--- /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_coafter.ma".
+include "static_2/relocation/drops_drops.ma".
+include "static_2/static/frees_fqup.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ →
+ ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f.
+#b #L elim L -L /2 width=1 by frees_atom/
+#L #I #IH *
+[ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
+| /4 width=3 by frees_lref, drops_inv_drop1/
+]
+qed.
+
+lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≘ f →
+ ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+#f #K #V #Hf #i elim i -i
+[ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/
+| #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
+]
+qed.
+
+lemma frees_unit_drops: ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
+ L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+#f #Hf #I #K #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/
+| #i #IH #Y #H elim (drops_inv_succ … H) -H
+ #J #L #HLK #H destruct /3 width=1 by frees_lref/
+]
+qed.
+(*
+lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃⋆s⦄ ≘ ⫯*[i] f.
+#f #K #s #Hf #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
+]
+qed.
+*)
+lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≘ ⫯*[i] f.
+#f #K #j #Hf #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+| #i #IH #L #H elim (drops_inv_succ … H) -H
+ #I #Y #HYK #H destruct /3 width=1 by frees_lref/
+]
+qed.
+(*
+lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃§l⦄ ≘ ⫯*[i] f.
+#f #K #l #Hf #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
+]
+qed.
+*)
+(* Advanced inversion lemmas ************************************************)
+
+lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f →
+ ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
+ | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g &
+ ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
+ | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
+#L elim L -L
+[ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H
+[ elim (frees_inv_atom … H) -H #f #Hf #H destruct
+ /3 width=3 by or3_intro0, ex3_intro/
+| elim (frees_inv_unit … H) -H #f #Hf #H destruct
+ /4 width=3 by drops_refl, or3_intro2, ex3_3_intro/
+| elim (frees_inv_pair … H) -H #f #Hf #H destruct
+ /4 width=7 by drops_refl, or3_intro1, ex3_4_intro/
+| elim (frees_inv_lref … H) -H #f #Hf #H destruct
+ elim (IH … Hf) -IH -Hf *
+ [ /4 width=3 by drops_drop, or3_intro0, ex3_intro/
+ | /4 width=7 by drops_drop, or3_intro1, ex3_4_intro/
+ | /4 width=3 by drops_drop, or3_intro2, ex3_3_intro/
+ ]
+]
+qed-.
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
+ ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
+ ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
+#b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
+[ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+ >(lifts_inv_sort1 … H2) -U /2 width=1 by frees_sort/
+| #f1 #i #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
+ elim (lifts_inv_lref1 … H2) -H2 #j #Hij #H destruct
+ elim (coafter_fwd_xnx_pushs … Hij H3) -H3 #g2 #Hg2 #H2 destruct
+ lapply (coafter_isid_inv_dx … Hg2 … Hf1) -f1 #Hf2
+ elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf
+ elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i
+ lapply (at_inv_uni … Hj) -Hj #H destruct
+ /3 width=8 by frees_atom_drops, drops_trans/
+| #f1 #I #K #V #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+ lapply (isfin_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+ lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
+ elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #H
+ elim (liftsb_inv_pair_sn … H) -H #W #HVW #H destruct
+ elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
+ lapply (IH … HYK … HVW … H3) -IH -H3 -HYK -HVW //
+ /2 width=5 by frees_pair_drops/
+| #f1 #I #K #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
+ lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
+ elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hg2
+ elim (drops_split_trans_bind2 … H1 … Hf) -H1 -Hf #Z #Y #HLY #_ #H
+ lapply (liftsb_inv_unit_sn … H) -H #H destruct
+ /2 width=3 by frees_unit_drops/
+| #f1 #I #K #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+ lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+ lapply (lifts_inv_lref1 … H2) -H2 * #x #Hf #H destruct
+ elim (at_inv_nxx … Hf) -Hf [ |*: // ] #j #Hf #H destruct
+ elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #_
+ elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H3 #H2 destruct
+ lapply (drops_isuni_fwd_drop2 … HLY) -HLY // #HLY
+ lapply (IH … HYK … H3) -IH -H3 -HYK [4: |*: /2 width=2 by lifts_lref/ ]
+ >plus_S1 /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *)
+| #f1 #K #l #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+ >(lifts_inv_gref1 … H2) -U /2 width=1 by frees_gref/
+| #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
+ elim (sor_inv_isfin3 … H1f1) // #Hf1V #H
+ lapply (isfin_inv_tl … H) -H
+ elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H
+ elim (coafter_inv_tl1 … H) -H
+ /5 width=5 by frees_bind, drops_skip, ext2_pair/
+| #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
+ elim (sor_inv_isfin3 … H1f1) //
+ elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ elim (coafter_sor … H3 … H1f1)
+ /3 width=5 by coafter_isfin2_fwd, frees_flat/
+]
+qed-.
