lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
Q 0 T2 →
- (â\88\80n1,n2,T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h,n1] T â\86\92 â\9dªG,Lâ\9d« ⊢ T ➡*[h,n2] T2 → Q n2 T → Q (n1+n2) T1) →
- â\88\80n,T1. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n] T2 → Q n T1.
+ (â\88\80n1,n2,T1,T. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n1] T â\86\92 â\9d¨G,Lâ\9d© ⊢ T ➡*[h,n2] T2 → Q n2 T → Q (n1+n2) T1) →
+ â\88\80n,T1. â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n] T2 → Q n T1.
#h #G #L #T2 #Q @ltc_ind_sn_refl //
qed-.
lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
Q 0 T1 →
- (â\88\80n1,n2,T,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h,n1] T â\86\92 Q n1 T â\86\92 â\9dªG,Lâ\9d« ⊢ T ➡[h,n2] T2 → Q (n1+n2) T2) →
- â\88\80n,T2. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n] T2 → Q n T2.
+ (â\88\80n1,n2,T,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,n1] T â\86\92 Q n1 T â\86\92 â\9d¨G,Lâ\9d© ⊢ T ➡[h,n2] T2 → Q (n1+n2) T2) →
+ â\88\80n,T2. â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n] T2 → Q n T2.
#h #G #L #T1 #Q @ltc_ind_dx_refl //
qed-.
(* Basic_1: uses: pr3_pr2 *)
(* Basic_2A1: includes: cpr_cprs *)
lemma cpm_cpms (h) (G) (L):
- â\88\80n,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n] T2.
+ â\88\80n,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n] T2.
/2 width=1 by ltc_rc/ qed.
lemma cpms_step_sn (h) (G) (L):
- â\88\80n1,T1,T. â\9dªG,Lâ\9d« ⊢ T1 ➡[h,n1] T →
- â\88\80n2,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[h,n2] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n1+n2] T2.
+ â\88\80n1,T1,T. â\9d¨G,Lâ\9d© ⊢ T1 ➡[h,n1] T →
+ â\88\80n2,T2. â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡*[h,n2] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n1+n2] T2.
/2 width=3 by ltc_sn/ qed-.
lemma cpms_step_dx (h) (G) (L):
- â\88\80n1,T1,T. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n1] T →
- â\88\80n2,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[h,n2] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n1+n2] T2.
+ â\88\80n1,T1,T. â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n1] T →
+ â\88\80n2,T2. â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡[h,n2] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n1+n2] T2.
/2 width=3 by ltc_dx/ qed-.
(* Basic_2A1: uses: cprs_bind_dx *)
lemma cpms_bind_dx (h) (n) (G) (L):
- â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h,0] V2 →
- â\88\80I,T1,T2. â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ➡*[h,n] T2 →
- â\88\80p. â\9dªG,Lâ\9d« ⊢ ⓑ[p,I]V1.T1 ➡*[h,n] ⓑ[p,I]V2.T2.
+ â\88\80V1,V2. â\9d¨G,Lâ\9d© ⊢ V1 ➡[h,0] V2 →
+ â\88\80I,T1,T2. â\9d¨G,L.â\93\91[I]V1â\9d© ⊢ T1 ➡*[h,n] T2 →
+ â\88\80p. â\9d¨G,Lâ\9d© ⊢ ⓑ[p,I]V1.T1 ➡*[h,n] ⓑ[p,I]V2.T2.
#h #n #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
lemma cpms_appl_dx (h) (n) (G) (L):
- â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h,0] V2 →
- â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n] T2 →
- â\9dªG,Lâ\9d« ⊢ ⓐV1.T1 ➡*[h,n] ⓐV2.T2.
+ â\88\80V1,V2. â\9d¨G,Lâ\9d© ⊢ V1 ➡[h,0] V2 →
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n] T2 →
+ â\9d¨G,Lâ\9d© ⊢ ⓐV1.T1 ➡*[h,n] ⓐV2.T2.
#h #n #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
qed.
lemma cpms_zeta (h) (n) (G) (L):
∀T1,T. ⇧[1] T ≘ T1 →
- â\88\80V,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[h,n] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ +ⓓV.T1 ➡*[h,n] T2.
