+++ /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/lrsubeqc_2.ma".
-include "basic_2/syntax/lenv.ma".
-
-(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Basic_2A1: just tpr_cpr and tprs_cprs require the extended lsubr_atom *)
-(* Basic_2A1: includes: lsubr_pair *)
-inductive lsubr: relation lenv ≝
-| lsubr_atom: lsubr (⋆) (⋆)
-| lsubr_bind: ∀I,L1,L2. lsubr L1 L2 → lsubr (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubr_beta: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
-| lsubr_unit: ∀I1,I2,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I1}V) (L2.ⓤ{I2})
-.
-
-interpretation
- "restricted refinement (local environment)"
- 'LRSubEqC L1 L2 = (lsubr L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsubr_refl: ∀L. L ⫃ L.
-#L elim L -L /2 width=1 by lsubr_atom, lsubr_bind/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⫃ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2 //
-[ #I #L1 #L2 #_ #H destruct
-| #L1 #L2 #V #W #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_atom1: ∀L2. ⋆ ⫃ L2 → L2 = ⋆.
-/2 width=3 by lsubr_inv_atom1_aux/ qed-.
-
-fact lsubr_inv_bind1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
- I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
- I = BPair J1 V.
-#L1 #L2 * -L1 -L2
-[ #J #K1 #H destruct
-| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by or3_intro0, ex2_intro/
-| #L1 #L2 #V #W #HL12 #J #K1 #H destruct /3 width=6 by or3_intro1, ex3_3_intro/
-| #I1 #I2 #L1 #L2 #V #HL12 #J #K1 #H destruct /3 width=4 by or3_intro2, ex3_4_intro/
-]
-qed-.
-
-(* Basic_2A1: uses: lsubr_inv_pair1 *)
-lemma lsubr_inv_bind1: ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
- I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
- I = BPair J1 V.
-/2 width=3 by lsubr_inv_bind1_aux/ qed-.
-
-fact lsubr_inv_atom2_aux: ∀L1,L2. L1 ⫃ L2 → L2 = ⋆ → L1 = ⋆.
-#L1 #L2 * -L1 -L2 //
-[ #I #L1 #L2 #_ #H destruct
-| #L1 #L2 #V #W #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_atom2: ∀L1. L1 ⫃ ⋆ → L1 = ⋆.
-/2 width=3 by lsubr_inv_atom2_aux/ qed-.
-
-fact lsubr_inv_bind2_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
-#L1 #L2 * -L1 -L2
-[ #J #K2 #H destruct
-| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or3_intro0/
-| #L1 #L2 #V1 #V2 #HL12 #J #K2 #H destruct /3 width=6 by ex3_3_intro, or3_intro1/
-| #I1 #I2 #L1 #L2 #V #HL12 #J #K2 #H destruct /3 width=5 by ex3_4_intro, or3_intro2/
-]
-qed-.
-
-lemma lsubr_inv_bind2: ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
-/2 width=3 by lsubr_inv_bind2_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
- | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
-#K1 #L2 #W #H elim (lsubr_inv_bind1 … H) -H *
-/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
-#K2 #V2 #W2 #_ #_ #H destruct
-qed-.
-
-lemma lsubr_inv_unit1: ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
- ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
-#I #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
-[ #K2 #HK12 #H destruct /2 width=3 by ex2_intro/
-| #K2 #V #W #_ #_ #H destruct
-| #I1 #I2 #K2 #V #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_pair2: ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
- | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
-#I #L1 #K2 #W #H elim (lsubr_inv_bind2 … H) -H *
-[ /3 width=3 by ex2_intro, or_introl/
-| #K2 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
-| #I1 #I1 #K2 #V #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_abbr2: ∀L1,K2,V. L1 ⫃ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
-#L1 #K2 #V #H elim (lsubr_inv_pair2 … H) -H *
-[ /2 width=3 by ex2_intro/
-| #K1 #X #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_abst2: ∀L1,K2,W. L1 ⫃ K2.ⓛW →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓛW
- | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V.
-#L1 #K2 #W #H elim (lsubr_inv_pair2 … H) -H *
-/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
-qed-.
-
-lemma lsubr_inv_unit2: ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
- | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
-#I #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
-[ /3 width=3 by ex2_intro, or_introl/
-| #K1 #W #V #_ #_ #H destruct
-| #I1 #I2 #K1 #V #HK12 #H1 #H2 destruct /3 width=5 by ex2_3_intro, or_intror/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsubr_fwd_bind1: ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
- ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
-#I1 #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
-[ #K2 #HK12 #H destruct /3 width=4 by ex2_2_intro/
-| #K2 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-| #I1 #I2 #K2 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lsubr_fwd_bind2: ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
-#I2 #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
-[ #K1 #HK12 #H destruct /3 width=4 by ex2_2_intro/
-| #K1 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-| #I1 #I2 #K1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-]
-qed-.
projects / helm.git / blobdiff