lfpx_fle_comp, lfdeq_fle_comp, cpx_tdeq_conf_lexs, lfxs_conf/ qed-.
(* Basic_2A1: uses: lleq_lpx_trans *)
lemma lfdeq_lfpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
"degree-based equivalence on referred entries (closure)"
'StarEqSn h o G1 L1 T1 G2 L2 T2 = (ffdeq h o G1 L1 T1 G2 L2 T2).
-(* Basic properties *********************************************************)
-
-lemma ffdeq_sym: ∀h,o. tri_symmetric … (ffdeq h o).
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1 /3 width=1 by ffdeq_intro, lfdeq_sym/
-qed-.
-
(* Basic inversion lemmas ***************************************************)
lemma ffdeq_inv_gen: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+(* Advanced properties ******************************************************)
+
+lemma ffdeq_sym: ∀h,o. tri_symmetric … (ffdeq h o).
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1 /3 width=1 by ffdeq_intro, lfdeq_sym/
+qed-.
+
(* Main properties **********************************************************)
theorem ffdeq_trans: ∀h,o. tri_transitive … (ffdeq h o).
(**************************************************************************)
include "basic_2/syntax/lveq_lveq.ma".
-include "basic_2/static/frees_frees.ma".
include "basic_2/static/fle_fqup.ma".
(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
/2 width=6 by ex3_3_intro/
qed-.
+lemma fle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
+ ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
+ ∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & f1 ⊆ f2.
+#L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
+elim (fle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
+elim (lveq_inj_length … H2L) // -L2 #H1 #H2 destruct
+/2 width=3 by ex2_intro/
+qed-.
+
(* Main properties **********************************************************)
(*
theorem fle_trans: bi_transitive … fle.
@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
/4 width=5 by frees_flat, sor_inv_sle_dx, sor_tls, sle_trans/
qed.
+
+(* Advanced forward lemmas ***************************************************)
+
+lemma fle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
+ ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄.
+#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 *
+#n1 #n2 #f1 #f2 #Hf1 #Hf2 #HL12 #Hf12
+elim (lveq_inv_pair_pair … HL12) -HL12 #HL12 #H1 #H2 destruct
+elim (frees_total (L1.ⓧ) T1) #g1 #Hg1
+lapply (lsubr_lsubf … Hg1 … Hf1) -Hf1 /2 width=1 by lsubr_unit/ #Hfg1
+/5 width=10 by lsubf_fwd_sle, lveq_bind, sle_trans, ex4_4_intro/ (**) (* full auto too slow *)
+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/frees_fqup.ma".
-include "basic_2/static/lsubf_lsubr.ma".
-include "basic_2/static/fle.ma".
-
-(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
-
-(* Advanced forward lemmas ***************************************************)
-
-lemma fle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄.
-#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 *
-#n1 #n2 #f1 #f2 #Hf1 #Hf2 #HL12 #Hf12
-elim (lveq_inv_pair_pair … HL12) -HL12 #HL12 #H1 #H2 destruct
-elim (frees_total (L1.ⓧ) T1) #g1 #Hg1
-lapply (lsubr_lsubf … Hg1 … Hf1) -Hf1 /2 width=1 by lsubr_unit/ #Hfg1
-/5 width=10 by lsubf_fwd_sle, lveq_bind, sle_trans, ex4_4_intro/ (**) (* full auto too slow *)
-qed-.
(* Basic properties ***********************************************************)
-lemma frees_tdeq_conf_lexs: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≛[h, o] T2 →
- ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f.
+lemma frees_tdeq_conf_lfdeq: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≛[h, o] T2 →
+ ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f.
#h #o #f #L1 #T1 #H elim H -f -L1 -T1
[ #f #L1 #s1 #Hf #X #H1 #L2 #_
elim (tdeq_inv_sort1 … H1) -H1 #s2 #d #_ #_ #H destruct
lemma frees_tdeq_conf: ∀h,o,f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f →
∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≡ f.
-/4 width=7 by frees_tdeq_conf_lexs, lexs_refl, ext2_refl/ qed-.
+/4 width=7 by frees_tdeq_conf_lfdeq, lexs_refl, ext2_refl/ qed-.
-lemma frees_lexs_conf: ∀h,o,f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f →
- ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f.
-/2 width=7 by frees_tdeq_conf_lexs, tdeq_refl/ qed-.
