(* *)
(**************************************************************************)
-include "basic_2/relocation/drops_drops.ma".
-include "basic_2/s_computation/fqup_weight.ma".
-include "basic_2/s_computation/fqup_drops.ma".
+include "ground_2/xoa/ex_5_5.ma".
+include "static_2/relocation/drops_drops.ma".
+include "static_2/s_computation/fqup_weight.ma".
+include "static_2/s_computation/fqup_drops.ma".
include "basic_2/rt_transition/cpg.ma".
(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
(* Advanced properties ******************************************************)
-lemma cpg_delta_drops: â\88\80Rt,c,h,G,K,V,V2,i,L,T2. â¬\87*[i] L â\89\98 K.â\93\93V â\86\92 â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[Rt, c, h] V2 →
- â¬\86*[â\86\91i] V2 â\89\98 T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â¬\88[Rt, c, h] T2.
+lemma cpg_delta_drops: â\88\80Rt,c,h,G,K,V,V2,i,L,T2. â\87©*[i] L â\89\98 K.â\93\93V â\86\92 â\9dªG,Kâ\9d« â\8a¢ V â¬\88[Rt,c,h] V2 →
+ â\87§*[â\86\91i] V2 â\89\98 T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ #i â¬\88[Rt,c,h] T2.
#Rt #c #h #G #K #V #V2 #i elim i -i
[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
| #i #IH #L0 #T0 #H0 #HV2 #HVT2
elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
- elim (lifts_split_trans â\80¦ HVT2 (ð\9d\90\94â\9d´â\86\91iâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ) ?) -HVT2 /3 width=3 by cpg_lref/
+ elim (lifts_split_trans â\80¦ HVT2 (ð\9d\90\94â\9d¨â\86\91iâ\9d©) (ð\9d\90\94â\9d¨1â\9d©) ?) -HVT2 /3 width=3 by cpg_lref/
]
qed.
-lemma cpg_ell_drops: â\88\80Rt,c,h,G,K,V,V2,i,L,T2. â¬\87*[i] L â\89\98 K.â\93\9bV â\86\92 â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[Rt,c, h] V2 →
- â¬\86*[â\86\91i] V2 â\89\98 T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â¬\88[Rt, c+ð\9d\9f\98ð\9d\9f\99, h] T2.
+lemma cpg_ell_drops: â\88\80Rt,c,h,G,K,V,V2,i,L,T2. â\87©*[i] L â\89\98 K.â\93\9bV â\86\92 â\9dªG,Kâ\9d« â\8a¢ V â¬\88[Rt,c,h] V2 →
+ â\87§*[â\86\91i] V2 â\89\98 T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ #i â¬\88[Rt,c+ð\9d\9f\98ð\9d\9f\99,h] T2.
#Rt #c #h #G #K #V #V2 #i elim i -i
[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
| #i #IH #L0 #T0 #H0 #HV2 #HVT2
elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
- elim (lifts_split_trans â\80¦ HVT2 (ð\9d\90\94â\9d´â\86\91iâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ) ?) -HVT2 /3 width=3 by cpg_lref/
+ elim (lifts_split_trans â\80¦ HVT2 (ð\9d\90\94â\9d¨â\86\91iâ\9d©) (ð\9d\90\94â\9d¨1â\9d©) ?) -HVT2 /3 width=3 by cpg_lref/
]
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpg_inv_lref1_drops: â\88\80Rt,c,h,G,i,L,T2. â¦\83G, Lâ¦\84 â\8a¢ #i â¬\88[Rt,c, h] T2 →
+lemma cpg_inv_lref1_drops: â\88\80Rt,c,h,G,i,L,T2. â\9dªG,Lâ\9d« â\8a¢ #i â¬\88[Rt,c,h] T2 →
∨∨ T2 = #i ∧ c = 𝟘𝟘
- | â\88\83â\88\83cV,K,V,V2. â¬\87*[i] L â\89\98 K.â\93\93V & â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[Rt, cV, h] V2 &
- â¬\86*[↑i] V2 ≘ T2 & c = cV
- | â\88\83â\88\83cV,K,V,V2. â¬\87*[i] L â\89\98 K.â\93\9bV & â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[Rt, cV, h] V2 &
- â¬\86*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
+ | â\88\83â\88\83cV,K,V,V2. â\87©*[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« â\8a¢ V â¬\88[Rt,cV,h] V2 &
+ â\87§*[↑i] V2 ≘ T2 & c = cV
+ | â\88\83â\88\83cV,K,V,V2. â\87©*[i] L â\89\98 K.â\93\9bV & â\9dªG,Kâ\9d« â\8a¢ V â¬\88[Rt,cV,h] V2 &
+ â\87§*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
#Rt #c #h #G #i elim i -i
[ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
/4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/
]
qed-.
