(* Advanced inversion lemmas ************************************************)
+lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≡ Y →
+ ∨∨ ∃∃s. I = Sort s & Y = ⋆s
+ | ∃∃i,j. @⦃i, f⦄ ≡ j & I = LRef i & Y = #j
+ | ∃∃l. I = GRef l & Y = §l.
+#f * #n #Y #H
+[ lapply (lifts_inv_sort1 … H)
+| elim (lifts_inv_lref1 … H)
+| lapply (lifts_inv_gref1 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
+lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≡ ⓪{I} →
+ ∨∨ ∃∃s. X = ⋆s & I = Sort s
+ | ∃∃i,j. @⦃i, f⦄ ≡ j & X = #i & I = LRef j
+ | ∃∃l. X = §l & I = GRef l.
+#f * #n #X #H
+[ lapply (lifts_inv_sort2 … H)
+| elim (lifts_inv_lref2 … H)
+| lapply (lifts_inv_gref2 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
(* Basic_2A1: includes: lift_inv_pair_xy_x *)
lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≡ V → ⊥.
#f #J #V elim V -V
(* Basic forward lemmas *****************************************************)
-lemma lifts_fwd_atom2: ∀f,I,X. ⬆*[f] X ≡ ⓪{I} → ∃J. X = ⓪{J}.
-#f * #n #X #H
-[ lapply (lifts_inv_sort2 … H)
-| elim (lifts_inv_lref2 … H)
-| lapply (lifts_inv_gref2 … H)
-] -H /2 width=2 by ex_intro/
-qed-.
-
(* Basic_2A1: includes: lift_inv_O2 *)
lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≡ T2 → 𝐈⦃f⦄ → T1 = T2.
#f #T1 #T2 #H elim H -f -T1 -T2
include "ground_2/relocation/rtmap_pushs.ma".
include "ground_2/relocation/rtmap_coafter.ma".
-include "basic_2/relocation/lifts_lifts.ma".
-include "basic_2/relocation/drops.ma".
+include "basic_2/relocation/drops_drops.ma".
include "basic_2/static/frees.ma".
(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
(* Advanced properties ******************************************************)
-lemma frees_lref_atom: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ → L ⊢ 𝐅*⦃#i⦄ ≡ 𝐈𝐝.
-#L elim L -L /2 width=1 by frees_atom/
+lemma frees_lref_atom: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≡ ⋆ →
+ ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≡ f.
+#b #L elim L -L /2 width=1 by frees_atom/
#L #I #V #IH *
[ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
-| /4 width=3 by frees_lref, drops_inv_drop1/
+| /5 width=3 by frees_eq_repl_back, frees_lref, drops_inv_drop1, eq_push_inv_isid/
]
qed.
(* Properties with generic slicing for local environments *******************)
+axiom coafter_inv_xpx: ∀g2,f1,g. g2 ~⊚ ↑f1 ≡ g → ∀n. @⦃0, g2⦄ ≡ n →
+ ∃∃f2,f. f2 ~⊚ f1 ≡ f & ⫱*[n]g2 = ↑f2 & ⫱*[n]g = ↑f.
+(*
+#g2 #g1 #g #Hg #n #Hg2
+lapply (coafter_tls … Hg2 … Hg) -Hg #Hg
+lapply (at_pxx_tls … Hg2) -Hg2 #H
+elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
+elim (coafter_inv_pxx … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0 destruct
+<tls_rew_S <tls_rew_S <H2 <H0 -g2 -g -n //
+qed.
+*)
+
+lemma coafter_tls_succ: ∀g2,g1,g. g2 ~⊚ g1 ≡ g →
+ ∀n. @⦃0, g2⦄ ≡ n → ⫱*[⫯n]g2 ~⊚ ⫱g1 ≡ ⫱*[⫯n]g.
+#g2 #g1 #g #Hg #n #Hg2
+lapply (coafter_tls … Hg2 … Hg) -Hg #Hg
+lapply (at_pxx_tls … Hg2) -Hg2 #H
+elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
+elim (coafter_inv_pxx … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0 destruct
+<tls_rew_S <tls_rew_S <H2 <H0 -g2 -g -n //
+qed.
