lemma tps_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ≫ T2.
-#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
+#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: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
#T elim T -T //
-#I elim I -I /2/
+#I elim I -I /2 width=1/
qed.
(* 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/
- elim (lt_or_eq_or_gt i d) #Hid [ /3/ |3: /3/ ]
- destruct -d;
+[ * #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 ? ? ?) // <minus_plus_m_m_comm /3/
+ elim (lift_split … HVW i i ? ? ?) // <minus_plus_m_m_comm /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/ -HLK /3 width=8/
- | elim (IHU1 … HLK) -IHU1 HLK /3 width=8/
+ [ 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 ⊢ T1 [d2, e2] ≫ T2.
-#L #T1 #T2 #d1 #e1 #H elim H -H L T1 T2 d1 e1
+#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 ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+#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 (plus_minus_m_m_comm (|L|) d ?) /2 width=1/ /2 width=4/
+| normalize /2 width=1/
+| /2 width=1/
]
qed.
lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
-[ /2/
+#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_comm 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_comm … 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_c1 >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.
↓[O, i] L ≡ K. 𝕓{Abbr} V &
↑[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
]
↓[O, i] L ≡ K. 𝕓{Abbr} V &
↑[O, i + 1] V ≡ T2 &
I = LRef i.
-/2/ qed-.
+/2 width=3/ qed-.
(* Basic_1: was: subst1_gen_sort *)
↓[O, i] L ≡ K. 𝕓{Abbr} V &
↑[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/
+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 →
∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ 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/
∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 &
U2 = 𝕓{I} V2. T2.
-/2/ qed-.
+/2 width=3/ qed-.
fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ 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
lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
U2 = 𝕗{I} V2. T2.
-/2/ qed-.
+/2 width=3/ qed-.
fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+#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.
(* 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
-/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3/ is too slow *)
+#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:
projects / helm.git / blobdiff