-
-lemma frees_lfdeq_conf_lexs: ∀h,o. lexs_frees_confluent (cdeq_ext h o) cfull.
-/3 width=7 by frees_tdeq_conf_lexs, ex2_intro/ qed-.
+lemma frees_lfdeq_conf: ∀h,o,f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f →
+ ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f.
+/2 width=7 by frees_tdeq_conf_lfdeq, tdeq_refl/ qed-.
lemma tdeq_lfdeq_conf_sn: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
#h #o #L1 #T1 #T2 #HT12 #L2 *
/3 width=5 by frees_tdeq_conf, ex2_intro/
qed-.
-(* Basic_2A1: uses: lleq_sym *)
-lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T).
-#h #o #T #L1 #L2 *
-/4 width=7 by frees_tdeq_conf_lexs, lfxs_sym, tdeq_sym, ex2_intro/
-qed-.
-
lemma lfdeq_atom: ∀h,o,I. ⋆ ≛[h, o, ⓪{I}] ⋆.
/2 width=1 by lfxs_atom/ qed.
(* *)
(**************************************************************************)
+include "basic_2/syntax/lveq_length.ma".
include "basic_2/relocation/lifts_tdeq.ma".
include "basic_2/static/lfxs_length.ma".
include "basic_2/static/lfdeq.ma".
(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-(* Forward lemmas with length for local environments ************************)
+(* Advanved properties ******************************************************)
-(* Basic_2A1: lleq_fwd_length *)
-lemma lfdeq_fwd_length: ∀h,o,L1,L2. ∀T:term. L1 ≛[h, o, T] L2 → |L1| = |L2|.
-/2 width=3 by lfxs_fwd_length/ qed-.
+lemma lfdeq_fle_comp: ∀h,o. lfxs_fle_compatible (cdeq h o).
+#h #o #L1 #L2 #T * #f1 #Hf1 #HL12
+lapply (frees_lfdeq_conf h o … Hf1 … HL12)
+lapply (lexs_fwd_length … HL12)
+/3 width=8 by lveq_length_eq, ex4_4_intro/ (**) (* full auto fails *)
+qed-.
(* Properties with length for local environments ****************************)
∀b,f. ⬇*[b, f] L1 ≡ K1 → ⬇*[b, f] L2 ≡ K2 →
∀U. ⬆*[f] T ≡ U → L1 ≛[h, o, U] L2.
/3 width=9 by lfxs_lifts_bi, tdeq_lifts_sn/ qed-.
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: lleq_fwd_length *)
+lemma lfdeq_fwd_length: ∀h,o,L1,L2. ∀T:term. L1 ≛[h, o, T] L2 → |L1| = |L2|.
+/2 width=3 by lfxs_fwd_length/ qed-.
+
include "basic_2/syntax/ext2_ext2.ma".
include "basic_2/syntax/tdeq_tdeq.ma".
include "basic_2/static/lfxs_lfxs.ma".
-include "basic_2/static/lfdeq.ma".
+include "basic_2/static/lfdeq_length.ma".
(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
(* Advanced properties ******************************************************)
+(* Basic_2A1: uses: lleq_sym *)
+lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T).
+/3 width=3 by lfdeq_fle_comp, lfxs_sym, tdeq_sym/ qed-.
+
(* Basic_2A1: uses: lleq_dec *)
lemma lfdeq_dec: ∀h,o,L1,L2. ∀T:term. Decidable (L1 ≛[h, o, T] L2).
/3 width=1 by lfxs_dec, tdeq_dec/ qed-.
(* Basic_2A1: uses: lleq_trans *)
theorem lfdeq_trans: ∀h,o,T. Transitive … (lfdeq h o T).
#h #o #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
-lapply (frees_tdeq_conf_lexs … Hf1 T … HL1) // #H0
+lapply (frees_tdeq_conf_lfdeq … Hf1 T … HL1) // #H0
lapply (frees_mono … Hf2 … H0) -Hf2 -H0
/5 width=7 by lexs_trans, lexs_eq_repl_back, tdeq_trans, ext2_trans, ex2_intro/
qed-.
(* *)
(**************************************************************************)
-include "ground_2/relocation/rtmap_id.ma".
include "basic_2/notation/relations/relationstar_4.ma".
include "basic_2/syntax/cext2.ma".
include "basic_2/relocation/lexs.ma".
