(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_3_5.ma".
include "ground_2/xoa/ex_5_7.ma".
include "basic_2/rt_transition/cpm_lsubr.ma".
include "basic_2/rt_computation/cpms_drops.ma".
(* Main properties **********************************************************)
(* Basic_2A1: includes: cprs_bind *)
-theorem cpms_bind (n) (h) (G) (L):
- â\88\80I,V1,T1,T2. â¦\83G,L.â\93\91{I}V1â¦\84 â\8a¢ T1 â\9e¡*[n,h] T2 →
- â\88\80V2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡*[h] V2 →
- 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.
-#n #h #G #L #I #V1 #T1 #T2 #HT12 #V2 #H @(cprs_ind_dx … H) -V2
+theorem cpms_bind (h) (n) (G) (L):
+ â\88\80I,V1,T1,T2. â\9dªG,L.â\93\91[I]V1â\9d« â\8a¢ T1 â\9e¡*[h,n] T2 →
+ â\88\80V2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡*[h,0] V2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â\9e¡*[h,n] â\93\91[p,I]V2.T2.
+#h #n #G #L #I #V1 #T1 #T2 #HT12 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_bind_dx/
| #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
/3 width=3 by cpr_pair_sn, cpms_step_dx/
]
qed.
-theorem cpms_appl (n) (h) (G) (L):
- â\88\80T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[n,h] T2 →
- â\88\80V1,V2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡*[h] V2 →
- â¦\83G,Lâ¦\84 â\8a¢ â\93\90V1.T1 â\9e¡*[n,h] ⓐV2.T2.
-#n #h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
+theorem cpms_appl (h) (n) (G) (L):
+ â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h,n] T2 →
+ â\88\80V1,V2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡*[h,0] V2 →
+ â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.T1 â\9e¡*[h,n] ⓐV2.T2.
+#h #n #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_appl_dx/
| #V #V2 #_ #HV2 #IH >(plus_n_O … n) -HT12
/3 width=3 by cpr_pair_sn, cpms_step_dx/
qed.
(* Basic_2A1: includes: cprs_beta_rc *)
-theorem cpms_beta_rc (n) (h) (G) (L):
- â\88\80V1,V2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V2 →
- â\88\80W1,T1,T2. â¦\83G,L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡*[n,h] T2 →
- â\88\80W2. â¦\83G,Lâ¦\84 â\8a¢ W1 â\9e¡*[h] W2 →
- 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.
-#n #h #G #L #V1 #V2 #HV12 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
+theorem cpms_beta_rc (h) (n) (G) (L):
+ â\88\80V1,V2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h,0] V2 →
+ â\88\80W1,T1,T2. â\9dªG,L.â\93\9bW1â\9d« â\8a¢ T1 â\9e¡*[h,n] T2 →
+ â\88\80W2. â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡*[h,0] W2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\9b[p]W1.T1 â\9e¡*[h,n] â\93\93[p]ⓝW2.V2.T2.
+#h #n #G #L #V1 #V2 #HV12 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
[ /2 width=1 by cpms_beta_dx/
| #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
/4 width=3 by cpr_pair_sn, cpms_step_dx/
qed.
(* Basic_2A1: includes: cprs_beta *)
-theorem cpms_beta (n) (h) (G) (L):
- â\88\80W1,T1,T2. â¦\83G,L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡*[n,h] T2 →
- â\88\80W2. â¦\83G,Lâ¦\84 â\8a¢ W1 â\9e¡*[h] W2 →
- â\88\80V1,V2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡*[h] V2 →
- 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.
-#n #h #G #L #W1 #T1 #T2 #HT12 #W2 #HW12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
+theorem cpms_beta (h) (n) (G) (L):
+ â\88\80W1,T1,T2. â\9dªG,L.â\93\9bW1â\9d« â\8a¢ T1 â\9e¡*[h,n] T2 →
+ â\88\80W2. â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡*[h,0] W2 →
+ â\88\80V1,V2. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡*[h,0] V2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\9b[p]W1.T1 â\9e¡*[h,n] â\93\93[p]ⓝW2.V2.T2.
