(* *)
(**************************************************************************)
-include "ground_2/xoa/ex_3_4.ma".
-include "ground_2/xoa/ex_4_1.ma".
+include "ground/xoa/ex_3_1.ma".
+include "ground/xoa/ex_3_4.ma".
+include "ground/xoa/ex_4_1.ma".
+include "ground/arith/nat_succ.ma".
include "static_2/notation/relations/voidstareq_4.ma".
include "static_2/syntax/lenv.ma".
(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
inductive lveq: bi_relation nat lenv ≝
-| lveq_atom : lveq 0 (⋆) 0 (⋆)
-| lveq_bind : ∀I1,I2,K1,K2. lveq 0 K1 0 K2 →
- lveq 0 (K1.ⓘ{I1}) 0 (K2.ⓘ{I2})
-| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 0 K2 →
- lveq (↑n1) (K1.ⓧ) 0 K2
-| lveq_void_dx: ∀K1,K2,n2. lveq 0 K1 n2 K2 →
- lveq 0 K1 (↑n2) (K2.ⓧ)
+| lveq_atom : lveq 𝟎 (⋆) 𝟎 (⋆)
+| lveq_bind : ∀I1,I2,K1,K2. lveq 𝟎 K1 𝟎 K2 →
+ lveq 𝟎 (K1.ⓘ[I1]) 𝟎 (K2.ⓘ[I2])
+| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 𝟎 K2 →
+ lveq (↑n1) (K1.ⓧ) 𝟎 K2
+| lveq_void_dx: ∀K1,K2,n2. lveq 𝟎 K1 n2 K2 →
+ lveq 𝟎 K1 (↑n2) (K2.ⓧ)
.
interpretation "equivalence up to exclusion binders (local environment)"
(* Basic properties *********************************************************)
-lemma lveq_refl: ∀L. L ≋ⓧ*[0,0] L.
+lemma lveq_refl: ∀L. L ≋ⓧ*[𝟎,𝟎] L.
#L elim L -L /2 width=1 by lveq_atom, lveq_bind/
qed.
(* Basic inversion lemmas ***************************************************)
fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
- 0 = n1 → 0 = n2 →
+ (𝟎 = n1) → 𝟎 = n2 →
∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
- | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[𝟎,𝟎] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: /3 width=1 by or_introl, conj/
|2: /3 width=7 by ex3_4_intro, or_intror/
-|*: #K1 #K2 #n #_ #H1 #H2 destruct
+|*: #K1 #K2 #n #_ [ #H #_ | #_ #H ]
+ elim (eq_inv_zero_nsucc … H)
]
qed-.
-lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 →
+lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[𝟎,𝟎] L2 →
∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
- | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[𝟎,𝟎] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2.
/2 width=5 by lveq_inv_zero_aux/ qed-.
fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∀m1. ↑m1 = n1 →
- ∃∃K1. K1 ≋ⓧ*[m1,0] L2 & K1.ⓧ = L1 & 0 = n2.
+ ∃∃K1. K1 ≋ⓧ*[m1,𝟎] L2 & K1.ⓧ = L1 & 𝟎 = n2.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[1: #m #H destruct
-|2: #I1 #I2 #K1 #K2 #_ #m #H destruct
-|*: #K1 #K2 #n #HK #m #H destruct /2 width=3 by ex3_intro/
+[1: #m #H elim (eq_inv_nsucc_zero … H)
+|2: #I1 #I2 #K1 #K2 #_ #m #H elim (eq_inv_nsucc_zero … H)
+|*: #K1 #K2 #n #HK #m #H
+ [ >(eq_inv_nsucc_bi … H) -m /2 width=3 by ex3_intro/
+ | elim (eq_inv_nsucc_zero … H)
+ ]
]
qed-.
lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1,n2] K2 →
- ∃∃K1. K1 ≋ⓧ*[n1,0] K2 & K1.ⓧ = L1 & 0 = n2.
+ ∃∃K1. K1 ≋ⓧ*[n1,𝟎] K2 & K1.ⓧ = L1 & 𝟎 = n2.
/2 width=3 by lveq_inv_succ_sn_aux/ qed-.
lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1,↑n2] L2 →
- ∃∃K2. K1 ≋ⓧ*[0,n2] K2 & K2.ⓧ = L2 & 0 = n1.
+ ∃∃K2. K1 ≋ⓧ*[𝟎,n2] K2 & K2.ⓧ = L2 & 𝟎 = n1.
