lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
Q 0 T2 →
- (â\88\80n1,n2,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n1, h] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
- â\88\80n,T1. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 → Q n T1.
+ (â\88\80n1,n2,T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n1,h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[n2,h] T2 → Q n2 T → Q (n1+n2) T1) →
+ â\88\80n,T1. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n,h] 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¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n1, h] T â\86\92 Q n1 T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[n2, h] T2 → Q (n1+n2) T2) →
- â\88\80n,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 → Q n T2.
+ (â\88\80n1,n2,T,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n1,h] T â\86\92 Q n1 T â\86\92 â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[n2,h] T2 → Q (n1+n2) T2) →
+ â\88\80n,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n,h] T2 → Q n T2.
#h #G #L #T1 #Q @ltc_ind_dx_refl //
qed-.
(* Basic_1: includes: pr1_pr0 *)
(* Basic_1: uses: pr3_pr2 *)
(* Basic_2A1: includes: cpr_cprs *)
-lemma cpm_cpms (h) (G) (L): â\88\80n,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n, h] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T2.
+lemma cpm_cpms (h) (G) (L): â\88\80n,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n,h] T2.
/2 width=1 by ltc_rc/ qed.
-lemma cpms_step_sn (h) (G) (L): â\88\80n1,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[n1, h] T →
- â\88\80n2,T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[n2, h] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n1+n2, h] T2.
+lemma cpms_step_sn (h) (G) (L): â\88\80n1,T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[n1,h] T →
+ â\88\80n2,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[n2,h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n1+n2,h] T2.
/2 width=3 by ltc_sn/ qed-.
-lemma cpms_step_dx (h) (G) (L): â\88\80n1,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n1, h] T →
- â\88\80n2,T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[n2, h] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n1+n2, h] T2.
+lemma cpms_step_dx (h) (G) (L): â\88\80n1,T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n1,h] T →
+ â\88\80n2,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡[n2,h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n1+n2,h] T2.
/2 width=3 by ltc_dx/ qed-.
(* Basic_2A1: uses: cprs_bind_dx *)
lemma cpms_bind_dx (n) (h) (G) (L):
- â\88\80V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ➡[h] V2 →
- â\88\80I,T1,T2. â¦\83G, L.â\93\91{I}V1â¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 →
- â\88\80p. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â\9e¡*[n, h] â\93\91{p,I}V2.T2.
+ â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h] V2 →
+ â\88\80I,T1,T2. â\9dªG,L.â\93\91[I]V1â\9d« â\8a¢ T1 â\9e¡*[n,h] T2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â\9e¡*[n,h] â\93\91[p,I]V2.T2.
#n #h #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 (n) (h) (G) (L):
- â\88\80V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ➡[h] V2 →
- â\88\80T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 →
- â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.T1 â\9e¡*[n, h] ⓐV2.T2.
+ â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h] V2 →
+ â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n,h] T2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.T1 â\9e¡*[n,h] ⓐV2.T2.
#n #h #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 (n) (h) (G) (L):
- â\88\80T1,T. â¬\86*[1] T ≘ T1 →
- â\88\80V,T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[n, h] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ +â\93\93V.T1 â\9e¡*[n, h] T2.
+ â\88\80T1,T. â\87§*[1] T ≘ T1 →
+ â\88\80V,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[n,h] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ +â\93\93V.T1 â\9e¡*[n,h] T2.
#n #h #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 (n) (h) (G) (L):
- â\88\80T2,T. â¬\86*[1] T2 ≘ T →
- â\88\80V,T1. â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ +â\93\93V.T1 â\9e¡*[n, h] T2.
+ â\88\80T2,T. â\87§*[1] T2 ≘ T →
+ â\88\80V,T1. â\9dªG,L.â\93\93Vâ\9d« â\8a¢ T1 â\9e¡*[n,h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ +â\93\93V.T1 â\9e¡*[n,h] T2.
#n #h #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 (n) (h) (G) (L):
- â\88\80T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 →
- â\88\80V. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dV.T1 â\9e¡*[n, h] T2.
+ â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[n,h] T2 →
+ â\88\80V. â\9dªG,Lâ\9d« â\8a¢ â\93\9dV.T1 â\9e¡*[n,h] T2.
#n #h #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 (n) (h) (G) (L):
- â\88\80U1,U2. â¦\83G, Lâ¦\84 â\8a¢ U1 â\9e¡*[n, h] U2 →
- â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dU1.T â\9e¡*[â\86\91n, h] U2.
+ â\88\80U1,U2. â\9dªG,Lâ\9d« â\8a¢ U1 â\9e¡*[n,h] U2 →
+ â\88\80T. â\9dªG,Lâ\9d« â\8a¢ â\93\9dU1.T â\9e¡*[â\86\91n,h] U2.
