inductive tps: nat → nat → lenv → relation term ≝
| tps_atom : ∀L,I,d,e. tps d e L (𝕒{I}) (𝕒{I})
| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
- â\86\93[0, i] L â\89¡ K. ð\9d\95\93{Abbr} V â\86\92 â\86\91[0, i + 1] V ≡ W → tps d e L (#i) W
+ â\87©[0, i] L â\89¡ K. ð\9d\95\93{Abbr} V â\86\92 â\87§[0, i + 1] V ≡ W → tps d e L (#i) W
| tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
tps d e L V1 V2 → tps (d + 1) e (L. 𝕓{I} V2) T1 T2 →
tps d e L (𝕓{I} V1. T1) (𝕓{I} V2. T2)
(* Basic properties *********************************************************)
-lemma tps_lsubs_conf: â\88\80L1,T1,T2,d,e. L1 â\8a¢ T1 [d, e] â\89« T2 →
- â\88\80L2. L1 [d, e] â\89¼ L2 â\86\92 L2 â\8a¢ T1 [d, e] â\89« T2.
-#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
+lemma tps_lsubs_conf: â\88\80L1,T1,T2,d,e. L1 â\8a¢ T1 [d, e] â\96¶ T2 →
+ â\88\80L2. L1 [d, e] â\89¼ L2 â\86\92 L2 â\8a¢ T1 [d, e] â\96¶ T2.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
[ //
| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
- elim (ldrop_lsubs_ldrop1_abbr … HL12 … HLK1 ? ?) -HL12 HLK1 // /2/
-| /4/
-| /3/
+ elim (ldrop_lsubs_ldrop1_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
+| /4 width=1/
+| /3 width=1/
]
qed.
-lemma tps_refl: â\88\80T,L,d,e. L â\8a¢ T [d, e] â\89« T.
+lemma tps_refl: â\88\80T,L,d,e. L â\8a¢ T [d, e] â\96¶ T.
#T elim T -T //
-#I elim I -I /2/
+#I elim I -I /2 width=1/
qed.
-lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
+(* Basic_1: was: subst1_ex *)
+lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. 𝕓{Abbr} V) →
+ ∃∃T2,T. L ⊢ T1 [d, 1] ▶ T2 & ⇧[d, 1] T ≡ T2.
+#K #V #T1 elim T1 -T1
+[ * #i #L #d #HLK /2 width=4/
+ elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i ? ? ?) // /3 width=4/
+| * #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. 𝕓{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=8/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
+ ]
+]
+qed.
+
+lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ▶ T2 →
∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
- L â\8a¢ T1 [d2, e2] â\89« T2.
-#L #T1 #T2 #d1 #e1 #H elim H -H L T1 T2 d1 e1
+ L â\8a¢ T1 [d2, e2] â\96¶ T2.
+#L #T1 #T2 #d1 #e1 #H elim H -L -T1 -T2 -d1 -e1
[ //
| #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12
- lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
- lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2/
-| /4/
-| /4/
+ lapply (transitive_le … Hd12 … Hid1) -Hd12 -Hid1 #Hid2
+ lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2 width=4/
+| /4 width=3/
+| /4 width=1/
]
qed.
lemma tps_weak_top: ∀L,T1,T2,d,e.
- L â\8a¢ T1 [d, e] â\89« T2 â\86\92 L â\8a¢ T1 [d, |L| - d] â\89« T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+ L â\8a¢ T1 [d, e] â\96¶ T2 â\86\92 L â\8a¢ T1 [d, |L| - d] â\96¶ T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
[ //
| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
- lapply (le_to_lt_to_lt … Hdi Hi) #Hd
- lapply (plus_minus_m_m_comm (|L|) d ?) /2/
-| normalize /2/
-| /2/
+ lapply (le_to_lt_to_lt … Hdi Hi) /3 width=4/
+| normalize /2 width=1/
+| /2 width=1/
]
qed.
lemma tps_weak_all: ∀L,T1,T2,d,e.
- L â\8a¢ T1 [d, e] â\89« T2 â\86\92 L â\8a¢ T1 [0, |L|] â\89« T2.
+ L â\8a¢ T1 [d, e] â\96¶ T2 â\86\92 L â\8a¢ T1 [0, |L|] â\96¶ T2.
#L #T1 #T2 #d #e #HT12
lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
lapply (tps_weak_top … HT12) //
qed.
