+++ /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/lsubeqx_5.ma".
-include "basic_2/rt_computation/rdsx.ma".
-
-(* CLEAR OF STRONGLY NORMALIZING ENTRIES FOR UNBOUND RT-TRANSITION **********)
-
-(* Note: this should be an instance of a more general sex *)
-(* Basic_2A1: uses: lcosx *)
-inductive lsubsx (h) (G): rtmap → relation lenv ≝
-| lsubsx_atom: ∀f. lsubsx h G f (⋆) (⋆)
-| lsubsx_push: ∀f,I,K1,K2. lsubsx h G f K1 K2 →
- lsubsx h G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})
-| lsubsx_unit: ∀f,I,K1,K2. lsubsx h G f K1 K2 →
- lsubsx h G (↑f) (K1.ⓤ{I}) (K2.ⓧ)
-| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ →
- lsubsx h G f K1 K2 → lsubsx h G (↑f) (K1.ⓑ{I}V) (K2.ⓧ)
-.
-
-interpretation
- "local environment refinement (clear)"
- 'LSubEqX h f G L1 L2 = (lsubsx h G f L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
- L1 = ⋆ → L2 = ⋆.
-#h #g #G #L1 #L2 * -g -L1 -L2 //
-[ #f #I #K1 #K2 #_ #H destruct
-| #f #I #K1 #K2 #_ #H destruct
-| #f #I #K1 #K2 #V #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h,g] L2 → L2 = ⋆.
-/2 width=7 by lsubsx_inv_atom_sn_aux/ qed-.
-
-fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
- ∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}.
-#h #g #G #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #_ #H destruct
-| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
- <(injective_push … H1) -g /2 width=3 by ex2_intro/
-| #f #I #K1 #K2 #_ #g #J #L1 #H
- elim (discr_next_push … H)
-| #f #I #K1 #K2 #V #_ #_ #g #J #L1 #H
- elim (discr_next_push … H)
-]
-qed-.
-
-lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h,⫯f] L2 →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}.
-/2 width=5 by lsubsx_inv_push_sn_aux/ qed-.
-
-fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
- ∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
-#h #g #G #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #_ #H destruct
-| #f #I #K1 #K2 #_ #g #J #L1 #H
- elim (discr_push_next … H)
-| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
- <(injective_next … H1) -g /2 width=3 by ex2_intro/
-| #f #I #K1 #K2 #V #_ #_ #g #J #L1 #_ #H destruct
-]
-qed-.
-
-lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h,↑f] L2 →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
-/2 width=6 by lsubsx_inv_unit_sn_aux/ qed-.
-
-fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 →
- ∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V →
- ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
-#h #g #G #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #W #_ #H destruct
-| #f #I #K1 #K2 #_ #g #J #L1 #W #H
- elim (discr_push_next … H)
-| #f #I #K1 #K2 #_ #g #J #L1 #W #_ #H destruct
-| #f #I #K1 #K2 #V #HV #HK12 #g #J #L1 #W #H1 #H2 destruct
- <(injective_next … H1) -g /2 width=4 by ex3_intro/
-]
-qed-.
-
-(* Basic_2A1: uses: lcosx_inv_pair *)
-lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,↑f] L2 →
- ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ.
-/2 width=6 by lsubsx_inv_pair_sn_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,g] L2 →
- ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
- | ∃∃f,K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ↑f & L2 = K2.ⓧ.
-#h #g #I #G #K1 #L2 #V #H
-elim (pn_split g) * #f #Hf destruct
-[ elim (lsubsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/
-| elim (lsubsx_inv_pair_sn … H) -H /3 width=6 by ex4_2_intro, or_intror/
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h,g] L2 →
- ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h,⫱g] K2 & L2 = K2.ⓘ{I2}.
-#h #g #I1 #G #K1 #L2
-elim (pn_split g) * #f #Hf destruct
-[ #H elim (lsubsx_inv_push_sn … H) -H
-| cases I1 -I1 #I1
- [ #H elim (lsubsx_inv_unit_sn … H) -H
- | #V #H elim (lsubsx_inv_pair_sn … H) -H
- ]
-]
-/2 width=4 by ex2_2_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2).
-#h #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-[ #f #I #L1 #L2 #_ #IH #x #H
- elim (eq_inv_px … H) -H /3 width=3 by lsubsx_push/
-| #f #I #L1 #L2 #_ #IH #x #H
- elim (eq_inv_nx … H) -H /3 width=3 by lsubsx_unit/
-| #f #I #L1 #L2 #V #HV #_ #IH #x #H
- elim (eq_inv_nx … H) -H /3 width=3 by lsubsx_pair/
-]
-qed-.
-
-lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2).
-#h #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: lcosx_O *)
-lemma lsubsx_refl: ∀h,f,G. 𝐈⦃f⦄ → reflexive … (lsubsx h G f).
-#h #f #G #Hf #L elim L -L
-/3 width=3 by lsubsx_eq_repl_back, lsubsx_push, eq_push_inv_isid/
-qed.
-
-(* Basic_2A1: removed theorems 2:
- lcosx_drop_trans_lt lcosx_inv_succ
-*)
projects / helm.git / blobdiff