-include "basic_2/static/frees.ma".
+include "basic_2/static/fle.ma".
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
interpretation "generic extension on referred entries (local environment)"
'RelationStar R T L1 L2 = (lfxs R T L1 L2).
-definition R_frees_confluent: predicate (relation3 …) ≝
- λRN.
- ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 →
- ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
+definition R_fle_compatible: predicate (relation3 …) ≝ λRN.
+ ∀L,T1,T2. RN L T1 T2 → ⦃L, T2⦄ ⊆ ⦃L, T1⦄.
-definition lexs_frees_confluent: relation (relation3 …) ≝
- λRN,RP.
- ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
- ∀L2. L1 ⪤*[RN, RP, f1] L2 →
- ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
+definition lfxs_fle_compatible: predicate (relation3 …) ≝ λRN.
+ ∀L1,L2,T. L1 ⪤*[RN, T] L2 → ⦃L2, T⦄ ⊆ ⦃L1, T⦄.
definition R_confluent2_lfxs: relation4 (relation3 lenv term term)
(relation3 lenv term term) … ≝
/3 width=5 by lexs_pair_repl, ex2_intro/
qed-.
-lemma lfxs_sym: ∀R. lexs_frees_confluent (cext2 R) cfull →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (lfxs R T).
-#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1
-/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
-qed-.
-
(* Basic_2A1: uses: llpx_sn_co *)
lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
+++ /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/syntax/lveq_length.ma".
-include "basic_2/static/fle.ma".
-include "basic_2/static/lfxs_lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-(* Properties with free variables inclusion for restricted closures *********)
-
-(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
-lemma fle_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
- ∀L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
-#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
-elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
-/4 width=5 by lfxs_inv_frees, sle_lexs_trans, ex2_intro/
-qed-.
(* *)
(**************************************************************************)
+include "basic_2/relocation/lexs_length.ma".
include "basic_2/relocation/lexs_lexs.ma".
include "basic_2/static/frees_drops.ma".
+include "basic_2/static/fle_fle.ma".
include "basic_2/static/lfxs.ma".
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-(* Advanced properties ******************************************************)
+(* Advanced inversion lemmas ************************************************)
lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 →
∀f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ⪤*[cext2 R, cfull, f] L2.
#R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/
qed-.
+lemma frees_lexs_conf: ∀R. lfxs_fle_compatible R →
+ ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
+ ∀L2. L1 ⪤*[cext2 R, cfull, f1] L2 →
+ ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
+#R #HR #L1 #T #f1 #Hf1 #L2 #H1L
+lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L
+@(fle_frees_trans_eq … H2L … Hf1) /3 width=4 by lexs_fwd_length, sym_eq/
+qed-.
+
+(* Properties with free variables inclusion for restricted closures *********)
+
+(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
+lemma fle_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
+ ∀L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
+#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
+elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
+/4 width=5 by lfxs_inv_frees, sle_lexs_trans, ex2_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lfxs_sym: ∀R. lfxs_fle_compatible R →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (lfxs R T).
+#R #H1R #H2R #T #L1 #L2
+* #f1 #Hf1 #HL12
+elim (frees_lexs_conf … Hf1 … HL12) -Hf1 //
+/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
+qed-.
+
(* Basic_2A1: uses: llpx_sn_dec *)
lemma lfxs_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
∀L1,L2,T. Decidable (L1 ⪤*[R, T] L2).
qed-.
lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lexs_frees_confluent … (cext2 R1) cfull →
+ lfxs_fle_compatible R1 →
∀L1,L2,V. L1 ⪤*[R1, V] L2 → ∀I,T.
∃∃L. L1 ⪤*[R1, ②{I}V.T] L & L ⪤*[R2, V] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
lapply (sor_inv_sle_sn … Hy) -y2 #Hfg
elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2
lapply (sle_lexs_trans … HL1 … Hfg) // #H
-elim (HR … Hf … H) -HR -Hf -H
+elim (frees_lexs_conf … Hf … H) -Hf -H
/4 width=7 by sle_lexs_trans, ex2_intro/
qed-.
lemma lfxs_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lexs_frees_confluent … (cext2 R1) cfull →
+ lfxs_fle_compatible R1 →
∀L1,L2,T. L1 ⪤*[R1, T] L2 → ∀I,V.
