(* Basic_1: includes: pr0_lift pr2_lift *)
(* Basic_2A1: includes: cpr_lift *)
-lemma cpm_lifts_sn: ∀h,n,G. d_liftable2_sn … lifts (λL. cpm h G L n).
+lemma cpm_lifts_sn (h) (n) (G): d_liftable2_sn … lifts (λL. cpm h G L n).
#h #n #G #K #T1 #T2 * #c #Hc #HT12 #b #f #L #HLK #U1 #HTU1
elim (cpg_lifts_sn … HT12 … HLK … HTU1) -K -T1
/3 width=5 by ex2_intro/
qed-.
-lemma cpm_lifts_bi: ∀h,n,G. d_liftable2_bi … lifts (λL. cpm h G L n).
+lemma cpm_lifts_bi (h) (n) (G): d_liftable2_bi … lifts (λL. cpm h G L n).
#h #n #G #K #T1 #T2 * /3 width=11 by cpg_lifts_bi, ex2_intro/
qed-.
(* Basic_1: includes: pr0_gen_lift pr2_gen_lift *)
(* Basic_2A1: includes: cpr_inv_lift1 *)
-lemma cpm_inv_lifts_sn: ∀h,n,G. d_deliftable2_sn … lifts (λL. cpm h G L n).
+lemma cpm_inv_lifts_sn (h) (n) (G): d_deliftable2_sn … lifts (λL. cpm h G L n).
#h #n #G #L #U1 #U2 * #c #Hc #HU12 #b #f #K #HLK #T1 #HTU1
elim (cpg_inv_lifts_sn … HU12 … HLK … HTU1) -L -U1
/3 width=5 by ex2_intro/
qed-.
-lemma cpm_inv_lifts_bi: ∀h,n,G. d_deliftable2_bi … lifts (λL. cpm h G L n).
+lemma cpm_inv_lifts_bi (h) (n) (G): d_deliftable2_bi … lifts (λL. cpm h G L n).
#h #n #G #L #U1 #U2 * /3 width=11 by cpg_inv_lifts_bi, ex2_intro/
qed-.
(* Basic_1: includes: pr2_delta1 *)
(* Basic_2A1: includes: cpr_delta *)
-lemma cpm_delta_drops: ∀h,n,G,L,K,V,V2,W2,i.
- ⇩[i] L ≘ K.ⓓV → ❪G,K❫ ⊢ V ➡[h,n] V2 →
- ⇧[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #i ➡[h,n] W2.
+lemma cpm_delta_drops (h) (n) (G) (L):
+ ∀K,V,V2,W2,i.
+ ⇩[i] L ≘ K.ⓓV → ❪G,K❫ ⊢ V ➡[h,n] V2 →
+ ⇧[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #i ➡[h,n] W2.
#h #n #G #L #K #V #V2 #W2 #i #HLK *
/3 width=8 by cpg_delta_drops, ex2_intro/
qed.
-lemma cpm_ell_drops: ∀h,n,G,L,K,V,V2,W2,i.
- ⇩[i] L ≘ K.ⓛV → ❪G,K❫ ⊢ V ➡[h,n] V2 →
- ⇧[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #i ➡[h,↑n] W2.
+lemma cpm_ell_drops (h) (n) (G) (L):
+ ∀K,V,V2,W2,i.
+ ⇩[i] L ≘ K.ⓛV → ❪G,K❫ ⊢ V ➡[h,n] V2 →
+ ⇧[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #i ➡[h,↑n] W2.
#h #n #G #L #K #V #V2 #W2 #i #HLK *
/3 width=8 by cpg_ell_drops, isrt_succ, ex2_intro/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpm_inv_atom1_drops: ∀h,n,I,G,L,T2. ❪G,L❫ ⊢ ⓪[I] ➡[h,n] T2 →
- ∨∨ T2 = ⓪[I] ∧ n = 0
- | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & n = 1
- | ∃∃K,V,V2,i. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ➡[h,n] V2 &
- ⇧[↑i] V2 ≘ T2 & I = LRef i
- | ∃∃m,K,V,V2,i. ⇩[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ➡[h,m] V2 &
- ⇧[↑i] V2 ≘ T2 & I = LRef i & n = ↑m.
