(* Forward lemmas with context-sensitive parallel reduction for terms *******)
lemma cpr_fwd_deq (h) (M): is_model M → is_extensional M →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ≗{M} T2.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ≗{M} T2.
#h #M #H1M #H2M #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ /2 width=2 by deq_refl/
| #G #K #V1 #V2 #W2 #_ #IH #HVW2 #gv #v #H
@(mq … H1M) [4,5: @(ti_comp … H) /2 width=2 by veq_refl/ |1,2: skip ] -v
@(mq … H1M)
[4: /3 width=1 by seq_sym, ml/ | skip
- |5: /2 width=2 by lifts_SO_fwd_vlift/ | skip ] -W2
- >vlift_eq /2 width=1 by/
+ |5: /2 width=2 by lifts_SO_fwd_vpush/ | skip ] -W2
+ >vpush_eq /2 width=1 by/
| #I #G #K #T #U #i #_ #IH #HTU #gv #v #H
elim (li_fwd_bind … H) // -H #lv #d #HK #H
@(mq … H1M) [4,5: @(ti_comp … H) /2 width=2 by veq_refl/ |1,2: skip ] -v
@(mq … H1M)
[4: /3 width=1 by seq_sym, ml/ | skip
- |5: /2 width=2 by lifts_SO_fwd_vlift/ | skip ] -U
- >vlift_gt /3 width=5 by ml, mq, mr/
+ |5: /2 width=2 by lifts_SO_fwd_vpush/ | skip ] -U
+ >vpush_gt /3 width=5 by ml, mq, mr/
| #p * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #gv #lv #Hlv
- [ @(mq … H1M) [4,5: /3 width=2 by seq_sym, md/ |1,2: skip ] -p
+ [ @(mq … H1M) [4,5: /3 width=2 by seq_sym, md/ |1,2: skip ]
+ @mc [3:|*: /2 width=1 by/ ] -p
@(seq_trans … H1M) [2: @IHT /2 width=1 by li_abbr/ | skip ] -T1
- /4 width=1 by ti_comp, vlift_comp, (* 2x *) veq_refl/
+ /4 width=1 by ti_comp, vpush_comp, (* 2x *) veq_refl/
| /4 width=1 by li_abst, mx/
]
| * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #gv #lv #Hlv
| @(mq … H1M) [4,5: /3 width=2 by seq_sym, me/ |1,2: skip ]
/2 width=1 by/
]
-| #G #L #V #U1 #U2 #T2 #_ #IH #HTU2 #gv #lv #Hlv
+| #G #L #V #U1 #T1 #T2 #HTU1 #_ #IH #gv #lv #Hlv
@(mq … H1M)
[4: /3 width=2 by seq_sym, md/ | skip
- |3: /3 width=1 by li_abbr/ | skip ] -L -U1
- /3 width=1 by lifts_SO_fwd_vlift, seq_sym/
+ |3: @(seq_trans … H1M) [2: @mz // | skip ]
+ ] /3 width=3 by lifts_SO_fwd_vpush, seq_sym/
| #G #L #V #T1 #T2 #_ #IH #gv #lv #Hlv
@(seq_trans … H1M) [2: @(me … H1M) | skip ]
/2 width=1 by/
| #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV #_ #IHT #gv #lv #Hlv
@(mq … H1M) [4,5: /3 width=2 by seq_sym, ma, md/ |1,2: skip ]
- @(mq … H1M)
- [4: /3 width=2 by seq_sym, mb/ | skip
- |5: @IHT /2 width=1 by li_abst/ | skip ] -T2
+ @(seq_trans … H1M) [3:|*: /2 width=2 by mb/ ]
+ @mc // -p [ /4 width=5 by seq_trans, seq_sym, me/ ]
+ @(seq_trans … H1M) [2: @IHT /2 width=1 by li_abst/ | skip ] -T1
@ti_comp /2 width=1 by veq_refl/
- @vlift_comp /2 width=1 by veq_refl/
- @(mq … H1M) [4,5: /3 width=2 by seq_sym, me/ |1,2: skip ] -L -V1
- /2 width=1 by mr/
+ @vpush_comp /2 width=1 by veq_refl/
+ /4 width=5 by seq_trans, seq_sym, me/
| #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV #IHW #IHT #HV2 #gv #lv #Hlv
@(mq … H1M) [4,5: /3 width=2 by seq_sym, ma, md/ |1,2: skip ]
- @(seq_trans … H1M) [3: /3 width=1 by seq_sym, ma/ | skip ]
- @mp // [ @(seq_trans … H1M) /3 width=3 by lifts_SO_fwd_vlift/ ] -V1 -V -V2
@(mq … H1M)
- [4: /3 width=2 by seq_sym, md/ | skip
- |3: @IHT /2 width=1 by li_abbr/ | skip ] -T1
- /4 width=1 by ti_comp, vlift_comp, (* 2x *) veq_refl/
+ [4: /4 width=2 by seq_sym, md, mp/ |1: skip
+ |5: /4 width=2 by seq_sym, ma, mc/ |2: skip
+ ]
+ @(seq_trans … H1M) [2: @mh // | skip ]
+ @mc [3:|*: /2 width=1 by mr/ ]
+ @mp [3:|*: /2 width=1 by lifts_SO_fwd_vpush/ ]
+ @(seq_trans … H1M) [2: @IHT /2 width=1 by li_abbr/ | skip ] -T1
+ /4 width=1 by ti_comp, vpush_comp, (* 2x *) veq_refl/
]
qed-.