+++ /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 "ground_2/relocation/nstream_sle.ma".
-include "basic_2/notation/relations/relationstar_5.ma".
-include "basic_2/grammar/lenv.ma".
-
-(* GENERAL ENTRYWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *)
-inductive lexs (RS,RO:relation3 lenv term term): rtmap → relation lenv ≝
-| lexs_atom: ∀f. lexs RS RO f (⋆) (⋆)
-| lexs_next: ∀I,L1,L2,V1,V2,f.
- lexs RS RO f L1 L2 → RS L1 V1 V2 →
- lexs RS RO (⫯f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
-| lexs_push: ∀I,L1,L2,V1,V2,f.
- lexs RS RO f L1 L2 → RO L1 V1 V2 →
- lexs RS RO (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
-.
-
-interpretation "general entrywise extension (local environment)"
- 'RelationStar RS RO f L1 L2 = (lexs RS RO f L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lexs_inv_atom1_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → X = ⋆ → Y = ⋆.
-#RS #RO #X #Y #f * -X -Y -f //
-#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
-qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma lexs_inv_atom1: ∀RS,RO,Y,f. ⋆ ⦻*[RS, RO, f] Y → Y = ⋆.
-/2 width=6 by lexs_inv_atom1_aux/ qed-.
-
-fact lexs_inv_next1_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ⫯g →
- ∃∃K2,W2. K1 ⦻*[RS, RO, g] K2 & RS K1 W1 W2 & Y = K2.ⓑ{J}W2.
-#RS #RO #X #Y #f * -X -Y -f
-[ #f #J #K1 #W1 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #HL #HS #J #K1 #W1 #g #H1 #H2 <(injective_next … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K1 #W1 #g #_ #H elim (discr_push_next … H)
-]
-qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: ∀RS,RO,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RS, RO, ⫯g] Y →
- ∃∃K2,W2. K1 ⦻*[RS, RO, g] K2 & RS K1 W1 W2 & Y = K2.ⓑ{J}W2.
-/2 width=7 by lexs_inv_next1_aux/ qed-.
-
-
-fact lexs_inv_push1_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ↑g →
- ∃∃K2,W2. K1 ⦻*[RS, RO, g] K2 & RO K1 W1 W2 & Y = K2.ⓑ{J}W2.
-#RS #RO #X #Y #f * -X -Y -f
-[ #f #J #K1 #W1 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K1 #W1 #g #_ #H elim (discr_next_push … H)
-| #I #L1 #L2 #V1 #V2 #f #HL #HO #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lexs_inv_push1: ∀RS,RO,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RS, RO, ↑g] Y →
- ∃∃K2,W2. K1 ⦻*[RS, RO, g] K2 & RO K1 W1 W2 & Y = K2.ⓑ{J}W2.
-/2 width=7 by lexs_inv_push1_aux/ qed-.
-
-fact lexs_inv_atom2_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → Y = ⋆ → X = ⋆.
-#RS #RO #X #Y #f * -X -Y -f //
-#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
-qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma lexs_inv_atom2: ∀RS,RO,X,f. X ⦻*[RS, RO, f] ⋆ → X = ⋆.
-/2 width=6 by lexs_inv_atom2_aux/ qed-.
-
-fact lexs_inv_next2_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ⫯g →
- ∃∃K1,W1. K1 ⦻*[RS, RO, g] K2 & RS K1 W1 W2 & X = K1.ⓑ{J}W1.
-#RS #RO #X #Y #f * -X -Y -f
-[ #f #J #K2 #W2 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #HL #HS #J #K2 #W2 #g #H1 #H2 <(injective_next … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K2 #W2 #g #_ #H elim (discr_push_next … H)
-]
-qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: ∀RS,RO,J,X,K2,W2,g. X ⦻*[RS, RO, ⫯g] K2.ⓑ{J}W2 →
- ∃∃K1,W1. K1 ⦻*[RS, RO, g] K2 & RS K1 W1 W2 & X = K1.ⓑ{J}W1.
-/2 width=7 by lexs_inv_next2_aux/ qed-.
-
-fact lexs_inv_push2_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ↑g →
- ∃∃K1,W1. K1 ⦻*[RS, RO, g] K2 & RO K1 W1 W2 & X = K1.ⓑ{J}W1.
-#RS #RO #X #Y #f * -X -Y -f
-[ #f #J #K2 #W2 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K2 #W2 #g #_ #H elim (discr_next_push … H)
-| #I #L1 #L2 #V1 #V2 #f #HL #HO #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lexs_inv_push2: ∀RS,RO,J,X,K2,W2,g. X ⦻*[RS, RO, ↑g] K2.ⓑ{J}W2 →
- ∃∃K1,W1. K1 ⦻*[RS, RO, g] K2 & RO K1 W1 W2 & X = K1.ⓑ{J}W1.