+ â\88\80V,T2. â\9d¨G,Lâ\9d© â\8a¢ T â\9e¡*[h,n] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ +ⓓV.T1 ➡*[h,n] T2.
#h #n #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
/3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
qed.
(* Basic_2A1: uses: cprs_zeta *)
lemma cpms_zeta_dx (h) (n) (G) (L):
∀T2,T. ⇧[1] T2 ≘ T →
- â\88\80V,T1. â\9dªG,L.â\93\93Vâ\9d« â\8a¢ T1 â\9e¡*[h,n] T â\86\92 â\9dªG,Lâ\9d« ⊢ +ⓓV.T1 ➡*[h,n] T2.
+ â\88\80V,T1. â\9d¨G,L.â\93\93Vâ\9d© â\8a¢ T1 â\9e¡*[h,n] T â\86\92 â\9d¨G,Lâ\9d© ⊢ +ⓓV.T1 ➡*[h,n] T2.
#h #n #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
qed.
(* Basic_2A1: uses: cprs_eps *)
lemma cpms_eps (h) (n) (G) (L):
- â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n] T2 →
- â\88\80V. â\9dªG,Lâ\9d« ⊢ ⓝV.T1 ➡*[h,n] T2.
+ â\88\80T1,T2. â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n] T2 →
+ â\88\80V. â\9d¨G,Lâ\9d© ⊢ ⓝV.T1 ➡*[h,n] T2.
#h #n #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
qed.
lemma cpms_ee (h) (n) (G) (L):
- â\88\80U1,U2. â\9dªG,Lâ\9d« ⊢ U1 ➡*[h,n] U2 →
- â\88\80T. â\9dªG,Lâ\9d« ⊢ ⓝU1.T ➡*[h,↑n] U2.
+ â\88\80U1,U2. â\9d¨G,Lâ\9d© ⊢ U1 ➡*[h,n] U2 →
+ â\88\80T. â\9d¨G,Lâ\9d© ⊢ ⓝU1.T ➡*[h,↑n] U2.
#h #n #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
[ /3 width=1 by cpm_cpms, cpm_ee/
| #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
(* Basic_2A1: uses: cprs_beta_dx *)
lemma cpms_beta_dx (h) (n) (G) (L):
- â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h,0] V2 →
- â\88\80W1,W2. â\9dªG,Lâ\9d« ⊢ W1 ➡[h,0] W2 →
- â\88\80T1,T2. â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ➡*[h,n] T2 →
- â\88\80p. â\9dªG,Lâ\9d« ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[h,n] ⓓ[p]ⓝW2.V2.T2.
+ â\88\80V1,V2. â\9d¨G,Lâ\9d© ⊢ V1 ➡[h,0] V2 →
+ â\88\80W1,W2. â\9d¨G,Lâ\9d© ⊢ W1 ➡[h,0] W2 →
+ â\88\80T1,T2. â\9d¨G,L.â\93\9bW1â\9d© ⊢ T1 ➡*[h,n] T2 →
+ â\88\80p. â\9d¨G,Lâ\9d© ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[h,n] ⓓ[p]ⓝW2.V2.T2.
#h #n #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
/4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
qed.
(* Basic_2A1: uses: cprs_theta_dx *)
lemma cpms_theta_dx (h) (n) (G) (L):
- â\88\80V1,V. â\9dªG,Lâ\9d« ⊢ V1 ➡[h,0] V →
+ â\88\80V1,V. â\9d¨G,Lâ\9d© ⊢ V1 ➡[h,0] V →
∀V2. ⇧[1] V ≘ V2 →
- â\88\80W1,W2. â\9dªG,Lâ\9d« ⊢ W1 ➡[h,0] W2 →
- â\88\80T1,T2. â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ➡*[h,n] T2 →
- â\88\80p. â\9dªG,Lâ\9d« ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[h,n] ⓓ[p]W2.ⓐV2.T2.
+ â\88\80W1,W2. â\9d¨G,Lâ\9d© ⊢ W1 ➡[h,0] W2 →
+ â\88\80T1,T2. â\9d¨G,L.â\93\93W1â\9d© ⊢ T1 ➡*[h,n] T2 →
+ â\88\80p. â\9d¨G,Lâ\9d© ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[h,n] ⓓ[p]W2.ⓐV2.T2.
