(* Main properties **********************************************************)
-theorem cpy_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶×[d1, e1] T1 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶×[d2, e2] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶×[d1, e1] T.
+(* Basic_1: was: subst1_confluence_eq *)
+theorem cpy_conf_eq: ∀G,L,T0,T1,d1,e1. ⦃G, L⦄ ⊢ T0 ▶[d1, e1] T1 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T2 ▶[d1, e1] T.
#G #L #T0 #T1 #d1 #e1 #H elim H -G -L -T0 -T1 -d1 -e1
[ /2 width=3 by ex2_intro/
| #I1 #G #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
]
| #a #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- elim (IHV01 … HV02) -V0 #V #HV1 #HV2
+ elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
elim (IHT01 … HT02) -T0 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+ lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
+ /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
| #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
elim (IHV01 … HV02) -V0
]
qed-.
-theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶×[d1, e1] T1 →
- ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶×[d2, e2] T2 →
+(* Basic_1: was: subst1_confluence_neq *)
+theorem cpy_conf_neq: ∀G,L1,T0,T1,d1,e1. ⦃G, L1⦄ ⊢ T0 ▶[d1, e1] T1 →
+ ∀L2,T2,d2,e2. ⦃G, L2⦄ ⊢ T0 ▶[d2, e2] T2 →
(d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
- ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶×[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶×[d1, e1] T.
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L1⦄ ⊢ T2 ▶[d1, e1] T.
#G #L1 #T0 #T1 #d1 #e1 #H elim H -G -L1 -T0 -T1 -d1 -e1
[ /2 width=3 by ex2_intro/
| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
elim (cpy_inv_lref1 … H1) -H1
[ #H destruct /3 width=7 by cpy_subst, ex2_intro/
- | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
- [ -Hd1 -Hde2
- lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H
- lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H
- elim (lt_refl_false … H)
- | -Hd2 -Hde1
- lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H
- lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H
- elim (lt_refl_false … H)
+ | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded [ -Hd1 -Hde2 | -Hd2 -Hde1 ]
+ [ elim (ylt_yle_false … Hde1) -Hde1 /2 width=3 by yle_trans/
+ | elim (ylt_yle_false … Hde2) -Hde2 /2 width=3 by yle_trans/
]
]
| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
+ elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
elim (IHT01 … HT02) -T0
[ -H #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
- | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H
- /3 width=1 by monotonic_le_plus_l, or_intror, or_introl/
+ lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+ | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
]
| #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
qed-.
(* Note: the constant 1 comes from cpy_subst *)
-theorem cpy_trans_ge: ∀G,L,T1,T0,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T0 →
- ∀T2. ⦃G, L⦄ ⊢ T0 ▶×[d, 1] T2 → 1 ≤ e → ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2.
+(* Basic_1: was: subst1_trans *)
+theorem cpy_trans_ge: ∀G,L,T1,T0,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T0 →
+ ∀T2. ⦃G, L⦄ ⊢ T0 ▶[d, 1] T2 → 1 ≤ e → ⦃G, L⦄ ⊢ T1 ▶[d, e] T2.
#G #L #T1 #T0 #d #e #H elim H -G -L -T1 -T0 -d -e
[ #I #G #L #d #e #T2 #H #He
elim (cpy_inv_atom1 … H) -H
[ #H destruct //
| * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
- lapply (lt_to_le_to_lt … (d+e) Hide2 ?) /2 width=5 by cpy_subst, monotonic_lt_plus_r/
+ lapply (ylt_yle_trans … (d+e) … Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
]
| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
- lapply (cpy_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1 by le_S_S/ #HVT2
- <(cpy_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=5 by cpy_subst/
+ lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
+ >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V0) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- lapply (IHT10 … HT02 He) -T0 #HT12
- lapply (lsuby_cpy_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /3 width=1 by cpy_bind, lsuby_succ/
+ lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ lapply (IHT10 … HT02 He) -T0 /3 width=1 by cpy_bind/
| #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
]
qed-.
-theorem cpy_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶×[d1, e1] T0 →
- ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶×[d2, e2] T2 → d2 + e2 ≤ d1 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d2, e2] T & ⦃G, L⦄ ⊢ T ▶×[d1, e1] T2.
+theorem cpy_trans_down: ∀G,L,T1,T0,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T0 →
+ ∀T2,d2,e2. ⦃G, L⦄ ⊢ T0 ▶[d2, e2] T2 → d2 + e2 ≤ d1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[d2, e2] T & ⦃G, L⦄ ⊢ T ▶[d1, e1] T2.
#G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1
[ /2 width=3 by ex2_intro/
| #I #G #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
- lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
- lapply (cpy_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1 by le_S/ -Hde2i1 #HWT2
- <(cpy_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
+ lapply (yle_trans … Hde2d1 … Hdi1) -Hde2d1 #Hde2i1
+ lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hde2i1
+ >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (lsuby_cpy_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
- elim (IHT10 … HT02 ?) -T0 /2 width=1 by le_S_S/ #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
+ lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ elim (IHV10 … HV02) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
| #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
elim (IHV10 … HV02) -V0 //
projects / helm.git / blobdiff