include "ground_2/relocation/nstream_coafter.ma".
include "basic_2/relocation/drops_drops.ma".
include "basic_2/static/frees_fqup.ma".
-include "basic_2/static/frees_frees.ma".
(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
(* Advanced properties ******************************************************)
-lemma frees_lref_atom: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≡ ⋆ →
- ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≡ f.
+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 #V #IH *
+#L #I #IH *
[ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
-| /5 width=3 by frees_eq_repl_back, frees_lref, drops_inv_drop1, eq_push_inv_isid/
+| /4 width=3 by frees_lref, drops_inv_drop1/
]
qed.
-lemma frees_lref_pair: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≡ f →
- ∀i,I,L. ⬇*[i] L ≡ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≡ ↑*[i] ⫯f.
+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_zero/
+[ #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_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≡ f →
- ∀i,L. ⬇*[i] L ≡ K → L ⊢ 𝐅*⦃⋆s⦄ ≡ ↑*[i] f.
+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: â\88\80f,K,j. K â\8a¢ ð\9d\90\85*â¦\83#jâ¦\84 â\89¡ f →
- â\88\80i,L. â¬\87*[i] L â\89¡ K â\86\92 L â\8a¢ ð\9d\90\85*â¦\83#(i+j)â¦\84 â\89¡ â\86\91*[i] f.
+*)
+lemma frees_lref_pushs: â\88\80f,K,j. K â\8a¢ ð\9d\90\85*â¦\83#jâ¦\84 â\89\98 f →
+ â\88\80i,L. â¬\87*[i] L â\89\98 K â\86\92 L â\8a¢ ð\9d\90\85*â¦\83#(i+j)â¦\84 â\89\98 ⫯*[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 #V #HYK #H destruct /3 width=1 by frees_lref/
+ #I #Y #HYK #H destruct /3 width=1 by frees_lref/
]
qed.
-
-lemma frees_gref_pushs: â\88\80f,K,l. K â\8a¢ ð\9d\90\85*â¦\83§lâ¦\84 â\89¡ f →
- â\88\80i,L. â¬\87*[i] L â\89¡ K â\86\92 L â\8a¢ ð\9d\90\85*â¦\83§lâ¦\84 â\89¡ â\86\91*[i] f.
+(*
+lemma frees_gref_pushs: â\88\80f,K,l. K â\8a¢ ð\9d\90\85*â¦\83§lâ¦\84 â\89\98 f →
+ â\88\80i,L. â¬\87*[i] L â\89\98 K â\86\92 L â\8a¢ ð\9d\90\85*â¦\83§lâ¦\84 â\89\98 ⫯*[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: ∀i,f,L. L ⊢ 𝐅*⦃#i⦄ ≡ f →
- (⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ ∧ 𝐈⦃f⦄) ∨
- ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≡ g &
- ⬇*[i] L ≡ K.ⓑ{I}V & f = ↑*[i] ⫯g.
-#i elim i -i
-[ #f #L #H elim (frees_inv_zero … H) -H *
- /4 width=7 by ex3_4_intro, or_introl, or_intror, conj, drops_refl/
-| #i #IH #f #L #H elim (frees_inv_lref … H) -H * /3 width=1 by or_introl, conj/
- #g #I #K #V #Hg #H1 #H2 destruct
- elim (IH … Hg) -IH -Hg *
- [ /4 width=3 by or_introl, conj, isid_push, drops_drop/
- | /4 width=7 by drops_drop, ex3_4_intro, or_intror/
+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: â\88\80b,f1,K,T. K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f1 →
- â\88\80f,L. â¬\87*[b, f] L â\89¡ K â\86\92 â\88\80U. â¬\86*[f] T â\89¡ U →
- â\88\80f2. f ~â\8a\9a f1 â\89¡ f2 â\86\92 L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89¡ f2.