+#h #n #G #L #W1 #T1 #T2 #HT12 #W2 #HW12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_beta_rc/
| #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
/4 width=5 by cpms_step_dx, cpr_pair_sn, cpm_cast/
qed.
(* Basic_2A1: includes: cprs_theta_rc *)
-theorem cpms_theta_rc (n) (h) (G) (L):
- â\88\80V1,V. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V â\86\92 â\88\80V2. â\87§*[1] V ≘ V2 →
- â\88\80W1,T1,T2. â¦\83G,L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡*[n,h] T2 →
- â\88\80W2. â¦\83G,Lâ¦\84 â\8a¢ W1 â\9e¡*[h] W2 →
- 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.
-#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
+theorem cpms_theta_rc (h) (n) (G) (L):
+ â\88\80V1,V. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡[h,0] V â\86\92 â\88\80V2. â\87§[1] V ≘ V2 →
+ â\88\80W1,T1,T2. â\9dªG,L.â\93\93W1â\9d« â\8a¢ T1 â\9e¡*[h,n] T2 →
+ â\88\80W2. â\9dªG,Lâ\9d« â\8a¢ W1 â\9e¡*[h,0] W2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\93[p]W1.T1 â\9e¡*[h,n] â\93\93[p]W2.ⓐV2.T2.
+#h #n #G #L #V1 #V #HV1 #V2 #HV2 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
[ /2 width=3 by cpms_theta_dx/
| #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
/3 width=3 by cpr_pair_sn, cpms_step_dx/
qed.
(* Basic_2A1: includes: cprs_theta *)
-theorem cpms_theta (n) (h) (G) (L):
- ∀V,V2. ⇧*[1] V ≘ V2 → ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
- â\88\80T1,T2. â¦\83G,L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡*[n,h] T2 →
- â\88\80V1. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡*[h] V →
- 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.
-#n #h #G #L #V #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #HT12 #V1 #H @(cprs_ind_sn … H) -V1
+theorem cpms_theta (h) (n) (G) (L):
+ ∀V,V2. ⇧[1] V ≘ V2 → ∀W1,W2. ❪G,L❫ ⊢ W1 ➡*[h,0] W2 →
+ â\88\80T1,T2. â\9dªG,L.â\93\93W1â\9d« â\8a¢ T1 â\9e¡*[h,n] T2 →
+ â\88\80V1. â\9dªG,Lâ\9d« â\8a¢ V1 â\9e¡*[h,0] V →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\90V1.â\93\93[p]W1.T1 â\9e¡*[h,n] â\93\93[p]W2.ⓐV2.T2.
+#h #n #G #L #V #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #HT12 #V1 #H @(cprs_ind_sn … H) -V1
[ /2 width=3 by cpms_theta_rc/
| #V1 #V0 #HV10 #_ #IH #p >(plus_O_n … n) -HT12
/3 width=3 by cpr_pair_sn, cpms_step_sn/
(* Basic_2A1: uses: lstas_scpds_trans scpds_strap2 *)
theorem cpms_trans (h) (G) (L):
- ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T →
- ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2.
+ ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡*[h,n1] T →
+ ∀n2,T2. ❪G,L❫ ⊢ T ➡*[h,n2] T2 → ❪G,L❫ ⊢ T1 ➡*[h,n1+n2] T2.
/2 width=3 by ltc_trans/ qed-.
(* Basic_2A1: uses: scpds_cprs_trans *)
-theorem cpms_cprs_trans (n) (h) (G) (L):
- â\88\80T1,T. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[n,h] T →
- â\88\80T2. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[h] T2 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[n,h] T2.
-#n #h #G #L #T1 #T #HT1 #T2 #HT2 >(plus_n_O … n)
+theorem cpms_cprs_trans (h) (n) (G) (L):
+ â\88\80T1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h,n] T →
+ â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[h,0] T2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h,n] T2.
+#h #n #G #L #T1 #T #HT1 #T2 #HT2 >(plus_n_O … n)
/2 width=3 by cpms_trans/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma cpms_inv_appl_sn (n) (h) (G) (L):
- ∀V1,T1,X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] X2 →
- ∨∨ ∃∃V2,T2.
- ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 &
- X2 = ⓐV2.T2
- | ∃∃n1,n2,p,W,T.
