(* *)
(**************************************************************************)
-include "Basic-2/unfold/tpss_ltps.ma".
-include "Basic-2/unfold/ltpss.ma".
+include "Basic_2/unfold/tpss_ltps.ma".
+include "Basic_2/unfold/ltpss.ma".
(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
(* Properties concerning partial unfold on terms ****************************)
-lemma ltpss_tpss_trans_ge: â\88\80L1,L0,d1,e1. L1 [d1, e1] â\89«* L0 →
- â\88\80T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\89«* U2 →
- d1 + e1 â\89¤ d2 â\86\92 L1 â\8a¢ T2 [d2, e2] â\89«* U2.
+lemma ltpss_tpss_trans_ge: â\88\80L1,L0,d1,e1. L1 [d1, e1] â\96¶* L0 →
+ â\88\80T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\96¶* U2 →
+ d1 + e1 â\89¤ d2 â\86\92 L1 â\8a¢ T2 [d2, e2] â\96¶* U2.
#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0 //
#L #L0 #_ #HL0 #IHL #T2 #U2 #d2 #e2 #HTU2 #Hde1d2
-lapply (ltps_tpss_trans_ge … HL0 HTU2) -HL0 HTU2 /2/
+lapply (ltps_tpss_trans_ge … HL0 HTU2) -HL0 -HTU2 /2 width=1/
qed.
-lemma ltpss_tps_trans_ge: â\88\80L1,L0,d1,e1. L1 [d1, e1] â\89«* L0 →
- â\88\80T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\89« U2 →
- d1 + e1 â\89¤ d2 â\86\92 L1 â\8a¢ T2 [d2, e2] â\89«* U2.
+lemma ltpss_tps_trans_ge: â\88\80L1,L0,d1,e1. L1 [d1, e1] â\96¶* L0 →
+ â\88\80T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\96¶ U2 →
+ d1 + e1 â\89¤ d2 â\86\92 L1 â\8a¢ T2 [d2, e2] â\96¶* U2.
#L1 #L0 #d1 #e1 #HL10 #T2 #U2 #d2 #e2 #HTU2 #Hde1d2
-@(ltpss_tpss_trans_ge … HL10 … Hde1d2) /2/ (**) (* /3 width=6/ is too slow *)
+@(ltpss_tpss_trans_ge … HL10 … Hde1d2) /2 width=1/ (**) (* /3 width=6/ is too slow *)
qed.
-lemma ltpss_tpss_trans_eq: â\88\80L0,L1,d,e. L0 [d, e] â\89«* L1 →
- â\88\80T2,U2. L1 â\8a¢ T2 [d, e] â\89«* U2 â\86\92 L0 â\8a¢ T2 [d, e] â\89«* U2.
+lemma ltpss_tpss_trans_eq: â\88\80L0,L1,d,e. L0 [d, e] â\96¶* L1 →
+ â\88\80T2,U2. L1 â\8a¢ T2 [d, e] â\96¶* U2 â\86\92 L0 â\8a¢ T2 [d, e] â\96¶* U2.
#L0 #L1 #d #e #H @(ltpss_ind … H) -L1
-[ /2/
+[ /2 width=1/
| #L #L1 #_ #HL1 #IHL #T2 #U2 #HTU2
- lapply (ltps_tpss_trans_eq … HL1 HTU2) -HL1 HTU2 /2/
+ lapply (ltps_tpss_trans_eq … HL1 HTU2) -HL1 -HTU2 /2 width=1/
]
qed.
-lemma ltpss_tps_trans_eq: â\88\80L0,L1,d,e. L0 [d, e] â\89«* L1 →
- â\88\80T2,U2. L1 â\8a¢ T2 [d, e] â\89« U2 â\86\92 L0 â\8a¢ T2 [d, e] â\89«* U2.
-/3/ qed.
+lemma ltpss_tps_trans_eq: â\88\80L0,L1,d,e. L0 [d, e] â\96¶* L1 →
+ â\88\80T2,U2. L1 â\8a¢ T2 [d, e] â\96¶ U2 â\86\92 L0 â\8a¢ T2 [d, e] â\96¶* U2.
