(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predtysn_5.ma".
+include "basic_2/notation/relations/predtysn_4.ma".
include "static_2/static/rex.ma".
include "basic_2/rt_transition/cpx_ext.ma".
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+(* EXTENDED PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS **********)
-definition rpx (h) (G): relation3 term lenv lenv ≝
- rex (cpx h G).
+definition rpx (G): relation3 term lenv lenv ≝
+ rex (cpx G).
interpretation
- "unbound parallel rt-transition on referred entries (local environment)"
- 'PRedTySn h T G L1 L2 = (rpx h G T L1 L2).
+ "extended parallel rt-transition on referred entries (local environment)"
+ 'PRedTySn T G L1 L2 = (rpx G T L1 L2).
(* Basic properties ***********************************************************)
-lemma rpx_atom: ∀h,I,G. ❪G,⋆❫ ⊢ ⬈[h,⓪[I]] ⋆.
+lemma rpx_atom (G):
+ ∀I. ❪G,⋆❫ ⊢ ⬈[⓪[I]] ⋆.
/2 width=1 by rex_atom/ qed.
-lemma rpx_sort: ∀h,I1,I2,G,L1,L2,s.
- ❪G,L1❫ ⊢ ⬈[h,⋆s] L2 → ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[h,⋆s] L2.ⓘ[I2].
+lemma rpx_sort (G):
+ ∀I1,I2,L1,L2,s.
+ ❪G,L1❫ ⊢ ⬈[⋆s] L2 → ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[⋆s] L2.ⓘ[I2].
/2 width=1 by rex_sort/ qed.
-lemma rpx_pair: ∀h,I,G,L1,L2,V1,V2.
- ❪G,L1❫ ⊢ ⬈[h,V1] L2 → ❪G,L1❫ ⊢ V1 ⬈[h] V2 → ❪G,L1.ⓑ[I]V1❫ ⊢ ⬈[h,#0] L2.ⓑ[I]V2.
+lemma rpx_pair (G):
+ ∀I,L1,L2,V1,V2.
+ ❪G,L1❫ ⊢ ⬈[V1] L2 → ❪G,L1❫ ⊢ V1 ⬈ V2 → ❪G,L1.ⓑ[I]V1❫ ⊢ ⬈[#0] L2.ⓑ[I]V2.
/2 width=1 by rex_pair/ qed.
-lemma rpx_lref: ∀h,I1,I2,G,L1,L2,i.
- ❪G,L1❫ ⊢ ⬈[h,#i] L2 → ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[h,#↑i] L2.ⓘ[I2].
+lemma rpx_lref (G):
+ ∀I1,I2,L1,L2,i.
+ ❪G,L1❫ ⊢ ⬈[#i] L2 → ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[#↑i] L2.ⓘ[I2].
/2 width=1 by rex_lref/ qed.
-lemma rpx_gref: ∀h,I1,I2,G,L1,L2,l.
- ❪G,L1❫ ⊢ ⬈[h,§l] L2 → ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[h,§l] L2.ⓘ[I2].
+lemma rpx_gref (G):
+ ∀I1,I2,L1,L2,l.
+ ❪G,L1❫ ⊢ ⬈[§l] L2 → ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[§l] L2.ⓘ[I2].
/2 width=1 by rex_gref/ qed.
-lemma rpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
- ❪G,L1.ⓘ[I]❫ ⊢ ⬈[h,T] L2.ⓘ[I1] →
- ∀I2. ❪G,L1❫ ⊢ I ⬈[h] I2 →
- ❪G,L1.ⓘ[I]❫ ⊢ ⬈[h,T] L2.ⓘ[I2].
+lemma rpx_bind_repl_dx (G):
+ ∀I,I1,L1,L2,T. ❪G,L1.ⓘ[I]❫ ⊢ ⬈[T] L2.ⓘ[I1] →
+ ∀I2. ❪G,L1❫ ⊢ I ⬈ I2 → ❪G,L1.ⓘ[I]❫ ⊢ ⬈[T] L2.ⓘ[I2].
/2 width=2 by rex_bind_repl_dx/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ❪G,⋆❫ ⊢ ⬈[h,T] Y2 → Y2 = ⋆.
+lemma rpx_inv_atom_sn (G):
+ ∀Y2,T. ❪G,⋆❫ ⊢ ⬈[T] Y2 → Y2 = ⋆.
/2 width=3 by rex_inv_atom_sn/ qed-.
-lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ❪G,Y1❫ ⊢ ⬈[h,T] ⋆ → Y1 = ⋆.
+lemma rpx_inv_atom_dx (G):
+ ∀Y1,T. ❪G,Y1❫ ⊢ ⬈[T] ⋆ → Y1 = ⋆.
/2 width=3 by rex_inv_atom_dx/ qed-.
-lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ❪G,Y1❫ ⊢ ⬈[h,⋆s] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ❪G,L1❫ ⊢ ⬈[h,⋆s] L2 &
- Y1 = L1.ⓘ[I1] & Y2 = L2.ⓘ[I2].