#K1 #L2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/
fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[1: #m1 #m2 #H1 #H2 destruct
-|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H1 #H2 destruct
-|*: #K1 #K2 #n #_ #m1 #m2 #H1 #H2 destruct
+[1: #m1 #m2 #H #_ elim (eq_inv_nsucc_zero … H)
+|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H #_ elim (eq_inv_nsucc_zero … H)
+|*: #K1 #K2 #n #_ #m1 #m2 [ #_ #H | #H #_ ]
+ elim (eq_inv_nsucc_zero … H)
]
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0,0] K2.ⓘ{I2} → K1 ≋ⓧ*[0,0] K2.
+lemma lveq_inv_bind_O: ∀I1,I2,K1,K2. K1.ⓘ[I1] ≋ⓧ*[𝟎,𝟎] K2.ⓘ[I2] → K1 ≋ⓧ*[𝟎,𝟎] K2.
#I1 #I2 #K1 #K2 #H
elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct //
qed-.
-lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
-* [2: #n1 ] * [2,4: #n2 ] #H
+lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 𝟎 = n1 & 𝟎 = n2.
+#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ]
+#n2 @(nat_ind_succ … n2) -n2 [2,4: #n2 #_ ]
+#H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1,n2] ⋆ →
- ∃∃m1. K1 ≋ⓧ*[m1,0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2.
-#I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H
+lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ[I1] ≋ⓧ*[n1,n2] ⋆ →
+ ∃∃m1. K1 ≋ⓧ*[m1,𝟎] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 𝟎 = n2.
+#I1 #K1
+#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ]
+#n2 @(nat_ind_succ … n2) -n2 [2,4: #n2 #_ ]
+#H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
| elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=3 by ex4_intro/
]
qed-.
-lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1,n2] K2.ⓘ{I2} →
- ∃∃m2. ⋆ ≋ⓧ*[0,m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
+lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1,n2] K2.ⓘ[I2] →
+ ∃∃m2. ⋆ ≋ⓧ*[𝟎,m2] K2 & BUnit Void = I2 & 𝟎 = n1 & ↑m2 = n2.
#I2 #K2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_bind_atom … H) -H
/3 width=3 by lveq_sym, ex4_intro/
qed-.
-lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] K2.ⓑ{I2}V2 →
- ∧∧ K1 ≋ⓧ*[0,0] K2 & 0 = n1 & 0 = n2.
-#I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H
+lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2] K2.ⓑ[I2]V2 →
+ ∧∧ K1 ≋ⓧ*[𝟎,𝟎] K2 & 𝟎 = n1 & 𝟎 = n2.
+#I1 #I2 #K1 #K2 #V1 #V2
+#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ]
+#n2 @(nat_ind_succ … n2) -n2 [2,4: #n2 #_ ]
+#H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
qed-.
lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1,n2] L2 →
- ∧∧ L1 ≋ ⓧ*[n1,0] L2 & 0 = n2.
+ ∧∧ L1 ≋ ⓧ*[n1,𝟎] L2 & 𝟎 = n2.
#L1 #L2 #n1 #n2 #H
elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/
qed-.
lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,↑n2] L2.ⓧ →
- ∧∧ L1 ≋ ⓧ*[0,n2] L2 & 0 = n1.
+ ∧∧ L1 ≋ ⓧ*[𝟎,n2] L2 & 𝟎 = n1.
#L1 #L2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_void_succ_sn … H) -H
(* Advanced forward lemmas **************************************************)
lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
- ∨∨ 0 = n1 | 0 = n2.
-#L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H
+ ∨∨ 𝟎 = n1 | 𝟎 = n2.
+#L1 #L2
+#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ]
+#n2 @(nat_ind_succ … n2) -n2 [2,4: #n2 #_ ]
+#H
[ elim (lveq_inv_succ … H) ]
/2 width=1 by or_introl, or_intror/
qed-.
-lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] L2 → 0 = n1.
-#I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H
+lemma lveq_fwd_pair_sn:
+ ∀I1,K1,L2,V1,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2] L2 → 𝟎 = n1.
+#I1 #K1 #L2 #V1
+#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ] //
+#n2 @(nat_ind_succ … n2) -n2 [2: #n2 #_ ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ{I2}V2 → 0 = n2.
+lemma lveq_fwd_pair_dx:
+ ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ[I2]V2 → 𝟎 = n2.
/3 width=6 by lveq_fwd_pair_sn, lveq_sym/ qed-.