+/3 width=3/ qed.
lemma ltpss_tpss_conf_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2. d1 + e1 ≤ d2 →
- L0 â\8a¢ T2 [d2, e2] â\89«* U2 â\86\92 L0 [d1, e1] â\89«* L1 →
- L1 â\8a¢ T2 [d2, e2] â\89«* U2.
+ L0 â\8a¢ T2 [d2, e2] â\96¶* U2 â\86\92 L0 [d1, e1] â\96¶* L1 →
+ L1 â\8a¢ T2 [d2, e2] â\96¶* U2.
#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HTU2 #H @(ltpss_ind … H) -L1
[ //
| -HTU2 #L #L1 #_ #HL1 #IHL
- lapply (ltps_tpss_conf_ge … HL1 IHL) -HL1 IHL //
+ lapply (ltps_tpss_conf_ge … HL1 IHL) -HL1 -IHL //
]
qed.
lemma ltpss_tps_conf_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2. d1 + e1 ≤ d2 →
- L0 â\8a¢ T2 [d2, e2] â\89« U2 â\86\92 L0 [d1, e1] â\89«* L1 →
- L1 â\8a¢ T2 [d2, e2] â\89«* U2.
+ L0 â\8a¢ T2 [d2, e2] â\96¶ U2 â\86\92 L0 [d1, e1] â\96¶* L1 →
+ L1 â\8a¢ T2 [d2, e2] â\96¶* U2.
#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HTU2 #HL01
-@(ltpss_tpss_conf_ge … Hde1d2 … HL01) /2/ (**) (* /3 width=6/ is too slow *)
+@(ltpss_tpss_conf_ge … Hde1d2 … HL01) /2 width=1/ (**) (* /3 width=6/ is too slow *)
qed.
lemma ltpss_tpss_conf_eq: ∀L0,L1,T2,U2,d,e.
- L0 â\8a¢ T2 [d, e] â\89«* U2 â\86\92 L0 [d, e] â\89«* L1 →
- â\88\83â\88\83T. L1 â\8a¢ T2 [d, e] â\89«* T & L1 â\8a¢ U2 [d, e] â\89«* T.
+ L0 â\8a¢ T2 [d, e] â\96¶* U2 â\86\92 L0 [d, e] â\96¶* L1 →
+ â\88\83â\88\83T. L1 â\8a¢ T2 [d, e] â\96¶* T & L1 â\8a¢ U2 [d, e] â\96¶* T.
#L0 #L1 #T2 #U2 #d #e #HTU2 #H @(ltpss_ind … H) -L1
-[ /2/
+[ /2 width=3/
| -HTU2 #L #L1 #_ #HL1 * #W2 #HTW2 #HUW2
elim (ltps_tpss_conf … HL1 HTW2) -HTW2 #T #HT2 #HW2T
- elim (ltps_tpss_conf … HL1 HUW2) -HL1 HUW2 #U #HU2 #HW2U
- elim (tpss_conf_eq … HW2T … HW2U) -HW2T HW2U #V #HTV #HUV
- lapply (tpss_trans_eq … HT2 … HTV) -T;
- lapply (tpss_trans_eq … HU2 … HUV) -U /2/
+ elim (ltps_tpss_conf … HL1 HUW2) -HL1 -HUW2 #U #HU2 #HW2U
+ elim (tpss_conf_eq … HW2T … HW2U) -HW2T -HW2U #V #HTV #HUV
+ lapply (tpss_trans_eq … HT2 … HTV) -T
+ lapply (tpss_trans_eq … HU2 … HUV) -U /2 width=3/
]
qed.
lemma ltpss_tps_conf_eq: ∀L0,L1,T2,U2,d,e.
- L0 â\8a¢ T2 [d, e] â\89« U2 â\86\92 L0 [d, e] â\89«* L1 →
- â\88\83â\88\83T. L1 â\8a¢ T2 [d, e] â\89«* T & L1 â\8a¢ U2 [d, e] â\89«* T.
-/3/ qed.
+ L0 â\8a¢ T2 [d, e] â\96¶ U2 â\86\92 L0 [d, e] â\96¶* L1 →
+ â\88\83â\88\83T. L1 â\8a¢ T2 [d, e] â\96¶* T & L1 â\8a¢ U2 [d, e] â\96¶* T.
+/3 width=3/ qed.
