(* *)
(**************************************************************************)
+include "ground/xoa/ex_2_3.ma".
+include "ground/xoa/ex_3_2.ma".
+include "ground/xoa/ex_3_3.ma".
+include "ground/xoa/ex_3_4.ma".
include "static_2/notation/relations/lrsubeqc_2.ma".
include "static_2/syntax/lenv.ma".
(* Basic_2A1: includes: lsubr_pair *)
inductive lsubr: relation lenv ≝
| lsubr_atom: lsubr (⋆) (⋆)
-| lsubr_bind: ∀I,L1,L2. lsubr L1 L2 → lsubr (L1.ⓘ{I}) (L2.ⓘ{I})
+| lsubr_bind: ∀I,L1,L2. lsubr L1 L2 → lsubr (L1.ⓘ[I]) (L2.ⓘ[I])
| lsubr_beta: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
-| lsubr_unit: ∀I1,I2,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I1}V) (L2.ⓤ{I2})
+| lsubr_unit: ∀I1,I2,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ[I1]V) (L2.ⓤ[I2])
.
interpretation
#L1 #L2 * -L1 -L2 //
[ #I #L1 #L2 #_ #H destruct
| #L1 #L2 #V #W #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #_ #H destruct
]
qed-.
/2 width=3 by lsubr_inv_atom1_aux/ qed-.
fact lsubr_inv_bind1_aux:
- ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ[I] →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ[I]
| ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} & I = BPair J1 V.
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ[J2] & I = BPair J1 V.
#L1 #L2 * -L1 -L2
[ #J #K1 #H destruct
| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by or3_intro0, ex2_intro/
(* Basic_2A1: uses: lsubr_inv_pair1 *)
lemma lsubr_inv_bind1:
- ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ ∀I,K1,L2. K1.ⓘ[I] ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ[I]
| ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} & I = BPair J1 V.
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ[J2] & I = BPair J1 V.
/2 width=3 by lsubr_inv_bind1_aux/ qed-.
fact lsubr_inv_atom2_aux: ∀L1,L2. L1 ⫃ L2 → L2 = ⋆ → L1 = ⋆.
/2 width=3 by lsubr_inv_atom2_aux/ qed-.
fact lsubr_inv_bind2_aux:
- ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ[I] →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ[I]
| ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ[J1]V & I = BUnit J2.
#L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or3_intro0/
qed-.
lemma lsubr_inv_bind2:
- ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ ∀I,L1,K2. L1 ⫃ K2.ⓘ[I] →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ[I]
| ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ[J1]V & I = BUnit J2.
/2 width=3 by lsubr_inv_bind2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
lemma lsubr_inv_abst1:
∀K1,L2,W. K1.ⓛW ⫃ L2 →
∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
- | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
+ | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ[I2].
#K1 #L2 #W #H elim (lsubr_inv_bind1 … H) -H *
-/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
+/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
#K2 #V2 #W2 #_ #_ #H destruct
qed-.
lemma lsubr_inv_unit1:
- ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
- ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
+ ∀I,K1,L2. K1.ⓤ[I] ⫃ L2 →
+ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ[I].
#I #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
[ #K2 #HK12 #H destruct /2 width=3 by ex2_intro/
| #K2 #V #W #_ #_ #H destruct
qed-.
lemma lsubr_inv_pair2:
- ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
+ ∀I,L1,K2,W. L1 ⫃ K2.ⓑ[I]W →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ[I]W
| ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
#I #L1 #K2 #W #H elim (lsubr_inv_bind2 … H) -H *
[ /3 width=3 by ex2_intro, or_introl/
| #K1 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
-| #J1 #J1 #K1 #V #_ #_ #H destruct
+| #J1 #J1 #K1 #V #_ #_ #H destruct
]
qed-.
qed-.
lemma lsubr_inv_unit2:
- ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
- | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
+ ∀I,L1,K2. L1 ⫃ K2.ⓤ[I] →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ[I]
+ | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ[J]V.
#I #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
[ /3 width=3 by ex2_intro, or_introl/
| #K1 #W #V #_ #_ #H destruct
(* Basic forward lemmas *****************************************************)
lemma lsubr_fwd_bind1:
- ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
- ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
+ ∀I1,K1,L2. K1.ⓘ[I1] ⫃ L2 →
+ ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ[I2].
#I1 #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
[ #K2 #HK12 #H destruct /3 width=4 by ex2_2_intro/
| #K2 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
qed-.
lemma lsubr_fwd_bind2:
- ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
+ ∀I2,L1,K2. L1 ⫃ K2.ⓘ[I2] →
+ ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ[I1].
#I2 #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
[ #K1 #HK12 #H destruct /3 width=4 by ex2_2_intro/
| #K1 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/