-lemma tps_split_up: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → ∀i. d ≤ i → i ≤ d + e →
- â\88\83â\88\83T. L â\8a¢ T1 [d, i - d] â\89« T & L â\8a¢ T [i, d + e - i] â\89« T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
-[ /2/
+lemma tps_split_up: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → ∀i. d ≤ i → i ≤ d + e →
+ â\88\83â\88\83T. L â\8a¢ T1 [d, i - d] â\96¶ T & L â\8a¢ T [i, d + e - i] â\96¶ T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ /2 width=3/
| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
elim (lt_or_ge i j)
- [ -Hide Hjde;
- >(plus_minus_m_m_comm j d) in ⊢ (% → ?) // -Hdj /3/
- | -Hdi Hdj; #Hid
- generalize in match Hide -Hide (**) (* rewriting in the premises, rewrites in the goal too *)
- >(plus_minus_m_m_comm … Hjde) in ⊢ (% → ?) -Hjde /4/
+ [ -Hide -Hjde
+ >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=4/
+ | -Hdi -Hdj #Hid
+ generalize in match Hide; -Hide (**) (* rewriting in the premises, rewrites in the goal too *)
+ >(plus_minus_m_m … Hjde) in ⊢ (% → ?); -Hjde /4 width=4/
]
| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
- -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
+ elim (IHT12 (i + 1) ? ?) -IHT12 /2 width=1/
+ -Hdi -Hide >arith_c1x #T #HT1 #HT2
lapply (tps_lsubs_conf … HT1 (L. 𝕓{I} V) ?) -HT1 /3 width=5/
| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi Hide /3 width=5/
+ -Hdi -Hide /3 width=5/
]
qed.
(* Basic inversion lemmas ***************************************************)
-fact tps_inv_atom1_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → ∀I. T1 = 𝕒{I} →
+fact tps_inv_atom1_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → ∀I. T1 = 𝕒{I} →
T2 = 𝕒{I} ∨
∃∃K,V,i. d ≤ i & i < d + e &
- â\86\93[O, i] L ≡ K. 𝕓{Abbr} V &
- â\86\91[O, i + 1] V ≡ T2 &
+ â\87©[O, i] L ≡ K. 𝕓{Abbr} V &
+ â\87§[O, i + 1] V ≡ T2 &
I = LRef i.
-#L #T1 #T2 #d #e * -L T1 T2 d e
-[ #L #I #d #e #J #H destruct -I /2/
-| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct -I /3 width=8/
+#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
+[ #L #I #d #e #J #H destruct /2 width=1/
+| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /3 width=8/
| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
]
qed.
-lemma tps_inv_atom1: â\88\80L,T2,I,d,e. L â\8a¢ ð\9d\95\92{I} [d, e] â\89« T2 →
+lemma tps_inv_atom1: â\88\80L,T2,I,d,e. L â\8a¢ ð\9d\95\92{I} [d, e] â\96¶ T2 →
T2 = 𝕒{I} ∨
∃∃K,V,i. d ≤ i & i < d + e &
- â\86\93[O, i] L ≡ K. 𝕓{Abbr} V &
- â\86\91[O, i + 1] V ≡ T2 &
+ â\87©[O, i] L ≡ K. 𝕓{Abbr} V &
+ â\87§[O, i + 1] V ≡ T2 &
I = LRef i.
-/2/ qed.
+/2 width=3/ qed-.
(* Basic_1: was: subst1_gen_sort *)
-lemma tps_inv_sort1: â\88\80L,T2,k,d,e. L â\8a¢ â\8b\86k [d, e] â\89« T2 → T2 = ⋆k.
+lemma tps_inv_sort1: â\88\80L,T2,k,d,e. L â\8a¢ â\8b\86k [d, e] â\96¶ T2 → T2 = ⋆k.
#L #T2 #k #d #e #H
elim (tps_inv_atom1 … H) -H //
* #K #V #i #_ #_ #_ #_ #H destruct
-qed.
+qed-.
(* Basic_1: was: subst1_gen_lref *)
-lemma tps_inv_lref1: â\88\80L,T2,i,d,e. L â\8a¢ #i [d, e] â\89« T2 →
+lemma tps_inv_lref1: â\88\80L,T2,i,d,e. L â\8a¢ #i [d, e] â\96¶ T2 →
T2 = #i ∨
∃∃K,V. d ≤ i & i < d + e &
- â\86\93[O, i] L ≡ K. 𝕓{Abbr} V &
- â\86\91[O, i + 1] V ≡ T2.
+ â\87©[O, i] L ≡ K. 𝕓{Abbr} V &
+ â\87§[O, i + 1] V ≡ T2.
