+++ /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/lib/star.ma".
-include "basic_2/notation/relations/relationstar_4.ma".
-include "basic_2/static/rex.ma".
-
-(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-
-definition rexs (R): term → relation lenv ≝ CTC … (rex R).
-
-interpretation "iterated extension on referred entries (local environment)"
- 'RelationStar R T L1 L2 = (rexs R T L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R, T] L →
- ∀L2. L ⪤[R, T] L2 → L1 ⪤*[R, T] L2.
-#R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
-qed-.
-
-lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R, T] L →
- ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2.
-#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
-qed-.
-
-lemma rexs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆.
-/2 width=1 by inj/ qed.
-
-lemma rexs_sort: ∀R,I,L1,L2,V1,V2,s.
- L1 ⪤*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
-/3 width=4 by rex_sort, rexs_step_dx, inj/
-qed.
-
-lemma rexs_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2,V. L1 ⪤*[R, V] L2 →
- L1.ⓑ{I}V ⪤*[R, #0] L2.ⓑ{I}V.
-#R #HR #I #L1 #L2 #V #H elim H -L2
-/3 width=5 by rex_pair, rexs_step_dx, inj/
-qed.
-
-lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 →
- L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
-/3 width=3 by rex_unit, inj/ qed.
-
-lemma rexs_lref: ∀R,I,L1,L2,V1,V2,i.
- L1 ⪤*[R, #i] L2 → L1.ⓑ{I}V1 ⪤*[R, #↑i] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
-/3 width=4 by rex_lref, rexs_step_dx, inj/
-qed.
-
-lemma rexs_gref: ∀R,I,L1,L2,V1,V2,l.
- L1 ⪤*[R, §l] L2 → L1.ⓑ{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
-/3 width=4 by rex_gref, rexs_step_dx, inj/
-qed.
-
-lemma rexs_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
-/4 width=5 by rex_co, rexs_step_dx, inj/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
-lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R, ⓪{I}] Y2 → Y2 = ⋆.
-#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, rex_inv_atom_sn/
-qed-.
-
-(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
-lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R, ⓪{I}] ⋆ → Y1 = ⋆.
-#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
-/3 width=3 by inj, rex_inv_atom_dx/
-qed-.
-
-lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-#R #Y1 #Y2 #s #H elim H -Y2
-[ #Y2 #H elim (rex_inv_sort … H) -H *
- /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/
-| #Y #Y2 #_ #H elim (rex_inv_sort … H) -H *
- [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
- | #I #I2 #L #L2 #HL2 #H #H2 * *
- [ #H1 #H0 destruct
- | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct
- /4 width=7 by ex3_4_intro, rexs_step_dx, or_intror/
- ]
- ]
-]
-qed-.
-
-lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-#R #Y1 #Y2 #l #H elim H -Y2
-[ #Y2 #H elim (rex_inv_gref … H) -H *
- /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/
-| #Y #Y2 #_ #H elim (rex_inv_gref … H) -H *
- [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
- | #I #I2 #L #L2 #HL2 #H #H2 * *
- [ #H1 #H0 destruct
- | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct
- /4 width=7 by ex3_4_intro, rexs_step_dx, or_intror/
- ]
- ]
-]
-qed-.
-
-lemma rexs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
-#R #HR #p #I #L1 #L2 #V #T #H elim H -L2
-[ #L2 #H elim (rex_inv_bind … V ? H) -H /3 width=1 by inj, conj/
-| #L #L2 #_ #H * elim (rex_inv_bind … V ? H) -H /3 width=3 by rexs_step_dx, conj/
-]
-qed-.
-
-lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1 ⪤*[R, T] L2.
-#R #I #L1 #L2 #V #T #H elim H -L2
-[ #L2 #H elim (rex_inv_flat … H) -H /3 width=1 by inj, conj/
-| #L #L2 #_ #H * elim (rex_inv_flat … H) -H /3 width=3 by rexs_step_dx, conj/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R, ⋆s] Y2 →
- ∃∃I2,L2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
-#R #I1 #Y2 #L1 #s #H elim (rexs_inv_sort … H) -H *
-[ #H destruct
-| #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R, ⋆s] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
-#R #I2 #Y1 #L2 #s #H elim (rexs_inv_sort … H) -H *
-[ #_ #H destruct
-| #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R, §l] Y2 →
- ∃∃I2,L2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓘ{I2}.
-#R #I1 #Y2 #L1 #l #H elim (rexs_inv_gref … H) -H *
-[ #H destruct
-| #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R, §l] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓘ{I1}.
-#R #I2 #Y1 #L2 #l #H elim (rexs_inv_gref … H) -H *
-[ #_ #H destruct
-| #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2.
-#R #I #L1 #L2 #V #T #H elim H -L2
-/3 width=5 by rex_fwd_pair_sn, rexs_step_dx, inj/
-qed-.
-
-lemma rexs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
-#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind … H) -H //
-qed-.
-
-lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2.
-#R #I #L1 #L2 #V #T #H elim (rexs_inv_flat … H) -H //
-qed-.
-
-(* Basic_2A1: removed theorems 2:
- TC_lpx_sn_inv_pair1 TC_lpx_sn_inv_pair2
-*)
projects / helm.git / blobdiff