#h #n #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
/4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
qed.
(* Advanced properties ******************************************************)
lemma cpms_sort (h) (G) (L):
- â\88\80n,s. â\9dªG,Lâ\9d« ⊢ ⋆s ➡*[h,n] ⋆((next h)^n s).
+ â\88\80n,s. â\9d¨G,Lâ\9d© ⊢ ⋆s ➡*[h,n] ⋆((next h)^n s).
#h #G #L #n elim n -n [ // ]
#n #IH #s <plus_SO_dx
/3 width=3 by cpms_step_dx, cpm_sort/
(* Basic inversion lemmas ***************************************************)
lemma cpms_inv_sort1 (h) (n) (G) (L):
- â\88\80X2,s. â\9dªG,Lâ\9d« ⊢ ⋆s ➡*[h,n] X2 → X2 = ⋆(((next h)^n) s).
+ â\88\80X2,s. â\9d¨G,Lâ\9d© ⊢ ⋆s ➡*[h,n] X2 → X2 = ⋆(((next h)^n) s).
#h #n #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
#n1 #n2 #X #X2 #_ #IH #HX2 destruct
elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
qed-.
lemma cpms_inv_lref1_ctop (h) (n) (G):
- â\88\80X2,i. â\9dªG,â\8b\86â\9d« ⊢ #i ➡*[h,n] X2 → ∧∧ X2 = #i & n = 0.
+ â\88\80X2,i. â\9d¨G,â\8b\86â\9d© ⊢ #i ➡*[h,n] X2 → ∧∧ X2 = #i & n = 0.
#h #n #G #X2 #i #H @(cpms_ind_dx … H) -X2
[ /2 width=1 by conj/
| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
qed-.
lemma cpms_inv_zero1_unit (h) (n) (I) (K) (G):
- â\88\80X2. â\9dªG,K.â\93¤[I]â\9d« ⊢ #0 ➡*[h,n] X2 → ∧∧ X2 = #0 & n = 0.
+ â\88\80X2. â\9d¨G,K.â\93¤[I]â\9d© ⊢ #0 ➡*[h,n] X2 → ∧∧ X2 = #0 & n = 0.
#h #n #I #G #K #X2 #H @(cpms_ind_dx … H) -X2
[ /2 width=1 by conj/
| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
qed-.
lemma cpms_inv_gref1 (h) (n) (G) (L):
- â\88\80X2,l. â\9dªG,Lâ\9d« ⊢ §l ➡*[h,n] X2 → ∧∧ X2 = §l & n = 0.
+ â\88\80X2,l. â\9d¨G,Lâ\9d© ⊢ §l ➡*[h,n] X2 → ∧∧ X2 = §l & n = 0.
#h #n #G #L #X2 #l #H @(cpms_ind_dx … H) -X2
[ /2 width=1 by conj/
| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
qed-.
lemma cpms_inv_cast1 (h) (n) (G) (L):
- â\88\80W1,T1,X2. â\9dªG,Lâ\9d« ⊢ ⓝW1.T1 ➡*[h,n] X2 →
- â\88¨â\88¨ â\88\83â\88\83W2,T2. â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡*[h,n] W2 & â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n] T2 & X2 = ⓝW2.T2
- | â\9dªG,Lâ\9d« ⊢ T1 ➡*[h,n] X2
- | â\88\83â\88\83m. â\9dªG,Lâ\9d« ⊢ W1 ➡*[h,m] X2 & n = ↑m.
+ â\88\80W1,T1,X2. â\9d¨G,Lâ\9d© ⊢ ⓝW1.T1 ➡*[h,n] X2 →
+ â\88¨â\88¨ â\88\83â\88\83W2,T2. â\9d¨G,Lâ\9d© â\8a¢ W1 â\9e¡*[h,n] W2 & â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n] T2 & X2 = ⓝW2.T2
+ | â\9d¨G,Lâ\9d© ⊢ T1 ➡*[h,n] X2
+ | â\88\83â\88\83m. â\9d¨G,Lâ\9d© ⊢ W1 ➡*[h,m] X2 & n = ↑m.
#h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2
[ /3 width=5 by or3_intro0, ex3_2_intro/
| #n1 #n2 #X #X2 #_ * [ * || * ]