(* Inversion lemmas with restricted rt-transition for terms *****************)
lemma cnv_cpr_teqx_fwd_refl (h) (a) (G) (L):
- ∀T1,T2. ❪G,L❫ ⊢ T1 ➡[h] T2 → T1 ≛ T2 → ❪G,L❫ ⊢ T1 ![h,a] → T1 = T2.
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡[h,0] T2 → T1 ≛ T2 → ❪G,L❫ ⊢ T1 ![h,a] → T1 = T2.
#h #a #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ //
| #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
lemma cpm_teqx_inv_bind_sn (h) (a) (n) (p) (I) (G) (L):
∀V,T1. ❪G,L❫ ⊢ ⓑ[p,I]V.T1 ![h,a] →
- ∀X. ❪G,L❫ ⊢ ⓑ[p,I]V.T1 ➡[n,h] X → ⓑ[p,I]V.T1 ≛ X →
- ∃∃T2. ❪G,L❫ ⊢ V ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ[p,I]V.T2.
+ ∀X. ❪G,L❫ ⊢ ⓑ[p,I]V.T1 ➡[h,n] X → ⓑ[p,I]V.T1 ≛ X →
+ ∃∃T2. ❪G,L❫ ⊢ V ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ➡[h,n] T2 & T1 ≛ T2 & X = ⓑ[p,I]V.T2.
#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_bind1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
lemma cpm_teqx_inv_appl_sn (h) (a) (n) (G) (L):
∀V,T1. ❪G,L❫ ⊢ ⓐV.T1 ![h,a] →
- ∀X. ❪G,L❫ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛ X →
- ∃∃m,q,W,U1,T2. ad a m & ❪G,L❫ ⊢ V ![h,a] & ❪G,L❫ ⊢ V ➡*[1,h] W & ❪G,L❫ ⊢ T1 ➡*[m,h] ⓛ[q]W.U1
- & ❪G,L❫⊢ T1 ![h,a] & ❪G,L❫ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
+ ∀X. ❪G,L❫ ⊢ ⓐV.T1 ➡[h,n] X → ⓐV.T1 ≛ X →
+ ∃∃m,q,W,U1,T2. ad a m & ❪G,L❫ ⊢ V ![h,a] & ❪G,L❫ ⊢ V ➡*[h,1] W & ❪G,L❫ ⊢ T1 ➡*[h,m] ⓛ[q]W.U1
+ & ❪G,L❫⊢ T1 ![h,a] & ❪G,L❫ ⊢ T1 ➡[h,n] T2 & T1 ≛ T2 & X = ⓐV.T2.
#h #a #n #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_appl1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
lemma cpm_teqx_inv_cast_sn (h) (a) (n) (G) (L):
∀U1,T1. ❪G,L❫ ⊢ ⓝU1.T1 ![h,a] →
- ∀X. ❪G,L❫ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
- ∃∃U0,U2,T2. ❪G,L❫ ⊢ U1 ➡*[h] U0 & ❪G,L❫ ⊢ T1 ➡*[1,h] U0
- & ❪G,L❫ ⊢ U1 ![h,a] & ❪G,L❫ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
- & ❪G,L❫ ⊢ T1 ![h,a] & ❪G,L❫ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
+ ∀X. ❪G,L❫ ⊢ ⓝU1.T1 ➡[h,n] X → ⓝU1.T1 ≛ X →
+ ∃∃U0,U2,T2. ❪G,L❫ ⊢ U1 ➡*[h,0] U0 & ❪G,L❫ ⊢ T1 ➡*[h,1] U0
+ & ❪G,L❫ ⊢ U1 ![h,a] & ❪G,L❫ ⊢ U1 ➡[h,n] U2 & U1 ≛ U2
+ & ❪G,L❫ ⊢ T1 ![h,a] & ❪G,L❫ ⊢ T1 ➡[h,n] T2 & T1 ≛ T2 & X = ⓝU2.T2.
#h #a #n #G #L #U1 #T1 #H0 #X #H1 #H2
elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
[ #U2 #T2 #HU12 #HT12 #H destruct
lemma cpm_teqx_inv_bind_dx (h) (a) (n) (p) (I) (G) (L):
∀X. ❪G,L❫ ⊢ X ![h,a] →
- ∀V,T2. ❪G,L❫ ⊢ X ➡[n,h] ⓑ[p,I]V.T2 → X ≛ ⓑ[p,I]V.T2 →
- ∃∃T1. ❪G,L❫ ⊢ V ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ[p,I]V.T1.
+ ∀V,T2. ❪G,L❫ ⊢ X ➡[h,n] ⓑ[p,I]V.T2 → X ≛ ⓑ[p,I]V.T2 →
+ ∃∃T1. ❪G,L❫ ⊢ V ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ➡[h,n] T2 & T1 ≛ T2 & X = ⓑ[p,I]V.T1.