-lemma cpg_inv_atom1_drops: â\88\80Rt,c,h,I,G,L,T2. â¦\83G, Lâ¦\84 â\8a¢ â\93ª{I} â¬\88[Rt, c, h] T2 →
- ∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘
- | ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙
- | â\88\83â\88\83cV,i,K,V,V2. â¬\87*[i] L â\89\98 K.â\93\93V & â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[Rt, cV, h] V2 &
- â¬\86*[↑i] V2 ≘ T2 & I = LRef i & c = cV
- | â\88\83â\88\83cV,i,K,V,V2. â¬\87*[i] L â\89\98 K.â\93\9bV & â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[Rt, cV, h] V2 &
- â¬\86*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
+lemma cpg_inv_atom1_drops: â\88\80Rt,c,h,I,G,L,T2. â\9dªG,Lâ\9d« â\8a¢ â\93ª[I] â¬\88[Rt,c,h] T2 →
+ ∨∨ T2 = ⓪[I] ∧ c = 𝟘𝟘
+ | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & c = 𝟘𝟙
+ | â\88\83â\88\83cV,i,K,V,V2. â\87©*[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« â\8a¢ V â¬\88[Rt,cV,h] V2 &
+ â\87§*[↑i] V2 ≘ T2 & I = LRef i & c = cV
+ | â\88\83â\88\83cV,i,K,V,V2. â\87©*[i] L â\89\98 K.â\93\9bV & â\9dªG,Kâ\9d« â\8a¢ V â¬\88[Rt,cV,h] V2 &
+ â\87§*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
#Rt #c #h * #n #G #L #T2 #H
[ elim (cpg_inv_sort1 … H) -H *
/3 width=3 by or4_intro0, or4_intro1, ex3_intro, conj/
elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct
elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct
elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
- elim (lifts_total W2 (ð\9d\90\94â\9d´â\86\91i2â\9dµ)) #U2 #HWU2
+ elim (lifts_total W2 (ð\9d\90\94â\9d¨â\86\91i2â\9d©)) #U2 #HWU2
lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
/4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
elim (IH … HV12 … HLK … HVW1) -HV12 //
elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
/3 width=5 by cpg_bind, lifts_bind, ex2_intro/
- | #cT #T2 #HT12 #HXT2 #H1 #H2 #H3 destruct
- elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] #U2 #HTU2 #HU12
- lapply (lifts_trans … HXT2 … HTU2 ??) -T2 [3: |*: // ] #HXU2
- elim (lifts_split_trans … HXU2 f (𝐔❴↑O❵)) [2: /2 width=1 by after_uni_one_dx/ ]
+ | #cT #T2 #HT21 #HTX2 #H1 #H2 #H3 destruct
+ elim (lifts_trans4_one … HT21 … HTU1) -HTU1 #U2 #HTU2 #HU21
+ elim (IH … HTX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -K -V1 -T1 -T2
/3 width=5 by cpg_zeta, ex2_intro/
]
| * #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
elim (IH … HY12 … HLK … HYZ1) -HY12 //
elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
- elim (lifts_total W2 (ð\9d\90\94â\9d´1â\9dµ)) #W20 #HW20
+ elim (lifts_total W2 (ð\9d\90\94â\9d¨1â\9d©)) #W20 #HW20
lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
lapply (lifts_conf … HV20 … H (⫯f) ?) -V2 /2 width=3 by after_uni_one_sn/
/4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
[ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
#cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
- lapply (drops_split_div â\80¦ HLK (ð\9d\90\94â\9d´i1â\9dµ) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
+ lapply (drops_split_div â\80¦ HLK (ð\9d\90\94â\9d¨i1â\9d©) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct
elim (IH … HW12 … HLK … HVW1) -HW12 //
elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
/3 width=5 by cpg_bind, lifts_bind, ex2_intro/
- | #cU #U2 #HU12 #HXU2 #H1 #H2 #H3 destruct
- elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ] #T2 #HTU2 #HT12
- elim (lifts_div4_one … HTU2 … HXU2) -U2 /3 width=5 by cpg_zeta, ex2_intro/
+ | #cU #U2 #HU21 #HUX2 #H1 #H2 #H3 destruct
+ elim (lifts_div4_one … HTU1 … HU21) -HTU1 #T2 #HT21 #HTU2
+ elim (IH … HUX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -L -W1 -U1 -U2
+ /3 width=5 by cpg_zeta, ex2_intro/
]
| * #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
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