+
+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 #I #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+ elim (lifts_inv_atom1 … H2) -H2 *
+ /2 width=1 by frees_sort_gen, frees_gref_gen/
+ #i #j #Hij #H #H0 destruct
+ 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_lref_atom, drops_trans/
+| #f1 #I #K #V #s #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+ lapply (isfin_fwd_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+ lapply (lifts_inv_sort1 … H2) -H2 #H destruct
+ lapply (at_total 0 f) #H
+ elim (drops_split_trans … H1) -H1
+ [5: @(after_uni_dx … H) /2 width=1 by after_isid_dx/ |2,3: skip
+ |4: // ] #X #HLX #HXK
+ lapply (drops_inv_tls_at … H … HXK) -HXK #HXK
+ elim (drops_inv_skip2 … HXK) -HXK
+ #Y #W #HYK #HVW #H0 destruct
+(*
+
+ elim (coafter_inv_xpx … H3 ??) -H3 [ |*: // ] #g2 #g #Hg #H2 #H0
+ lapply (IH … Hg) -IH -Hg
+ [1,5: // | skip
+ |
+ |6: #H
+*)
+
+ lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ] #H3
+ lapply (IH … HYK … H3) -IH -H3 -HYK
+ [1,3: // | skip ]
+ #H lapply (frees_sort … H)
+
+ ]
+
+
+ elim (coafter_inv_xxp … H3) -H3 [1,3: * |*: // ]
+ [ #g #g1 #Hf2 #H #H0 destruct
+ elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
+ | #g #Hf2 #H destruct
+ lapply (drops_inv_drop1 … H1) -H1
+ ] /3 width=4 by frees_sort/
+
+|
+|
+|
+| #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 /4 width=5 by frees_bind, drops_skip/
+| #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/
+]
+
(* Inversion lemmas with generic slicing for local environments *************)
lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
[ #f2 #I #Hf2 #_ #f #K #H1 #T #H2 #f1 #H3
lapply (coafter_fwd_isid2 … H3 … Hf2) -H3 // -Hf2 #Hf1
elim (drops_inv_atom1 … H1) -H1 #H #_ destruct
- elim (lifts_fwd_atom2 … H2) -H2
- /2 width=3 by frees_atom/
+ elim (lifts_inv_atom2 … H2) -H2 * /2 width=3 by frees_atom/
| #f2 #I #L #W #s #_ #IH #Hf2 #f #Y #H1 #T #H2 #f1 #H3
lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
lapply (lifts_inv_sort2 … H2) -H2 #H destruct
elim (coafter_inv_sor … H3 … H1f2) -H3 -H1f2 /3 width=5 by frees_flat/
]
qed-.
+
+lemma frees_inv_drops: ∀f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
+ ∀f,K. ⬇*[Ⓣ, f] L ≡ K → ∀f1. f ~⊚ f1 ≡ f2 →
+ ∃∃T. K ⊢ 𝐅*⦃T⦄ ≡ f1 & ⬆*[f] T ≡ U.