#L #T2 #i #d #e #H
-elim (tps_inv_atom1 … H) -H /2/
-* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct -i /3/
-qed.
+elim (tps_inv_atom1 … H) -H /2 width=1/
+* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
+qed-.
-fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶ T2 → T2 = §p.
+#L #T2 #p #d #e #H
+elim (tps_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ▶ U2 →
∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
- â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\89« V2 &
- L. ð\9d\95\93{I} V2 â\8a¢ T1 [d + 1, e] â\89« T2 &
+ â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\96¶ V2 &
+ L. ð\9d\95\93{I} V2 â\8a¢ T1 [d + 1, e] â\96¶ T2 &
U2 = 𝕓{I} V2. T2.
-#d #e #L #U1 #U2 * -d e L U1 U2
+#d #e #L #U1 #U2 * -d -e -L -U1 -U2
[ #L #k #d #e #I #V1 #T1 #H destruct
| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
]
qed.
-lemma tps_inv_bind1: â\88\80d,e,L,I,V1,T1,U2. L â\8a¢ ð\9d\95\93{I} V1. T1 [d, e] â\89« U2 →
- â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\89« V2 &
- L. ð\9d\95\93{I} V2 â\8a¢ T1 [d + 1, e] â\89« T2 &
+lemma tps_inv_bind1: â\88\80d,e,L,I,V1,T1,U2. L â\8a¢ ð\9d\95\93{I} V1. T1 [d, e] â\96¶ U2 →
+ â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\96¶ V2 &
+ L. ð\9d\95\93{I} V2 â\8a¢ T1 [d + 1, e] â\96¶ T2 &
U2 = 𝕓{I} V2. T2.
-/2/ qed.
+/2 width=3/ qed-.
-fact tps_inv_flat1_aux: â\88\80d,e,L,U1,U2. L â\8a¢ U1 [d, e] â\89« U2 →
+fact tps_inv_flat1_aux: â\88\80d,e,L,U1,U2. L â\8a¢ U1 [d, e] â\96¶ U2 →
∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
- â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\89« V2 & L â\8a¢ T1 [d, e] â\89« T2 &
+ â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\96¶ V2 & L â\8a¢ T1 [d, e] â\96¶ T2 &
U2 = 𝕗{I} V2. T2.
-#d #e #L #U1 #U2 * -d e L U1 U2
+#d #e #L #U1 #U2 * -d -e -L -U1 -U2
[ #L #k #d #e #I #V1 #T1 #H destruct
| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct
| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
]
qed.
-lemma tps_inv_flat1: â\88\80d,e,L,I,V1,T1,U2. L â\8a¢ ð\9d\95\97{I} V1. T1 [d, e] â\89« U2 →
- â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\89« V2 & L â\8a¢ T1 [d, e] â\89« T2 &
+lemma tps_inv_flat1: â\88\80d,e,L,I,V1,T1,U2. L â\8a¢ ð\9d\95\97{I} V1. T1 [d, e] â\96¶ U2 →
+ â\88\83â\88\83V2,T2. L â\8a¢ V1 [d, e] â\96¶ V2 & L â\8a¢ T1 [d, e] â\96¶ T2 &
U2 = 𝕗{I} V2. T2.
-/2/ qed.
+/2 width=3/ qed-.
-fact tps_inv_refl_O2_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → e = 0 → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+fact tps_inv_refl_O2_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → e = 0 → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
[ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct -e;
- lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi Hide <plus_n_O #Hdd
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+ lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi
-Hide <plus_n_O #Hdd
elim (lt_refl_false … Hdd)
-| /3/
-| /3/
+| /3 width=1/
+| /3 width=1/
]
qed.
-lemma tps_inv_refl_O2: â\88\80L,T1,T2,d. L â\8a¢ T1 [d, 0] â\89« T2 → T1 = T2.
-/2 width=6/ qed.
+lemma tps_inv_refl_O2: â\88\80L,T1,T2,d. L â\8a¢ T1 [d, 0] â\96¶ T2 → T1 = T2.
+/2 width=6/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → #[T1] ≤ #[T2].
-#L #T1 #T2 #d #e #H elim H normalize -H L T1 T2 d e
-[ //
-| /2/
-| /3 by monotonic_le_plus_l, le_plus/ (**) (* just /3/ is too slow *)
-| /3 by monotonic_le_plus_l, le_plus/ (**) (* just /3/ is too slow *)
-]
-qed.
+lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → #[T1] ≤ #[T2].
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
+/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
+qed-.
(* Basic_1: removed theorems 25:
subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
projects / helm.git / blobdiff