(* Forward lemmas on atomic arity assignment for terms **********************)
(* Basic_2A1: uses: snv_fwd_aaa *)
-lemma cnv_fwd_aaa (h) (a): ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ∃A. ⦃G,L⦄ ⊢ T ⁝ A.
+lemma cnv_fwd_aaa (h) (a):
+ ∀G,L,T. ❨G,L❩ ⊢ T ![h,a] → ∃A. ❨G,L❩ ⊢ T ⁝ A.
#h #a #G #L #T #H elim H -G -L -T
[ /2 width=2 by aaa_sort, ex_intro/
| #I #G #L #V #_ * /3 width=2 by aaa_zero, ex_intro/
(* Forward lemmas with t_bound rt_transition for terms **********************)
lemma cnv_fwd_cpm_SO (h) (a) (G) (L):
- â\88\80T. â¦\83G,Lâ¦\84 â\8a¢ T ![h,a] â\86\92 â\88\83U. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡[1,h] U.
+ â\88\80T. â\9d¨G,Lâ\9d© â\8a¢ T ![h,a] â\86\92 â\88\83U. â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡[h,1] U.
#h #a #G #L #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by aaa_cpm_SO/
(* Forward lemmas with t_bound rt_computation for terms *********************)
lemma cnv_fwd_cpms_total (h) (a) (n) (G) (L):
- â\88\80T. â¦\83G,Lâ¦\84 â\8a¢ T ![h,a] â\86\92 â\88\83U. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n,h] U.
+ â\88\80T. â\9d¨G,Lâ\9d© â\8a¢ T ![h,a] â\86\92 â\88\83U. â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡*[h,n] U.
#h #a #n #G #L #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by cpms_total_aaa/
qed-.
lemma cnv_fwd_cpms_abst_dx_le (h) (a) (G) (L) (W) (p):
- â\88\80T. â¦\83G,Lâ¦\84 ⊢ T ![h,a] →
- â\88\80n1,U1. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n1,h] â\93\9b{p}W.U1 → ∀n2. n1 ≤ n2 →
- â\88\83â\88\83U2. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n2,h] â\93\9b{p}W.U2 & â¦\83G,L.â\93\9bWâ¦\84 â\8a¢ U1 â\9e¡*[n2-n1,h] U2.
+ â\88\80T. â\9d¨G,Lâ\9d© ⊢ T ![h,a] →
+ â\88\80n1,U1. â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡*[h,n1] â\93\9b[p]W.U1 → ∀n2. n1 ≤ n2 →
+ â\88\83â\88\83U2. â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡*[h,n2] â\93\9b[p]W.U2 & â\9d¨G,L.â\93\9bWâ\9d© â\8a¢ U1 â\9e¡*[h,n2-n1] U2.
#h #a #G #L #W #p #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by cpms_abst_dx_le_aaa/
lemma cnv_appl_ge (h) (a) (n1) (p) (G) (L):
∀n2. n1 ≤ n2 → ad a n2 →
- â\88\80V. â¦\83G,Lâ¦\84 â\8a¢ V ![h,a] â\86\92 â\88\80T. â¦\83G,Lâ¦\84 ⊢ T ![h,a] →
- â\88\80X. â¦\83G,Lâ¦\84 â\8a¢ V â\9e¡*[1,h] X â\86\92 â\88\80W. â¦\83G,Lâ¦\84 â\8a¢ W â\9e¡*[h] X →
- â\88\80U. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n1,h] â\93\9b{p}W.U â\86\92 â¦\83G,Lâ¦\84 ⊢ ⓐV.T ![h,a].
+ â\88\80V. â\9d¨G,Lâ\9d© â\8a¢ V ![h,a] â\86\92 â\88\80T. â\9d¨G,Lâ\9d© ⊢ T ![h,a] →
+ â\88\80X. â\9d¨G,Lâ\9d© â\8a¢ V â\9e¡*[h,1] X â\86\92 â\88\80W. â\9d¨G,Lâ\9d© â\8a¢ W â\9e¡*[h,0] X →
+ â\88\80U. â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡*[h,n1] â\93\9b[p]W.U â\86\92 â\9d¨G,Lâ\9d© ⊢ ⓐV.T ![h,a].
#h #a #n1 #p #G #L #n2 #Hn12 #Ha #V #HV #T #HT #X #HVX #W #HW #X #HTX
elim (cnv_fwd_cpms_abst_dx_le … HT … HTX … Hn12) #U #HTU #_ -n1
/4 width=11 by cnv_appl, cpms_bind, cpms_cprs_trans/