+#f2 #L #U #H lapply (frees_fwd_isfin … H) elim H -f2 -L -U
+[ #f2 #I #Hf2 #_ #f #K #H1 #f1 #H2
+ lapply (coafter_fwd_isid2 … H2 ??) -H2 // -Hf2 #Hf1
+ elim (drops_inv_atom1 … H1) -H1 #H #Hf destruct
+ /4 width=3 by frees_atom, lifts_refl, ex2_intro/
+| #f2 #I #L #W #s #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+ lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+ elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
+ [ #g #g1 #Hf2 #H #H0 destruct
+ elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
+ | #g #Hf2 #H destruct
+ lapply (drops_inv_drop1 … H1) -H1 #HLK
+ ]
+ elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
+ lapply (lifts_inv_sort2 … HX) -HX #H destruct
+ /3 width=3 by frees_sort, lifts_sort, ex2_intro/
+| #f2 #I #L #W #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+ lapply (isfin_inv_next … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+ elim (coafter_inv_xxn … H2) -H2 [ |*: // ] #g #g1 #Hf2 #H0 #H destruct
+ elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #HVW #H destruct
+ elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
+ lapply (lifts_inj … HX … HVW) -W #H destruct
+ /3 width=3 by frees_zero, lifts_lref, ex2_intro/
+| #f2 #I #L #W #j #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+ lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+ elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
+ [ #g #g1 #Hf2 #H #H0 destruct
+ elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
+ | #g #Hf2 #H destruct
+ lapply (drops_inv_drop1 … H1) -H1 #HLK
+ ]
+ elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
+ elim (lifts_inv_lref2 … HX) -HX #i #Hij #H destruct
+ /4 width=7 by frees_lref, lifts_lref, at_S1, at_next, ex2_intro/
+| #f2 #I #L #W #l #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+ lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+ elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
+ [ #g #g1 #Hf2 #H #H0 destruct
+ elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
+ | #g #Hf2 #H destruct
+ lapply (drops_inv_drop1 … H1) -H1 #HLK
+ ]
+ elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
+ lapply (lifts_inv_gref2 … HX) -HX #H destruct
+ /3 width=3 by frees_gref, lifts_gref, ex2_intro/
+| #f2W #f2U #f2 #p #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
+ elim (sor_inv_isfin3 … H1f2) // #H1f2W #H
+ lapply (isfin_inv_tl … H) -H #H1f2U
+ elim (coafter_inv_sor … H2 … H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H #Hf1
+ elim (coafter_inv_tl0 … H) -H #g1 #H2f2U #H destruct
+ elim (IHW … H1 … H2f2W) -IHW -H2f2W // -H1f2W #V #Hf1W #HVW
+ elim (IHU … H2f2U) -IHU -H2f2U
+ /3 width=5 by frees_bind, drops_skip, lifts_bind, ex2_intro/
+| #f2W #f2U #f2 #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
+ elim (sor_inv_isfin3 … H1f2) // #H1f2W #H1f2U
+ elim (coafter_inv_sor … H2 … H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H2f2U #Hf1
+ elim (IHW … H1 … H2f2W) -IHW -H2f2W // -H1f2W
+ elim (IHU … H1 … H2f2U) -L -H2f2U
+ /3 width=5 by frees_flat, lifts_flat, ex2_intro/
+]
+qed-.
definition H_coafter_fwd_isid2: predicate rtmap ≝
λf1. ∀f2,f. f1 ~⊚ f2 ≡ f → 𝐓⦃f1⦄ → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
+definition H_coafter_isfin2_fwd: predicate rtmap ≝
+ λf1. ∀f2. 𝐅⦃f2⦄ → 𝐓⦃f1⦄ → ∀f. f1 ~⊚ f2 ≡ f → 𝐅⦃f⦄.
+
(* Basic inversion lemmas ***************************************************)
lemma coafter_inv_ppx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
(* Inversion lemmas with tail ***********************************************)
+lemma coafter_inv_tl1: ∀g2,g1,g. g2 ~⊚ ⫱g1 ≡ g →
+ ∃∃f. ↑g2 ~⊚ g1 ≡ f & ⫱f = g.
+#g2 #g1 #g elim (pn_split g1) * #f1 #H1 #H destruct
+[ /3 width=7 by coafter_refl, ex2_intro/
+| @(ex2_intro … (⫯g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+]
+qed-.
+
lemma coafter_inv_tl0: ∀g2,g1,g. g2 ~⊚ g1 ≡ ⫱g →
∃∃f1. ↑g2 ~⊚ f1 ≡ g & ⫱f1 = g1.
-#g1 #g2 #g elim (pn_split g) * #f #H0 #H destruct
+#g2 #g1 #g elim (pn_split g) * #f #H0 #H destruct
[ /3 width=7 by coafter_refl, ex2_intro/
-| @(ex2_intro … (⫯g2)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+| @(ex2_intro … (⫯g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
]
qed-.