- ⦃G,L⦄ ⊢ T1 ➡*[n1,h] ⓛ{p}W.T & ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[n2,h] X2 &
- n1 + n2 = n
- | ∃∃n1,n2,p,V0,V2,V,T.
- ⦃G,L⦄ ⊢ V1 ➡*[h] V0 & ⇧*[1] V0 ≘ V2 &
- ⦃G,L⦄ ⊢ T1 ➡*[n1,h] ⓓ{p}V.T & ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[n2,h] X2 &
- n1 + n2 = n.
-#n #h #G #L #V1 #T1 #U2 #H
+lemma cpms_inv_appl_sn (h) (n) (G) (L):
+ ∀V1,T1,X2. ❪G,L❫ ⊢ ⓐV1.T1 ➡*[h,n] X2 →
+ ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡*[h,0] V2 & ❪G,L❫ ⊢ T1 ➡*[h,n] T2 & X2 = ⓐV2.T2
+ | ∃∃n1,n2,p,W,T. ❪G,L❫ ⊢ T1 ➡*[h,n1] ⓛ[p]W.T & ❪G,L❫ ⊢ ⓓ[p]ⓝW.V1.T ➡*[h,n2] X2 & n1 + n2 = n
+ | ∃∃n1,n2,p,V0,V2,V,T. ❪G,L❫ ⊢ V1 ➡*[h,0] V0 & ⇧[1] V0 ≘ V2 & ❪G,L❫ ⊢ T1 ➡*[h,n1] ⓓ[p]V.T & ❪G,L❫ ⊢ ⓓ[p]V.ⓐV2.T ➡*[h,n2] X2 & n1 + n2 = n.
+#h #n #G #L #V1 #T1 #U2 #H
@(cpms_ind_dx … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
#n1 #n2 #U #U2 #_ * *
[ #V0 #T0 #HV10 #HT10 #H #HU2 destruct
]
qed-.
-lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2 →
- ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G,L⦄ ⊢ T ➡*[n2,h] T2.
+lemma cpms_inv_plus (h) (G) (L):
+ ∀n1,n2,T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,n1+n2] T2 →
+ ∃∃T. ❪G,L❫ ⊢ T1 ➡*[h,n1] T & ❪G,L❫ ⊢ T ➡*[h,n2] T2.
#h #G #L #n1 elim n1 -n1 /2 width=3 by ex2_intro/
#n1 #IH #n2 #T1 #T2 <plus_S1 #H
elim (cpms_inv_succ_sn … H) -H #T0 #HT10 #HT02
(* Advanced main properties *************************************************)
-theorem cpms_cast (n) (h) (G) (L):
- ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
- ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 →
- ⦃G,L⦄ ⊢ ⓝU1.T1 ➡*[n,h] ⓝU2.T2.
-#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 -n
+theorem cpms_cast (h) (n) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,n] T2 →
+ ∀U1,U2. ❪G,L❫ ⊢ U1 ➡*[h,n] U2 →
+ ❪G,L❫ ⊢ ⓝU1.T1 ➡*[h,n] ⓝU2.T2.
+#h #n #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 -n
[ /3 width=3 by cpms_cast_sn/
| #n1 #n2 #T1 #T #HT1 #_ #IH #U1 #U2 #H
elim (cpms_inv_plus … H) -H #U #HU1 #HU2
qed.
theorem cpms_trans_swap (h) (G) (L) (T1):
- â\88\80n1,T. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[n1,h] T â\86\92 â\88\80n2,T2. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n2,h] T2 →
- â\88\83â\88\83T0. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[n2,h] T0 & â¦\83G,Lâ¦\84 â\8a¢ T0 â\9e¡*[n1,h] T2.
+ â\88\80n1,T. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h,n1] T â\86\92 â\88\80n2,T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[h,n2] T2 →
+ â\88\83â\88\83T0. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡*[h,n2] T0 & â\9dªG,Lâ\9d« â\8a¢ T0 â\9e¡*[h,n1] T2.
#h #G #L #T1 #n1 #T #HT1 #n2 #T2 #HT2
lapply (cpms_trans … HT1 … HT2) -T <commutative_plus #HT12
/2 width=1 by cpms_inv_plus/
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