+lemma rpx_inv_sort (G):
+ ∀Y1,Y2,s. ❪G,Y1❫ ⊢ ⬈[⋆s] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. ❪G,L1❫ ⊢ ⬈[⋆s] L2 & Y1 = L1.ⓘ[I1] & Y2 = L2.ⓘ[I2].
/2 width=1 by rex_inv_sort/ qed-.
-lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ❪G,Y1❫ ⊢ ⬈[h,#↑i] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ❪G,L1❫ ⊢ ⬈[h,#i] L2 &
- Y1 = L1.ⓘ[I1] & Y2 = L2.ⓘ[I2].
+lemma rpx_inv_lref (G):
+ ∀Y1,Y2,i. ❪G,Y1❫ ⊢ ⬈[#↑i] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. ❪G,L1❫ ⊢ ⬈[#i] L2 & Y1 = L1.ⓘ[I1] & Y2 = L2.ⓘ[I2].
/2 width=1 by rex_inv_lref/ qed-.
-lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ❪G,Y1❫ ⊢ ⬈[h,§l] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ❪G,L1❫ ⊢ ⬈[h,§l] L2 &
- Y1 = L1.ⓘ[I1] & Y2 = L2.ⓘ[I2].
+lemma rpx_inv_gref (G):
+ ∀Y1,Y2,l. ❪G,Y1❫ ⊢ ⬈[§l] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. ❪G,L1❫ ⊢ ⬈[§l] L2 & Y1 = L1.ⓘ[I1] & Y2 = L2.ⓘ[I2].
/2 width=1 by rex_inv_gref/ qed-.
-lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ❪G,L1❫ ⊢ ⬈[h,ⓑ[p,I]V.T] L2 →
- ∧∧ ❪G,L1❫ ⊢ ⬈[h,V] L2 & ❪G,L1.ⓑ[I]V❫ ⊢ ⬈[h,T] L2.ⓑ[I]V.
+lemma rpx_inv_bind (G):
+ ∀p,I,L1,L2,V,T. ❪G,L1❫ ⊢ ⬈[ⓑ[p,I]V.T] L2 →
+ ∧∧ ❪G,L1❫ ⊢ ⬈[V] L2 & ❪G,L1.ⓑ[I]V❫ ⊢ ⬈[T] L2.ⓑ[I]V.
/2 width=2 by rex_inv_bind/ qed-.
-lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ❪G,L1❫ ⊢ ⬈[h,ⓕ[I]V.T] L2 →
- ∧∧ ❪G,L1❫ ⊢ ⬈[h,V] L2 & ❪G,L1❫ ⊢ ⬈[h,T] L2.
+lemma rpx_inv_flat (G):
+ ∀I,L1,L2,V,T. ❪G,L1❫ ⊢ ⬈[ⓕ[I]V.T] L2 →
+ ∧∧ ❪G,L1❫ ⊢ ⬈[V] L2 & ❪G,L1❫ ⊢ ⬈[T] L2.
/2 width=2 by rex_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[h,⋆s] Y2 →
- ∃∃I2,L2. ❪G,L1❫ ⊢ ⬈[h,⋆s] L2 & Y2 = L2.ⓘ[I2].
+lemma rpx_inv_sort_bind_sn (G):
+ ∀I1,Y2,L1,s. ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[⋆s] Y2 →
+ ∃∃I2,L2. ❪G,L1❫ ⊢ ⬈[⋆s] L2 & Y2 = L2.ⓘ[I2].
/2 width=2 by rex_inv_sort_bind_sn/ qed-.
-lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ❪G,Y1❫ ⊢ ⬈[h,⋆s] L2.ⓘ[I2] →
- ∃∃I1,L1. ❪G,L1❫ ⊢ ⬈[h,⋆s] L2 & Y1 = L1.ⓘ[I1].
+lemma rpx_inv_sort_bind_dx (G):
+ ∀I2,Y1,L2,s. ❪G,Y1❫ ⊢ ⬈[⋆s] L2.ⓘ[I2] →
+ ∃∃I1,L1. ❪G,L1❫ ⊢ ⬈[⋆s] L2 & Y1 = L1.ⓘ[I1].
/2 width=2 by rex_inv_sort_bind_dx/ qed-.
-lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ❪G,L1.ⓑ[I]V1❫ ⊢ ⬈[h,#0] Y2 →
- ∃∃L2,V2. ❪G,L1❫ ⊢ ⬈[h,V1] L2 & ❪G,L1❫ ⊢ V1 ⬈[h] V2 &
- Y2 = L2.ⓑ[I]V2.
+lemma rpx_inv_zero_pair_sn (G):
+ ∀I,Y2,L1,V1. ❪G,L1.ⓑ[I]V1❫ ⊢ ⬈[#0] Y2 →
+ ∃∃L2,V2. ❪G,L1❫ ⊢ ⬈[V1] L2 & ❪G,L1❫ ⊢ V1 ⬈ V2 & Y2 = L2.ⓑ[I]V2.