-lemma ltpss_tpss_conf: â\88\80L1,T1,T2,d,e. L1 â\8a¢ T1 [d, e] â\89«* T2 →
- â\88\80L2,ds,es. L1 [ds, es] â\89«* L2 →
- â\88\83â\88\83T. L2 â\8a¢ T1 [d, e] â\89«* T & L1 â\8a¢ T2 [ds, es] â\89«* T.
+lemma ltpss_tpss_conf: â\88\80L1,T1,T2,d,e. L1 â\8a¢ T1 [d, e] â\96¶* T2 →
+ â\88\80L2,ds,es. L1 [ds, es] â\96¶* L2 →
+ â\88\83â\88\83T. L2 â\8a¢ T1 [d, e] â\96¶* T & L1 â\8a¢ T2 [ds, es] â\96¶* T.
#L1 #T1 #T2 #d #e #HT12 #L2 #ds #es #H @(ltpss_ind … H) -L2
-[ /3/
+[ /3 width=3/
| #L #L2 #HL1 #HL2 * #T #HT1 #HT2
elim (ltps_tpss_conf … HL2 HT1) -HT1 #T0 #HT10 #HT0
- lapply (ltps_tpss_trans_eq … HL2 … HT0) -HL2 HT0 #HT0
- lapply (ltpss_tpss_trans_eq … HL1 … HT0) -HL1 HT0 #HT0
- lapply (tpss_trans_eq … HT2 … HT0) -T /2/
+ lapply (ltps_tpss_trans_eq … HL2 … HT0) -HL2 -HT0 #HT0
+ lapply (ltpss_tpss_trans_eq … HL1 … HT0) -HL1 -HT0 #HT0
+ lapply (tpss_trans_eq … HT2 … HT0) -T /2 width=3/
]
qed.
-lemma ltpss_tps_conf: â\88\80L1,T1,T2,d,e. L1 â\8a¢ T1 [d, e] â\89« T2 →
- â\88\80L2,ds,es. L1 [ds, es] â\89«* L2 →
- â\88\83â\88\83T. L2 â\8a¢ T1 [d, e] â\89«* T & L1 â\8a¢ T2 [ds, es] â\89«* T.
-/3/ qed.
+lemma ltpss_tps_conf: â\88\80L1,T1,T2,d,e. L1 â\8a¢ T1 [d, e] â\96¶ T2 →
+ â\88\80L2,ds,es. L1 [ds, es] â\96¶* L2 →
+ â\88\83â\88\83T. L2 â\8a¢ T1 [d, e] â\96¶* T & L1 â\8a¢ T2 [ds, es] â\96¶* T.
+/3 width=1/ qed.
(* Advanced properties ******************************************************)
lemma ltpss_tps2: ∀L1,L2,I,e.
- L1 [0, e] â\89«* L2 â\86\92 â\88\80V1,V2. L2 â\8a¢ V1 [0, e] â\89« V2 →
- L1. 𝕓{I} V1 [0, e + 1] ≫* L2. 𝕓{I} V2.
+ L1 [0, e] â\96¶* L2 â\86\92 â\88\80V1,V2. L2 â\8a¢ V1 [0, e] â\96¶ V2 →
+ L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2.
#L1 #L2 #I #e #H @(ltpss_ind … H) -L2
-[ /3/
+[ /3 width=1/
| #L #L2 #_ #HL2 #IHL #V1 #V2 #HV12
elim (ltps_tps_trans … HV12 … HL2) -HV12 #V #HV1 #HV2
- lapply (IHL … HV1) -IHL HV1 #HL1
+ lapply (IHL … HV1) -IHL -HV1 #HL1
@step /2 width=5/ (**) (* /3 width=5/ is too slow *)
]
qed.
lemma ltpss_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 [0, e - 1] â\89«* L2 â\86\92 L2 â\8a¢ V1 [0, e - 1] â\89« V2 →
- 0 < e → L1. 𝕓{I} V1 [0, e] ≫* L2. 𝕓{I} V2.
+ L1 [0, e - 1] â\96¶* L2 â\86\92 L2 â\8a¢ V1 [0, e - 1] â\96¶ V2 →
+ 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) /2/
+>(plus_minus_m_m e 1) // /2 width=1/
qed.
