∀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
+ lapply (pr_coafter_isi_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
+ lapply (pr_coafter_isi_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
+ elim (pr_after_pat_des … Hij … Hf) -f #x #_ #Hj -g -i
+ lapply (pr_pat_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 (pr_isf_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
| #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
+ lapply (pr_coafter_isi_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 (pr_isf_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 (pr_pat_inv_succ_sn … 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 *)
+ >nplus_succ_sn /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
+ lapply (pr_coafter_isi_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 (pr_sor_inv_isf … H1f1) // #Hf1V #H
+ lapply (pr_isf_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
+ elim (pr_sor_coafter_dx_tans … H3 … H1f1) /2 width=5 by pr_coafter_des_ist_isf/ -H3 -H1f1 #f2V #f2T #Hf2V #H
+ elim (pr_coafter_inv_tl_dx … 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 (pr_sor_inv_isf … 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/
+ elim (pr_sor_coafter_dx_tans … H3 … H1f1)
+ /3 width=5 by pr_coafter_des_ist_isf, frees_flat/
]
qed-.
∀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-.
+/4 width=11 by frees_lifts, frees_mono, pr_coafter_eq_repl_back/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
∀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/
+#f1 #Hf #Hf1 elim (pr_coafter_inv_next_sn … Hf) -Hf
+/3 width=5 by frees_eq_repl_back, pr_coafter_isi_inv_sn/
qed-.
lemma frees_inv_lifts:
∀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/
+/3 width=7 by frees_eq_repl_back, pr_coafter_inj/
qed-.
(* Note: this is used by rex_conf and might be modified *)
∃∃g2. L2 ⊢ 𝐅+❪V2❫ ≘ g2 & g2 ⊆ g1.
#f1 #L1 #T1 #H elim H -f1 -L1 -T1
[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #j #_ #g1 #H1 -I2 -L1 -s
- lapply (isid_tls j … Hf1) -Hf1 <H1 -f1 #Hf1
- elim (isid_inv_next … Hf1) -Hf1 //
+ lapply (pr_isi_tls j … Hf1) -Hf1 <H1 -f1 #Hf1
+ elim (pr_isi_inv_next … Hf1) -Hf1 //
| #f1 #i #_ #I2 #L2 #V2 #j #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
+ #H destruct #g1 #Hgf1 >(eq_inv_pr_next_bi … Hgf1) -g1
/2 width=3 by ex2_intro/
| -Hf1 #j #HL12 lapply (drops_inv_drop1 … HL12) -HL12
- #HL12 #g1 <tls_xn <tl_next_rew #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
+ #HL12 #g1 <pr_tls_swap <pr_tl_next #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
- | #j #_ #g1 #Hgf1 elim (isid_inv_next … Hgf1) -Hgf1 <tls_xn /2 width=1 by isid_tls/
+ | #j #_ #g1 #Hgf1 elim (pr_isi_inv_next … Hgf1) -Hgf1 <pr_tls_swap /2 width=1 by pr_isi_tls/
]
| #f1 #I1 #L1 #i #_ #IH #I2 #L2 #V2 *
- [ -IH #_ #g1 #Hgf1 elim (discr_next_push … Hgf1)
+ [ -IH #_ #g1 #Hgf1 elim (eq_inv_pr_next_push … Hgf1)
| #j #HL12 lapply (drops_inv_drop1 … HL12) -HL12
- #HL12 #g1 <tls_xn #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
+ #HL12 #g1 <pr_tls_swap #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
/2 width=3 by ex2_intro/
]
| #f1 #L1 #l #Hf1 #I2 #L2 #V2 #j #_ #g1 #H1 -I2 -L1 -l
- lapply (isid_tls j … Hf1) -Hf1 <H1 -f1 #Hf1
- elim (isid_inv_next … Hf1) -Hf1 //
+ lapply (pr_isi_tls j … Hf1) -Hf1 <H1 -f1 #Hf1
+ elim (pr_isi_inv_next … Hf1) -Hf1 //
| #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #j #HL12 #g1 #Hgf1
- lapply (sor_tls … Hf1 j) -Hf1 <Hgf1 -Hgf1 #Hf1
- elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
+ lapply (pr_sor_tls … Hf1 j) -Hf1 <Hgf1 -Hgf1 #Hf1
+ elim (pr_sor_next_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/
+ /3 width=6 by pr_sor_inv_sle_sn_trans, ex2_intro/
+ | -IHV1 #_ >pr_tls_swap #H2 elim (IHT1 … H2) -IHT1 -H2
+ /3 width=6 by drops_drop, pr_sor_inv_sle_dx_trans, ex2_intro/
]
| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #j #HL12 #g1 #Hgf1
- lapply (sor_tls … Hf1 j) -Hf1 <Hgf1 -Hgf1 #Hf1
- elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
+ lapply (pr_sor_tls … Hf1 j) -Hf1 <Hgf1 -Hgf1 #Hf1
+ elim (pr_sor_next_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/
+ /3 width=6 by pr_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/
+ /3 width=6 by pr_sor_inv_sle_dx_trans, ex2_intro/
]
]
qed-.
projects / helm.git / blobdiff