/2 width=1 by rex_inv_zero_pair_sn/ qed-.
-lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ❪G,Y1❫ ⊢ ⬈[h,#0] L2.ⓑ[I]V2 →
- ∃∃L1,V1. ❪G,L1❫ ⊢ ⬈[h,V1] L2 & ❪G,L1❫ ⊢ V1 ⬈[h] V2 &
- Y1 = L1.ⓑ[I]V1.
+lemma rpx_inv_zero_pair_dx (G):
+ ∀I,Y1,L2,V2. ❪G,Y1❫ ⊢ ⬈[#0] L2.ⓑ[I]V2 →
+ ∃∃L1,V1. ❪G,L1❫ ⊢ ⬈[V1] L2 & ❪G,L1❫ ⊢ V1 ⬈ V2 & Y1 = L1.ⓑ[I]V1.
/2 width=1 by rex_inv_zero_pair_dx/ qed-.
-lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[h,#↑i] Y2 →
- ∃∃I2,L2. ❪G,L1❫ ⊢ ⬈[h,#i] L2 & Y2 = L2.ⓘ[I2].
+lemma rpx_inv_lref_bind_sn (G):
+ ∀I1,Y2,L1,i. ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[#↑i] Y2 →
+ ∃∃I2,L2. ❪G,L1❫ ⊢ ⬈[#i] L2 & Y2 = L2.ⓘ[I2].
/2 width=2 by rex_inv_lref_bind_sn/ qed-.
-lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ❪G,Y1❫ ⊢ ⬈[h,#↑i] L2.ⓘ[I2] →
- ∃∃I1,L1. ❪G,L1❫ ⊢ ⬈[h,#i] L2 & Y1 = L1.ⓘ[I1].
+lemma rpx_inv_lref_bind_dx (G):
+ ∀I2,Y1,L2,i. ❪G,Y1❫ ⊢ ⬈[#↑i] L2.ⓘ[I2] →
+ ∃∃I1,L1. ❪G,L1❫ ⊢ ⬈[#i] L2 & Y1 = L1.ⓘ[I1].
/2 width=2 by rex_inv_lref_bind_dx/ qed-.
-lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[h,§l] Y2 →
- ∃∃I2,L2. ❪G,L1❫ ⊢ ⬈[h,§l] L2 & Y2 = L2.ⓘ[I2].
+lemma rpx_inv_gref_bind_sn (G):
+ ∀I1,Y2,L1,l. ❪G,L1.ⓘ[I1]❫ ⊢ ⬈[§l] Y2 →
+ ∃∃I2,L2. ❪G,L1❫ ⊢ ⬈[§l] L2 & Y2 = L2.ⓘ[I2].
/2 width=2 by rex_inv_gref_bind_sn/ qed-.
-lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ❪G,Y1❫ ⊢ ⬈[h,§l] L2.ⓘ[I2] →
- ∃∃I1,L1. ❪G,L1❫ ⊢ ⬈[h,§l] L2 & Y1 = L1.ⓘ[I1].
+lemma rpx_inv_gref_bind_dx (G):
+ ∀I2,Y1,L2,l. ❪G,Y1❫ ⊢ ⬈[§l] L2.ⓘ[I2] →
+ ∃∃I1,L1. ❪G,L1❫ ⊢ ⬈[§l] L2 & Y1 = L1.ⓘ[I1].
/2 width=2 by rex_inv_gref_bind_dx/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma rpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
- ❪G,L1❫ ⊢ ⬈[h,②[I]V.T] L2 → ❪G,L1❫ ⊢ ⬈[h,V] L2.
+lemma rpx_fwd_pair_sn (G):
+ ∀I,L1,L2,V,T. ❪G,L1❫ ⊢ ⬈[②[I]V.T] L2 → ❪G,L1❫ ⊢ ⬈[V] L2.
/2 width=3 by rex_fwd_pair_sn/ qed-.
-lemma rpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
- ❪G,L1❫ ⊢ ⬈[h,ⓑ[p,I]V.T] L2 → ❪G,L1.ⓑ[I]V❫ ⊢ ⬈[h,T] L2.ⓑ[I]V.
+lemma rpx_fwd_bind_dx (G):
+ ∀p,I,L1,L2,V,T. ❪G,L1❫ ⊢ ⬈[ⓑ[p,I]V.T] L2 → ❪G,L1.ⓑ[I]V❫ ⊢ ⬈[T] L2.ⓑ[I]V.
/2 width=2 by rex_fwd_bind_dx/ qed-.
-lemma rpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
- ❪G,L1❫ ⊢ ⬈[h,ⓕ[I]V.T] L2 → ❪G,L1❫ ⊢ ⬈[h,T] L2.
+lemma rpx_fwd_flat_dx (G):
+ ∀I,L1,L2,V,T. ❪G,L1❫ ⊢ ⬈[ⓕ[I]V.T] L2 → ❪G,L1❫ ⊢ ⬈[T] L2.
/2 width=3 by rex_fwd_flat_dx/ qed-.
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