+lemma frees_lifts: â\88\80b,f1,K,T. K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f1 →
+ â\88\80f,L. â¬\87*[b, f] L â\89\98 K â\86\92 â\88\80U. â¬\86*[f] T â\89\98 U →
+ â\88\80f2. f ~â\8a\9a f1 â\89\98 f2 â\86\92 L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89\98 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
+[ #f1 #K #s #Hf1 #_ #f #L #HLK #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
+ >(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_lref_atom, drops_trans/
-| #f1 #I #K #V #s #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
- lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
- lapply (lifts_inv_sort1 … H2) -H2 #H destruct
- elim (drops_split_trans_pair2 … H1) -H1 [ |*: // ] #Y #W #HLY #HYK #_
- elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H2 destruct
- lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ] #H3
- lapply (IH … HYK … H3) -IH -H3 -HYK [1,3: // | skip ] #Hf2 #H destruct
- /3 width=5 by drops_isuni_fwd_drop2, frees_sort_pushs/
+ /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_pair2 … H1) -H1 [ |*: // ] #Y #W #HLY #HYK #HVW
- elim (coafter_fwd_xnx_pushs … 0 … H3) [ |*: // ] #g2 #H2 destruct
- lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ]
- <tls_S in ⊢ (???%→?); <tl_next_rew #H3 #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_lref_pair/
-| #f1 #I #K #V #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+ /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_pair2 … H1) -H1 [ |*: // ] #Y #W #HLY #HYK #_
- elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H2 destruct
- lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ] #H3 #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 #I #K #V #l #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
- lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
- lapply (lifts_inv_gref1 … H2) -H2 #H destruct
- elim (drops_split_trans_pair2 … H1) -H1 [ |*: // ] #Y #W #HLY #HYK #_
- elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H2 destruct
- lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ] #H3 #H destruct
- lapply (IH … HYK … H3) -IH -H3 -HYK [1,3: // | skip ]
- /3 width=5 by drops_isuni_fwd_drop2, frees_gref_pushs/
+| #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 /4 width=5 by frees_bind, drops_skip/
+ 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
]
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: â\88\80b,f2,L,U. L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89¡ f2 →
- â\88\80f,K. â¬\87*[b, f] L â\89¡ K â\86\92 â\88\80T. â¬\86*[f] T â\89¡ U →
- â\88\80f1. K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f1 â\86\92 f ~â\8a\9a f1 â\89¡ f2.
+lemma frees_fwd_coafter: â\88\80b,f2,L,U. L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89\98 f2 →
+ â\88\80f,K. â¬\87*[b, f] L â\89\98 K â\86\92 â\88\80T. â¬\86*[f] T â\89\98 U →
+ â\88\80f1. K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f1 â\86\92 f ~â\8a\9a f1 â\89\98 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: â\88\80b,f2,L,U. L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89¡ f2 →
- â\88\80f,K. â¬\87*[b, f] L â\89¡ K â\86\92 â\88\80T. â¬\86*[f] T â\89¡ U →
- â\88\83â\88\83f1. f ~â\8a\9a f1 â\89¡ f2 & K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f1.
+lemma frees_inv_lifts_ex: â\88\80b,f2,L,U. L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89\98 f2 →
+ â\88\80f,K. â¬\87*[b, f] L â\89\98 K â\86\92 â\88\80T. â¬\86*[f] T â\89\98 U →
+ â\88\83â\88\83f1. f ~â\8a\9a f1 â\89\98 f2 & K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 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: â\88\80b,f,L,U. L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89¡ f →
- â\88\80K. â¬\87*[b, ð\9d\90\94â\9d´1â\9dµ] L â\89¡ K â\86\92 â\88\80T. â¬\86*[1] T â\89¡ U →
- K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ ⫱f.
+lemma frees_inv_lifts_SO: â\88\80b,f,L,U. L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89\98 f →
+ â\88\80K. â¬\87*[b, ð\9d\90\94â\9d´1â\9dµ] L â\89\98 K â\86\92 â\88\80T. â¬\86*[1] T â\89\98 U →
+ K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 ⫱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: â\88\80b,f2,L,U. L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89¡ f2 →
- â\88\80f,K. â¬\87*[b, f] L â\89¡ K â\86\92 â\88\80T. â¬\86*[f] T â\89¡ U →
- â\88\80f1. f ~â\8a\9a f1 â\89¡ f2 â\86\92 K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f1.
+lemma frees_inv_lifts: â\88\80b,f2,L,U. L â\8a¢ ð\9d\90\85*â¦\83Uâ¦\84 â\89\98 f2 →
+ â\88\80f,K. â¬\87*[b, f] L â\89\98 K â\86\92 â\88\80T. â¬\86*[f] T â\89\98 U →
+ â\88\80f1. f ~â\8a\9a f1 â\89\98 f2 â\86\92 K â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 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-.
-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_inv_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
+(* 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/
]
- 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_inv_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
+| #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/
]
- 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_inv_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
+| #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/
]
- 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-.
projects / helm.git / blobdiff