#n #h #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 (n) (h) (G) (L):
- â\88\80V1,V2. â¦\83G, Lâ¦\84 ⊢ V1 ➡[h] V2 →
- â\88\80W1,W2. â¦\83G, Lâ¦\84 ⊢ W1 ➡[h] W2 →
- â\88\80T1,T2. â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 →
- â\88\80p. â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{p}W1.T1 â\9e¡*[n, h] â\93\93{p}ⓝW2.V2.T2.
+ â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ➡[h] V2 →
+ â\88\80W1,W2. â\9dªG,Lâ\9d« ⊢ W1 ➡[h] W2 →
+ â\88\80T1,T2. â\9dªG,L.â\93\9bW1â\9d« â\8a¢ T1 â\9e¡*[n,h] T2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\9b[p]W1.T1 â\9e¡*[n,h] â\93\93[p]ⓝW2.V2.T2.
#n #h #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 (n) (h) (G) (L):
- â\88\80V1,V. â¦\83G, Lâ¦\84 ⊢ V1 ➡[h] V →
- â\88\80V2. â¬\86*[1] V ≘ V2 →
- â\88\80W1,W2. â¦\83G, Lâ¦\84 ⊢ W1 ➡[h] W2 →
- â\88\80T1,T2. â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡*[n, h] T2 →
- â\88\80p. â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{p}W1.T1 â\9e¡*[n, h] â\93\93{p}W2.ⓐV2.T2.
+ â\88\80V1,V. â\9dªG,Lâ\9d« ⊢ V1 ➡[h] V →
+ â\88\80V2. â\87§*[1] V ≘ V2 →
+ â\88\80W1,W2. â\9dªG,Lâ\9d« ⊢ W1 ➡[h] W2 →
+ â\88\80T1,T2. â\9dªG,L.â\93\93W1â\9d« â\8a¢ T1 â\9e¡*[n,h] T2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\93[p]W1.T1 â\9e¡*[n,h] â\93\93[p]W2.ⓐV2.T2.
#n #h #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) (n):
- â\88\80s. â¦\83G,Lâ¦\84 ⊢ ⋆s ➡*[n,h] ⋆((next h)^n s).
+ â\88\80s. â\9dªG,Lâ\9d« ⊢ ⋆s ➡*[n,h] ⋆((next h)^n s).
#h #G #L #n elim n -n [ // ]
-#n #IH #s <plus_SO
+#n #IH #s <plus_SO_dx
/3 width=3 by cpms_step_dx, cpm_sort/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpms_inv_sort1 (n) (h) (G) (L): â\88\80X2,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â\9e¡*[n, h] X2 → X2 = ⋆(((next h)^n) s).
+lemma cpms_inv_sort1 (n) (h) (G) (L): â\88\80X2,s. â\9dªG,Lâ\9d« â\8a¢ â\8b\86s â\9e¡*[n,h] X2 → X2 = ⋆(((next h)^n) s).
#n #h #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 (n) (h) (G):
+ ∀X2,i. ❪G,⋆❫ ⊢ #i ➡*[n,h] X2 → ∧∧ X2 = #i & n = 0.
+#n #h #G #X2 #i #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_lref1_ctop … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_inv_zero1_unit (n) (h) (I) (K) (G):
+ ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ➡*[n,h] X2 → ∧∧ X2 = #0 & n = 0.
+#n #h #I #G #K #X2 #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_zero1_unit … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_inv_gref1 (n) (h) (G) (L):
+ ∀X2,l. ❪G,L❫ ⊢ §l ➡*[n,h] X2 → ∧∧ X2 = §l & n = 0.
+#n #h #G #L #X2 #l #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_gref1 … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
lemma cpms_inv_cast1 (h) (n) (G) (L):
- â\88\80W1,T1,X2. â¦\83G, Lâ¦\84 ⊢ ⓝW1.T1 ➡*[n,h] X2 →
- â\88¨â\88¨ â\88\83â\88\83W2,T2. â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡*[n,h] W2 & â¦\83G, Lâ¦\84 ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
- | â¦\83G, Lâ¦\84 ⊢ T1 ➡*[n,h] X2
- | â\88\83â\88\83m. â¦\83G, Lâ¦\84 ⊢ W1 ➡*[m,h] X2 & n = ↑m.
+ â\88\80W1,T1,X2. â\9dªG,Lâ\9d« ⊢ ⓝW1.T1 ➡*[n,h] X2 →
+ â\88¨â\88¨ â\88\83â\88\83W2,T2. â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡*[n,h] W2 & â\9dªG,Lâ\9d« ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
+ | â\9dªG,Lâ\9d« ⊢ T1 ➡*[n,h] X2
+ | â\88\83â\88\83m. â\9dªG,Lâ\9d« ⊢ W1 ➡*[m,h] 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 #_ * [ * || * ]
projects / helm.git / blobdiff