/3 width=7 by coafter_fwd_isid2_aux, coafter_fwd_isid2_O_aux/
qed-.
+fact coafter_isfin2_fwd_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 →
+ H_coafter_isfin2_fwd f1.
+#f1 #Hf1 #f2 #H
+generalize in match Hf1; generalize in match f1; -f1
+@(isfin_ind … H) -f2
+[ /3 width=4 by coafter_isid_inv_dx, isfin_isid/ ]
+#f2 #_ #IH #f1 #H #Hf1 #f #Hf
+elim (at_inv_pxp … H) -H [ |*: // ] #g1 #H1
+lapply (istot_inv_push … Hf1 … H1) -Hf1 #Hg1
+elim (Hg1 0) #n #Hn
+[ elim (coafter_inv_ppx … Hf) | elim (coafter_inv_pnx … Hf)
+] -Hf [1,6: |*: // ] #g #Hg #H0 destruct
+/5 width=6 by isfin_next, isfin_push, isfin_inv_tls, istot_tls, at_pxx_tls, coafter_tls/
+qed-.
+
+fact coafter_isfin2_fwd_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_isfin2_fwd f1) →
+ ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_isfin2_fwd f1.
+#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
+#i2 #IH #f1 #H1f1 #f2 #Hf2 #H2f1 #f #Hf
+elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
+elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0
+lapply (IH … Hg1 … Hg) -i2 -Hg
+/2 width=4 by istot_inv_next, isfin_push/ (**) (* full auto fails *)
+qed-.
+
+lemma coafter_isfin2_fwd: ∀f1. H_coafter_isfin2_fwd f1.
+#f1 #f2 #Hf2 #Hf1 cases (Hf1 0)
+/3 width=7 by coafter_isfin2_fwd_aux, coafter_isfin2_fwd_O_aux/
+qed-.
+
lemma coafter_inv_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀fa,fb. fa ⋓ fb ≡ f →
∃∃f1a,f1b. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & f1a ⋓ f1b ≡ f1.
@isfin_ind
elim (IH … Hg2 … H1f … H2f) -f -Hg2
/3 width=11 by sor_np, sor_pn, sor_nn, ex3_2_intro, coafter_refl, coafter_push/
]
-qed-.
+qed-.
+
+(* Properties with istot ****************************************************)
+
+lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀f1a,f1b. f1a ⋓ f1b ≡ f1 →
+ ∃∃fa,fb. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & fa ⋓ fb ≡ f.
+@isfin_ind
+[ #f #Hf #f2 #Hf2 #f1 #Hf #f1a #f1b #Hf1
+ lapply (coafter_fwd_isid2 … Hf ??) -Hf // #H2f1
+ elim (sor_inv_isid3 … Hf1) -Hf1 //
+ /3 width=5 by coafter_isid_dx, sor_refl, ex3_2_intro/
+| #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
+ elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ]
+ [ #g2 #g1 #Hf #Hgf2 #Hgf1
+ elim (sor_inv_xxp … H2) -H2 [ |*: // ] #ga #gb #Hg1
+ lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
+ elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
+ /3 width=11 by coafter_refl, sor_pp, ex3_2_intro/
+ | #g2 #Hf #Hgf2
+ lapply (istot_inv_next … Hf2 … Hgf2) -Hf2 #Hg2
+ elim (IH … Hf … H2) // -f1 -IH -Hg2
+ /3 width=11 by coafter_next, sor_pp, ex3_2_intro/
+ ]
+| #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
+ elim (coafter_inv_xxn … H1) -H1 [ |*: // ] #g2 #g1 #Hf #Hgf2 #Hgf1
+ lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
+ elim (sor_inv_xxn … H2) -H2 [1,3,4: * |*: // ] #ga #gb #Hg1
+ elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
+ /3 width=11 by coafter_refl, coafter_push, sor_np, sor_pn, sor_nn, ex3_2_intro/
+]
+qed-.
projects / helm.git / commitdiff