-/2 width=7 by lexs_inv_push2_aux/ qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_pair *)
-lemma lexs_inv_next: ∀RS,RO,I1,I2,L1,L2,V1,V2,f.
- L1.ⓑ{I1}V1 ⦻*[RS, RO, ⫯f] (L2.ⓑ{I2}V2) →
- ∧∧ L1 ⦻*[RS, RO, f] L2 & RS L1 V1 V2 & I1 = I2.
-#RS #RO #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_next1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma lexs_inv_push: ∀RS,RO,I1,I2,L1,L2,V1,V2,f.
- L1.ⓑ{I1}V1 ⦻*[RS, RO, ↑f] (L2.ⓑ{I2}V2) →
- ∧∧ L1 ⦻*[RS, RO, f] L2 & RO L1 V1 V2 & I1 = I2.
-#RS #RO #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_push1 … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lexs_eq_repl_back: ∀RS,RO,L1,L2. eq_stream_repl_back … (λf. L1 ⦻*[RS, RO, f] L2).
-#RS #RO #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-[ #I #L1 #L2 #V1 #v2 #f1 #_ #HS #IH #f2 #H elim (next_inv_sn … H) -H /3 width=1 by lexs_next/
-| #I #L1 #L2 #V1 #v2 #f1 #_ #HO #IH #f2 #H elim (push_inv_sn … H) -H /3 width=1 by lexs_push/
-]
-qed-.
-
-lemma lexs_eq_repl_fwd: ∀RS,RO,L1,L2. eq_stream_repl_fwd … (λf. L1 ⦻*[RS, RO, f] L2).
-#RS #RO #L1 #L2 @eq_stream_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
-qed-.
-
-(* Basic_2A1: includes: lpx_sn_refl *)
-lemma lexs_refl: ∀RS,RO,f.
- (∀L. reflexive … (RS L)) →
- (∀L. reflexive … (RO L)) →
- reflexive … (lexs RS RO f).
-#RS #RO #f #HS #HO #L generalize in match f; -f elim L -L //
-#L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/
-qed.
-
-lemma sle_lexs_trans: ∀RS,RO. (∀L,T1,T2. RO L T1 T2 → RS L T1 T2) →
- ∀L1,L2,f2. L1 ⦻*[RS, RO, f2] L2 →
- ∀f1. f1 ⊆ f2 → L1 ⦻*[RS, RO, f1] L2.
-#RS #RO #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 //
-#I #L1 #L2 #V1 #V2 #f2 #_ #HV12 #IH
-[2: * * [2: #n1 ] ] #f1 #H
-[ <next_rew /4 width=5 by lexs_next, sle_inv_SO_aux/
-| /4 width=5 by lexs_push, sle_inv_OO_aux/
-| elim (sle_inv_xS_aux … H) -H [3: // |2: skip ]
- #g1 #H #H1 destruct /3 width=5 by lexs_next/
-]
-qed-.
-
-lemma sle_lexs_conf: ∀RS,RO. (∀L,T1,T2. RS L T1 T2 → RO L T1 T2) →
- ∀L1,L2,f1. L1 ⦻*[RS, RO, f1] L2 →
- ∀f2. f1 ⊆ f2 → L1 ⦻*[RS, RO, f2] L2.
-#RS #RO #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 //
-#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #IH
-[ * * [2: #n2 ] ] #f2 #H
-[ <next_rew /4 width=5 by lexs_next, sle_inv_SS_aux/
-| /4 width=5 by lexs_push, sle_inv_SO_aux/
-| elim (sle_inv_Ox_aux … H) -H [3: // |2: skip ]
- #g2 #H #H2 destruct /3 width=5 by lexs_push/
-]
-qed-.
-
-lemma lexs_co: ∀RS1,RS2,RO1,RO2.
- (∀L1,T1,T2. RS1 L1 T1 T2 → RS2 L1 T1 T2) →
- (∀L1,T1,T2. RO1 L1 T1 T2 → RO2 L1 T1 T2) →
- ∀L1,L2,f. L1 ⦻*[RS1, RO1, f] L2 → L1 ⦻*[RS2, RO2, f] L2.
-#RS1 #RS2 #RO1 #RO2 #HS #HO #L1 #L2 #f #H elim H -L1 -L2 -f
-/3 width=1 by lexs_atom, lexs_next, lexs_push/
-qed-.
-
-(* Basic_2A1: removed theorems 17:
- llpx_sn_inv_bind llpx_sn_inv_flat
- llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
- llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
- llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
- llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
-*)
projects / helm.git / blobdiff