∃∃L. L1 ⪤*[R1, ⓕ{I}V.T] L & L ⪤*[R2, T] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2
lapply (sle_lexs_trans … HL1 … Hfg) // #H
-elim (HR … Hf … H) -HR -Hf -H
+elim (frees_lexs_conf … Hf … H) -Hf -H
/4 width=7 by sle_lexs_trans, ex2_intro/
qed-.
lemma lfxs_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lexs_frees_confluent … (cext2 R1) cfull →
+ lfxs_fle_compatible R1 →
∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤*[R1, T] L2 → ∀p.
∃∃L,V. L1 ⪤*[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤*[R2, T] L2 & R1 L1 V1 V.
#R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p
lapply (sle_lexs_trans … H … Hfg) // #H0
elim (lexs_inv_next1 … H) -H #Z #L #HL1 #H
elim (ext2_inv_pair_sn … H) -H #V #HV #H1 #H2 destruct
-elim (HR … Hf … H0) -HR -Hf -H0
+elim (frees_lexs_conf … Hf … H0) -Hf -H0
/4 width=7 by sle_lexs_trans, ex3_2_intro, ex2_intro/
qed-.
lemma lfxs_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lexs_frees_confluent … (cext2 R1) cfull →
+ lfxs_fle_compatible R1 →
∀L1,L2,T. L1.ⓧ ⪤*[R1, T] L2 → ∀p,I,V.
∃∃L. L1 ⪤*[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤*[R2, T] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V
lapply (sle_lexs_trans … H … Hfg) // #H0
elim (lexs_inv_next1 … H) -H #Z #L #HL1 #H
elim (ext2_inv_unit_sn … H) -H #H destruct
-elim (HR … Hf … H0) -HR -Hf -H0
+elim (frees_lexs_conf … Hf … H0) -Hf -H0
/4 width=7 by sle_lexs_trans, ex2_intro/ (* note: 2 ex2_intro *)
qed-.
qed.
theorem lfxs_conf: ∀R1,R2.
- lexs_frees_confluent (cext2 R1) cfull →
- lexs_frees_confluent (cext2 R2) cfull →
+ lfxs_fle_compatible R1 →
+ lfxs_fle_compatible R2 →
R_confluent2_lfxs R1 R2 R1 R2 →
∀T. confluent2 … (lfxs R1 T) (lfxs R2 T).
#R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01
elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
[ #L #HL1 #HL2
- elim (HR1 … Hf … HL01) -HL01 #f1 #Hf1 #H1
- elim (HR2 … Hf … HL02) -HL02 #f2 #Hf2 #H2
+ elim (frees_lexs_conf … Hf … HL01) // -HR1 -HL01 #f1 #Hf1 #H1
+ elim (frees_lexs_conf … Hf … HL02) // -HR2 -HL02 #f2 #Hf2 #H2
lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1
lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2
/3 width=5 by ex2_intro/
}
]
[ { "generic extension of a context-sensitive relation" * } {
- [ [ "for lenvs on referred entries" ] "lfxs ( ? ⦻*[?,?] ? )" "lfxs_length" + "lfxs_lex" + "lfxs_drops" + "lfxs_fqup" + "flxs_fle" + "lfxs_lfxs" * ]
+ [ [ "for lenvs on referred entries" ] "lfxs ( ? ⦻*[?,?] ? )" "lfxs_length" + "lfxs_lex" + "lfxs_drops" + "lfxs_fqup" + "lfxs_lfxs" * ]
}
]
[ { "context-sensitive free variables" * } {
- [ [ "inclusion for restricted closures" ] "fle ( ⦃?,?⦄ ⊆ ⦃?,?⦄ )" "fle_drops" + "fle_fqup" + "fle_lsubf" + "fle_fle" * ]
+ [ [ "inclusion for restricted closures" ] "fle ( ⦃?,?⦄ ⊆ ⦃?,?⦄ )" "fle_drops" + "fle_fqup" + "fle_fle" * ]
[ [ "restricted refinement for lenvs" ] "lsubf ( ⦃?,?⦄ ⫃𝐅* ⦃?,?⦄ )" "lsubf_lsubr" + "lsubf_frees" + "lsubf_lsubf" * ]
[ [ "for terms" ] "frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )" "frees_drops" + "frees_fqup" + "frees_frees" * ]
}
projects / helm.git / commitdiff