(* Advanced properties ******************************************************)
lemma cpms_delta (n) (h) (G): ∀K,V1,V2. ❪G,K❫ ⊢ V1 ➡*[n,h] V2 →
- ∀W2. ⇧*[1] V2 ≘ W2 → ❪G,K.ⓓV1❫ ⊢ #0 ➡*[n,h] W2.
+ ∀W2. ⇧[1] V2 ≘ W2 → ❪G,K.ⓓV1❫ ⊢ #0 ➡*[n,h] W2.
#n #h #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=3 by cpm_cpms, cpm_delta/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
qed.
lemma cpms_ell (n) (h) (G): ∀K,V1,V2. ❪G,K❫ ⊢ V1 ➡*[n,h] V2 →
- ∀W2. ⇧*[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #0 ➡*[↑n,h] W2.
+ ∀W2. ⇧[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #0 ➡*[↑n,h] W2.
#n #h #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=3 by cpm_cpms, cpm_ell/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
qed.
lemma cpms_lref (n) (h) (I) (G): ∀K,T,i. ❪G,K❫ ⊢ #i ➡*[n,h] T →
- ∀U. ⇧*[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡*[n,h] U.
+ ∀U. ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡*[n,h] U.
#n #h #I #G #K #T #i #H @(cpms_ind_dx … H) -T
[ /3 width=3 by cpm_cpms, cpm_lref/
| #n1 #n2 #T #T2 #_ #IH #HT2 #U2 #HTU2
(* Note: apparently this was missing in basic_1 *)
(* Basic_2A1: uses: cprs_delta *)
lemma cpms_delta_drops (n) (h) (G):
- ∀L,K,V,i. ⇩*[i] L ≘ K.ⓓV →
+ ∀L,K,V,i. ⇩[i] L ≘ K.ⓓV →
∀V2. ❪G,K❫ ⊢ V ➡*[n,h] V2 →
- ∀W2. ⇧*[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #i ➡*[n,h] W2.
+ ∀W2. ⇧[↑i] V2 ≘ W2 → ❪G,L❫ ⊢ #i ➡*[n,h] W2.
#n #h #G #L #K #V #i #HLK #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=6 by cpm_cpms, cpm_delta_drops/
| #n1 #n2 #V1 #V2 #_ #IH #HV12 #W2 #HVW2
qed.
lemma cpms_ell_drops (n) (h) (G):
- ∀L,K,W,i. ⇩*[i] L ≘ K.ⓛW →
+ ∀L,K,W,i. ⇩[i] L ≘ K.ⓛW →
∀W2. ❪G,K❫ ⊢ W ➡*[n,h] W2 →
- ∀V2. ⇧*[↑i] W2 ≘ V2 → ❪G,L❫ ⊢ #i ➡*[↑n,h] V2.
+ ∀V2. ⇧[↑i] W2 ≘ V2 → ❪G,L❫ ⊢ #i ➡*[↑n,h] V2.
#n #h #G #L #K #W #i #HLK #W2 #H @(cpms_ind_dx … H) -W2
[ /3 width=6 by cpm_cpms, cpm_ell_drops/
| #n1 #n2 #W1 #W2 #_ #IH #HW12 #V2 #HWV2
lemma cpms_inv_lref1_drops (n) (h) (G):
∀L,T2,i. ❪G,L❫ ⊢ #i ➡*[n,h] T2 →
∨∨ ∧∧ T2 = #i & n = 0
- | ∃∃K,V,V2. ⇩*[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ➡*[n,h] V2 &
- ⇧*[↑i] V2 ≘ T2
- | ∃∃m,K,V,V2. ⇩*[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ➡*[m,h] V2 &
- ⇧*[↑i] V2 ≘ T2 & n = ↑m.
+ | ∃∃K,V,V2. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ➡*[n,h] V2 &
+ ⇧[↑i] V2 ≘ T2
+ | ∃∃m,K,V,V2. ⇩[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ➡*[m,h] V2 &
+ ⇧[↑i] V2 ≘ T2 & n = ↑m.
#n #h #G #L #T2 #i #H @(cpms_ind_dx … H) -T2
[ /3 width=1 by or3_intro0, conj/
| #n1 #n2 #T #T2 #_ #IH #HT2 cases IH -IH *
lemma cpms_inv_delta_sn (n) (h) (G) (K) (V):
∀T2. ❪G,K.ⓓV❫ ⊢ #0 ➡*[n,h] T2 →
∨∨ ∧∧ T2 = #0 & n = 0
- | ∃∃V2. ❪G,K❫ ⊢ V ➡*[n,h] V2 & ⇧*[1] V2 ≘ T2.
+ | ∃∃V2. ❪G,K❫ ⊢ V ➡*[n,h] V2 & ⇧[1] V2 ≘ T2.
#n #h #G #K #V #T2 #H
elim (cpms_inv_lref1_drops … H) -H *
[ /3 width=1 by or_introl, conj/
lemma cpms_inv_ell_sn (n) (h) (G) (K) (V):
∀T2. ❪G,K.ⓛV❫ ⊢ #0 ➡*[n,h] T2 →
∨∨ ∧∧ T2 = #0 & n = 0
- | ∃∃m,V2. ❪G,K❫ ⊢ V ➡*[m,h] V2 & ⇧*[1] V2 ≘ T2 & n = ↑m.
+ | ∃∃m,V2. ❪G,K❫ ⊢ V ➡*[m,h] V2 & ⇧[1] V2 ≘ T2 & n = ↑m.
#n #h #G #K #V #T2 #H
elim (cpms_inv_lref1_drops … H) -H *
[ /3 width=1 by or_introl, conj/
lemma cpms_inv_lref_sn (n) (h) (G) (I) (K):
∀U2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡*[n,h] U2 →
∨∨ ∧∧ U2 = #↑i & n = 0
- | ∃∃T2. ❪G,K❫ ⊢ #i ➡*[n,h] T2 & ⇧*[1] T2 ≘ U2.
+ | ∃∃T2. ❪G,K❫ ⊢ #i ➡*[n,h] T2 & ⇧[1] T2 ≘ U2.
#n #h #G #I #K #U2 #i #H
elim (cpms_inv_lref1_drops … H) -H *
[ /3 width=1 by or_introl, conj/