-lemma ltpss_tps1: â\88\80L1,L2,I,d,e. L1 [d, e] â\89«* L2 →
- â\88\80V1,V2. L2 â\8a¢ V1 [d, e] â\89« V2 →
- L1. 𝕓{I} V1 [d + 1, e] ≫* L2. 𝕓{I} V2.
+lemma ltpss_tps1: â\88\80L1,L2,I,d,e. L1 [d, e] â\96¶* L2 →
+ â\88\80V1,V2. L2 â\8a¢ V1 [d, e] â\96¶ V2 →
+ L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
#L1 #L2 #I #d #e #H @(ltpss_ind … H) -L2
-[ /3/
+[ /3 width=1/
| #L #L2 #_ #HL2 #IHL #V1 #V2 #HV12
elim (ltps_tps_trans … HV12 … HL2) -HV12 #V #HV1 #HV2
- lapply (IHL … HV1) -IHL HV1 #HL1
+ lapply (IHL … HV1) -IHL -HV1 #HL1
@step /2 width=5/ (**) (* /3 width=5/ is too slow *)
]
qed.
lemma ltpss_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 [d - 1, e] â\89«* L2 â\86\92 L2 â\8a¢ V1 [d - 1, e] â\89« V2 →
- 0 < d → L1. 𝕓{I} V1 [d, e] ≫* L2. 𝕓{I} V2.
+ L1 [d - 1, e] â\96¶* L2 â\86\92 L2 â\8a¢ V1 [d - 1, e] â\96¶ V2 →
+ 0 < d → L1. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) /2/
+>(plus_minus_m_m d 1) // /2 width=1/
qed.
(* Advanced forward lemmas **************************************************)
-lemma ltpss_fwd_tpsa21: ∀e,K1,I,V1,L2. 0 < e → K1. 𝕓{I} V1 [0, e] ≫* L2 →
- â\88\83â\88\83K2,V2. K1 [0, e - 1] â\89«* K2 & K1 â\8a¢ V1 [0, e - 1] â\89«* V2 &
- L2 = K2. 𝕓{I} V2.
+lemma ltpss_fwd_tpss21: ∀e,K1,I,V1,L2. 0 < e → K1. ⓑ{I} V1 [0, e] ▶* L2 →
+ â\88\83â\88\83K2,V2. K1 [0, e - 1] â\96¶* K2 & K1 â\8a¢ V1 [0, e - 1] â\96¶* V2 &
+ L2 = K2. ⓑ{I} V2.
#e #K1 #I #V1 #L2 #He #H @(ltpss_ind … H) -L2
[ /2 width=5/
-| #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct -L;
+| #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct
elim (ltps_inv_tps21 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H
lapply (ltps_tps_trans_eq … HV2 … HK2) -HV2 #HV2
lapply (ltpss_tpss_trans_eq … HK1 … HV2) -HV2 #HV2 /3 width=5/
]
-qed.
+qed-.
-lemma ltpss_fwd_tpss11: ∀d,e,I,K1,V1,L2. 0 < d → K1. 𝕓{I} V1 [d, e] ≫* L2 →
- â\88\83â\88\83K2,V2. K1 [d - 1, e] â\89«* K2 &
- K1 â\8a¢ V1 [d - 1, e] â\89«* V2 &
- L2 = K2. 𝕓{I} V2.
+lemma ltpss_fwd_tpss11: ∀d,e,I,K1,V1,L2. 0 < d → K1. ⓑ{I} V1 [d, e] ▶* L2 →
+ â\88\83â\88\83K2,V2. K1 [d - 1, e] â\96¶* K2 &
+ K1 â\8a¢ V1 [d - 1, e] â\96¶* V2 &
+ L2 = K2. ⓑ{I} V2.
#d #e #K1 #I #V1 #L2 #Hd #H @(ltpss_ind … H) -L2
[ /2 width=5/
-| #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct -L;
+| #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct
elim (ltps_inv_tps11 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H
lapply (ltps_tps_trans_eq … HV2 … HK2) -HV2 #HV2
lapply (ltpss_tpss_trans_eq … HK1 … HV2) -HV2 #HV2 /3 width=5/
]
-qed.
+qed-.
projects / helm.git / blobdiff