#h #a #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
elim (teqx_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
elim (cpm_teqx_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
(∀I,L. n = 0 → Q L (⓪[I]) (⓪[I])) →
(∀L,s. n = 1 → Q L (⋆s) (⋆(⫯[h]s))) →
(∀p,I,L,V,T1. ❪G,L❫⊢ V![h,a] → ❪G,L.ⓑ[I]V❫⊢T1![h,a] →
- ∀T2. ❪G,L.ⓑ[I]V❫ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀T2. ❪G,L.ⓑ[I]V❫ ⊢ T1 ➡[h,n] T2 → T1 ≛ T2 →
Q (L.ⓑ[I]V) T1 T2 → Q L (ⓑ[p,I]V.T1) (ⓑ[p,I]V.T2)
) →
(∀m. ad a m →
- ∀L,V. ❪G,L❫ ⊢ V ![h,a] → ∀W. ❪G,L❫ ⊢ V ➡*[1,h] W →
- ∀p,T1,U1. ❪G,L❫ ⊢ T1 ➡*[m,h] ⓛ[p]W.U1 → ❪G,L❫⊢ T1 ![h,a] →
- ∀T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀L,V. ❪G,L❫ ⊢ V ![h,a] → ∀W. ❪G,L❫ ⊢ V ➡*[h,1] W →
+ ∀p,T1,U1. ❪G,L❫ ⊢ T1 ➡*[h,m] ⓛ[p]W.U1 → ❪G,L❫⊢ T1 ![h,a] →
+ ∀T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → T1 ≛ T2 →
Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
) →
- (∀L,U0,U1,T1. ❪G,L❫ ⊢ U1 ➡*[h] U0 → ❪G,L❫ ⊢ T1 ➡*[1,h] U0 →
- ∀U2. ❪G,L❫ ⊢ U1 ![h,a] → ❪G,L❫ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
- ∀T2. ❪G,L❫ ⊢ T1 ![h,a] → ❪G,L❫ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ (∀L,U0,U1,T1. ❪G,L❫ ⊢ U1 ➡*[h,0] U0 → ❪G,L❫ ⊢ T1 ➡*[h,1] U0 →
+ ∀U2. ❪G,L❫ ⊢ U1 ![h,a] → ❪G,L❫ ⊢ U1 ➡[h,n] U2 → U1 ≛ U2 →
+ ∀T2. ❪G,L❫ ⊢ T1 ![h,a] → ❪G,L❫ ⊢ T1 ➡[h,n] T2 → T1 ≛ T2 →
Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
) →
∀L,T1. ❪G,L❫ ⊢ T1 ![h,a] →
- ∀T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → Q L T1 T2.
+ ∀T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → T1 ≛ T2 → Q L T1 T2.
#h #a #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
@(insert_eq_0 … G) #F
@(fqup_wf_ind_eq (Ⓣ) … F L T1) -L -T1 -F
lemma cpm_teqx_free (h) (a) (n) (G) (L):
∀T1. ❪G,L❫ ⊢ T1 ![h,a] →
- ∀T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
- ∀F,K. ❪F,K❫ ⊢ T1 ➡[n,h] T2.
+ ∀T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → T1 ≛ T2 →
+ ∀F,K. ❪F,K❫ ⊢ T1 ➡[h,n] T2.
#h #a #n #G #L #T1 #H0 #T2 #H1 #H2
@(cpm_teqx_ind … H0 … H1 H2) -L -T1 -T2
[ #I #L #H #F #K destruct //
lemma cpm_teqx_inv_bind_sn_void (h) (a) (n) (p) (I) (G) (L):
∀V,T1. ❪G,L❫ ⊢ ⓑ[p,I]V.T1 ![h,a] →
- ∀X. ❪G,L❫ ⊢ ⓑ[p,I]V.T1 ➡[n,h] X → ⓑ[p,I]V.T1 ≛ X →
- ∃∃T2. ❪G,L❫ ⊢ V ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ![h,a] & ❪G,L.ⓧ❫ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ[p,I]V.T2.
+ ∀X. ❪G,L❫ ⊢ ⓑ[p,I]V.T1 ➡[h,n] X → ⓑ[p,I]V.T1 ≛ X →
+ ∃∃T2. ❪G,L❫ ⊢ V ![h,a] & ❪G,L.ⓑ[I]V❫ ⊢ T1 ![h,a] & ❪G,L.ⓧ❫ ⊢ T1 ➡[h,n] T2 & T1 ≛ T2 & X = ⓑ[p,I]V.T2.
#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_teqx_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
/3 width=5 by ex5_intro, cpm_teqx_free/