-#h #n #I #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H *
+lemma cpm_inv_atom1_drops (h) (n) (G) (L):
+ ∀I,T2. ❪G,L❫ ⊢ ⓪[I] ➡[h,n] T2 →
+ ∨∨ ∧∧ T2 = ⓪[I] & n = 0
+ | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & n = 1
+ | ∃∃K,V,V2,i. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ➡[h,n] V2 & ⇧[↑i] V2 ≘ T2 & I = LRef i
+ | ∃∃m,K,V,V2,i. ⇩[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ➡[h,m] V2 & ⇧[↑i] V2 ≘ T2 & I = LRef i & n = ↑m.
+#h #n #G #L #I #T2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H *
[ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc
/3 width=1 by or4_intro0, conj/
-| #s #H1 #H2 #H3 destruct lapply (isrt_inv_01 … Hc) -Hc
+| #s1 #s2 #H1 #H2 #H3 #H4 destruct lapply (isrt_inv_01 … Hc) -Hc
/4 width=3 by or4_intro1, ex3_intro, sym_eq/ (**) (* sym_eq *)
| #cV #i #K #V1 #V2 #HLK #HV12 #HVT2 #H1 #H2 destruct
/4 width=8 by ex4_4_intro, ex2_intro, or4_intro2/
]
qed-.
-lemma cpm_inv_lref1_drops: ∀h,n,G,L,T2,i. ❪G,L❫ ⊢ #i ➡[h,n] T2 →
- ∨∨ T2 = #i ∧ n = 0
- | ∃∃K,V,V2. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ➡[h,n] V2 &
- ⇧[↑i] V2 ≘ T2
- | ∃∃m,K,V,V2. ⇩[i] L ≘ K. ⓛV & ❪G,K❫ ⊢ V ➡[h,m] V2 &
- ⇧[↑i] V2 ≘ T2 & n = ↑m.
+lemma cpm_inv_lref1_drops (h) (n) (G) (L):
+ ∀T2,i. ❪G,L❫ ⊢ #i ➡[h,n] T2 →
+ ∨∨ ∧∧ T2 = #i & n = 0
+ | ∃∃K,V,V2. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ➡[h,n] V2 & ⇧[↑i] V2 ≘ T2
+ | ∃∃m,K,V,V2. ⇩[i] L ≘ K. ⓛV & ❪G,K❫ ⊢ V ➡[h,m] V2 & ⇧[↑i] V2 ≘ T2 & n = ↑m.
#h #n #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1_drops … H) -H *
[ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc
/3 width=1 by or3_intro0, conj/
(* Advanced forward lemmas **************************************************)
-fact cpm_fwd_plus_aux (h) (n): ∀G,L,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
- ∀n1,n2. n1+n2 = n →
- ∃∃T. ❪G,L❫ ⊢ T1 ➡[h,n1] T & ❪G,L❫ ⊢ T ➡[h,n2] T2.
+fact cpm_fwd_plus_aux (h) (n) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 →
+ ∀n1,n2. n1+n2 = n →
+ ∃∃T. ❪G,L❫ ⊢ T1 ➡[h,n1] T & ❪G,L❫ ⊢ T ➡[h,n2] T2.
#h #n #G #L #T1 #T2 #H @(cpm_ind … H) -G -L -T1 -T2 -n
[ #I #G #L #n1 #n2 #H
elim (plus_inv_O3 … H) -H #H1 #H2 destruct
]
qed-.
-lemma cpm_fwd_plus (h) (G) (L): ∀n1,n2,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n1+n2] T2 →
- ∃∃T. ❪G,L❫ ⊢ T1 ➡[h,n1] T & ❪G,L❫ ⊢ T ➡[h,n2] T2.
+lemma cpm_fwd_plus (h) (G) (L):
+ ∀n1,n2,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n1+n2] T2 →
+ ∃∃T. ❪G,L❫ ⊢ T1 ➡[h,n1] T & ❪G,L❫ ⊢ T ➡[h,n2] T2.
/2 width=3 by cpm_fwd_plus_aux/ qed-.