syntactic components detached from basic_2 become static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / static / frees.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/frees.ma b/matita/matita/contribs/lambdadelta/static_2/static/frees.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/rtmap_sor.ma".
+include "static_2/notation/relations/freestar_3.ma".
+include "static_2/syntax/lenv.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+inductive frees: relation3 lenv term rtmap ≝
+| frees_sort: ∀f,L,s. 𝐈⦃f⦄ → frees L (⋆s) f
+| frees_atom: ∀f,i. 𝐈⦃f⦄ → frees (⋆) (#i) (⫯*[i]↑f)
+| frees_pair: ∀f,I,L,V. frees L V f →
+              frees (L.ⓑ{I}V) (#0) (↑f)
+| frees_unit: ∀f,I,L. 𝐈⦃f⦄ → frees (L.ⓤ{I}) (#0) (↑f)
+| frees_lref: ∀f,I,L,i. frees L (#i) f →
+              frees (L.ⓘ{I}) (#↑i) (⫯f)
+| frees_gref: ∀f,L,l. 𝐈⦃f⦄ → frees L (§l) f
+| frees_bind: ∀f1,f2,f,p,I,L,V,T. frees L V f1 → frees (L.ⓑ{I}V) T f2 →
+              f1 ⋓ ⫱f2 ≘ f → frees L (ⓑ{p,I}V.T) f
+| frees_flat: ∀f1,f2,f,I,L,V,T. frees L V f1 → frees L T f2 →
+              f1 ⋓ f2 ≘ f → frees L (ⓕ{I}V.T) f
+.
+
+interpretation
+   "context-sensitive free variables (term)"
+   'FreeStar L T f = (frees L T f).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
+#L #X #f #H elim H -f -L -X //
+[ #f #i #_ #x #H destruct
+| #f #_ #L #V #_ #_ #x #H destruct
+| #f #_ #L #_ #x #H destruct
+| #f #_ #L #i #_ #_ #x #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+]
+qed-.
+
+lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
+/2 width=5 by frees_inv_sort_aux/ qed-.
+
+fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
+                         ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+#f #L #X #H elim H -f -L -X
+[ #f #L #s #_ #j #_ #H destruct
+| #f #i #Hf #j #_ #H destruct /2 width=3 by ex2_intro/
+| #f #I #L #V #_ #_ #j #H destruct
+| #f #I #L #_ #j #H destruct
+| #f #I #L #i #_ #_ #j #H destruct
+| #f #L #l #_ #j #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #j #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #j #_ #H destruct
+]
+qed-.
+
+lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+/2 width=5 by frees_inv_atom_aux/ qed-.
+
+fact frees_inv_pair_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,V. L = K.ⓑ{I}V → X = #0 →
+                         ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #Z #Y #X #_ #H destruct
+| #f #i #_ #Z #Y #X #H destruct
+| #f #I #L #V #Hf #Z #Y #X #H #_ destruct /2 width=3 by ex2_intro/
+| #f #I #L #_ #Z #Y #X #H destruct
+| #f #I #L #i #_ #Z #Y #X #_ #H destruct
+| #f #L #l #_ #Z #Y #X #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #X #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #X #_ #H destruct
+]
+qed-.
+
+lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+/2 width=6 by frees_inv_pair_aux/ qed-.
+
+fact frees_inv_unit_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
+                         ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #Z #Y #_ #H destruct
+| #f #i #_ #Z #Y #H destruct
+| #f #I #L #V #_ #Z #Y #H destruct
+| #f #I #L #Hf #Z #Y #H destruct /2 width=3 by ex2_intro/
+| #f #I #L #i #_ #Z #Y #_ #H destruct
+| #f #L #l #_ #Z #Y #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #_ #H destruct
+]
+qed-.
+
+lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+/2 width=7 by frees_inv_unit_aux/ qed-.
+
+fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
+                         ∃∃g. K ⊢ 𝐅*⦃#j⦄ ≘ g & f = ⫯g.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #Z #Y #j #_ #H destruct
+| #f #i #_ #Z #Y #j #H destruct
+| #f #I #L #V #_ #Z #Y #j #_ #H destruct
+| #f #I #L #_ #Z #Y #j #_ #H destruct
+| #f #I #L #i #Hf #Z #Y #j #H1 #H2 destruct /2 width=3 by ex2_intro/
+| #f #L #l #_ #Z #Y #j #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #j #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #j #_ #H destruct
+]
+qed-.
+
+lemma frees_inv_lref: ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅*⦃#(↑i)⦄ ≘ f →
+                      ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≘ g & f = ⫯g.