+
+lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
+ ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
+#b #L #K #HLK #T #U #HTU #f #Hf
+@(frees_lifts b … Hf … HTU) // (**) (* auto fails *)
+qed.
+
+(* Forward lemmas with generic slicing for local environments ***************)
+
+lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
+/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1.
+#b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
+/3 width=9 by frees_fwd_coafter, ex2_intro/
+qed-.
+
+lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f →
+ ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
+ K ⊢ 𝐅*⦃T⦄ ≘ ⫱f.
+#b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
+#f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
+/3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/
+qed-.
+
+lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1.
+#b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
+/3 width=7 by frees_eq_repl_back, coafter_inj/
+qed-.
+
+(* Note: this is used by rex_conf and might be modified *)
+lemma frees_inv_drops_next: ∀f1,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 →
+ ∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 →
+ ∀g1. ↑g1 = ⫱*[n] f1 →
+ ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≘ g2 & g2 ⊆ g1.
+#f1 #L1 #T1 #H elim H -f1 -L1 -T1
+[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -s
+ lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
+ elim (isid_inv_next … Hf1) -Hf1 //
+| #f1 #i #_ #I2 #L2 #V2 #n #H
+ elim (drops_inv_atom1 … H) -H #H destruct
+| #f1 #I1 #L1 #V1 #Hf1 #IH #I2 #L2 #V2 *
+ [ -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
+ #H destruct #g1 #Hgf1 >(injective_next … Hgf1) -g1
+ /2 width=3 by ex2_intro/
+ | -Hf1 #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
+ #HL12 #g1 <tls_xn <tl_next_rew #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
+ /2 width=3 by ex2_intro/
+ ]
+| #f1 #I1 #L1 #Hf1 #I2 #L2 #V2 *
+ [ #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
+ | #n #_ #g1 #Hgf1 elim (isid_inv_next … Hgf1) -Hgf1 <tls_xn /2 width=1 by isid_tls/
+ ]
+| #f1 #I1 #L1 #i #_ #IH #I2 #L2 #V2 *
+ [ -IH #_ #g1 #Hgf1 elim (discr_next_push … Hgf1)
+ | #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
+ #HL12 #g1 <tls_xn #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
+ /2 width=3 by ex2_intro/
+ ]
+| #f1 #L1 #l #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -l
+ lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
+ elim (isid_inv_next … Hf1) -Hf1 //
+| #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
+ lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
+ elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
+ #gV1 #gT1 #Hg1
+ [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
+ /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
+ | -IHV1 #_ >tls_xn #H2 elim (IHT1 … H2) -IHT1 -H2
+ /3 width=6 by drops_drop, sor_inv_sle_dx_trans, ex2_intro/
+ ]
+| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
+ lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
+ elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
+ #gV1 #gT1 #Hg1
+ [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
+ /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
+ | -IHV1 #_ #H2 elim (IHT1 … HL12 … H2) -IHT1 -HL12 -H2
+ /3 width=6 by sor_inv_sle_dx_trans, ex2_intro/
+ ]
+]
+qed-.
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