+/2 width=6 by frees_inv_lref_aux/ qed-.
+
+fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃f⦄.
+#f #L #X #H elim H -f -L -X //
+[ #f #i #_ #x #H destruct
+| #f #_ #L #V #_ #_ #x #H destruct
+| #f #_ #L #_ #x #H destruct
+| #f #_ #L #i #_ #_ #x #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+]
+qed-.
+
+lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≘ f → 𝐈⦃f⦄.
+/2 width=5 by frees_inv_gref_aux/ qed-.
+
+fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
+                         ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #q #J #W #U #H destruct
+| #f #i #_ #q #J #W #U #H destruct
+| #f #I #L #V #_ #q #J #W #U #H destruct
+| #f #I #L #_ #q #J #W #U #H destruct
+| #f #I #L #i #_ #q #J #W #U #H destruct
+| #f #L #l #_ #q #J #W #U #H destruct
+| #f1 #f2 #f #p #I #L #V #T #HV #HT #Hf #q #J #W #U #H destruct /2 width=5 by ex3_2_intro/
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #q #J #W #U #H destruct
+]
+qed-.
+
+lemma frees_inv_bind: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
+                      ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+/2 width=4 by frees_inv_bind_aux/ qed-.
+
+fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,V,T. X = ⓕ{I}V.T →
+                         ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #J #W #U #H destruct
+| #f #i #_ #J #W #U #H destruct
+| #f #I #L #V #_ #J #W #U #H destruct
+| #f #I #L #_ #J #W #U #H destruct
+| #f #I #L #i #_ #J #W #U #H destruct
+| #f #L #l #_ #J #W #U #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #J #W #U #H destruct
+| #f1 #f2 #f #I #L #V #T #HV #HT #Hf #J #W #U #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma frees_inv_flat: ∀f,I,L,V,T. L ⊢ 𝐅*⦃ⓕ{I}V.T⦄ ≘ f →
+                      ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+/2 width=4 by frees_inv_flat_aux/ qed-.
+
+(* Basic properties ********************************************************)
+
+lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+#L #T #f1 #H elim H -f1 -L -T
+[ /3 width=3 by frees_sort, isid_eq_repl_back/
+| #f1 #i #Hf1 #g2 #H
+  elim (eq_inv_pushs_sn … H) -H #g #Hg #H destruct
+  elim (eq_inv_nx … Hg) -Hg
+  /3 width=3 by frees_atom, isid_eq_repl_back/
+| #f1 #I #L #V #_ #IH #g2 #H
+  elim (eq_inv_nx … H) -H
+  /3 width=3 by frees_pair/
+| #f1 #I #L #Hf1 #g2 #H
+  elim (eq_inv_nx … H) -H
+  /3 width=3 by frees_unit, isid_eq_repl_back/
+| #f1 #I #L #i #_ #IH #g2 #H
+  elim (eq_inv_px … H) -H /3 width=3 by frees_lref/
+| /3 width=3 by frees_gref, isid_eq_repl_back/
+| /3 width=7 by frees_bind, sor_eq_repl_back3/
+| /3 width=7 by frees_flat, sor_eq_repl_back3/
+]
+qed-.
+
+lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+#L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
+qed-.
+
+lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅*⦃#↑i⦄ ≘ ⫯f.
+#f #i #H
+elim (frees_inv_atom … H) -H #g #Hg #H destruct
+/2 width=1 by frees_atom/
+qed.
+
+(* Forward lemmas with test for finite colength *****************************)
+
+lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f → 𝐅⦃f⦄.
+#f #L #T #H elim H -f -L -T
+/4 width=5 by sor_isfin, isfin_isid, isfin_tl, isfin_pushs, isfin_push, isfin_next/
+qed-.
+
+(* Basic_2A1: removed theorems 30:
+              frees_eq frees_be frees_inv
+              frees_inv_sort frees_inv_gref frees_inv_lref frees_inv_lref_free
+              frees_inv_lref_skip frees_inv_lref_ge frees_inv_lref_lt
+              frees_inv_bind frees_inv_flat frees_inv_bind_O
+              frees_lref_eq frees_lref_be frees_weak
+              frees_bind_sn frees_bind_dx frees_flat_sn frees_flat_dx
+              frees_lift_ge frees_inv_lift_be frees_inv_lift_ge 
+              lreq_frees_trans frees_lreq_conf
+              llor_atom llor_skip llor_total
+              llor_tail_frees llor_tail_cofrees
+*)