(* EXAMPLES *****************************************************************)
-(* Extended validy (basic_2B) vs. restricted validity (basic_1A) ************)
+(* Extended validy (λδ-2A) vs. restricted validity (λδ-1B) ******************)
(* Note: extended validity of a closure, height of cnv_appl > 1 *)
-lemma cnv_extended (h) (p): ∀G,L,s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 !*[h].
+lemma cnv_extended (h) (p) (G) (L):
+ ∀s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 ![h,𝛚].
#h #p #G #L #s
@(cnv_appl … 2 p … (⋆s) … (⋆s))
[ //
qed.
(* Note: restricted validity of the η-expanded closure, height of cnv_appl = 1 **)
-lemma vnv_restricted (h) (p): ∀G,L,s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛⓛ{p}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ![h].
+lemma cnv_restricted (h) (p) (G) (L):
+ ∀s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛⓛ{p}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ![h,𝟐].
#h #p #G #L #s
@(cnv_appl … 1 p … (⋆s) … (ⓐ#0.#2))
[ //
include "static_2/syntax/ac.ma".
include "basic_2/notation/relations/exclaim_5.ma".
-include "basic_2/notation/relations/exclaim_4.ma".
-include "basic_2/notation/relations/exclaimstar_4.ma".
include "basic_2/rt_computation/cpms.ma".
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* activate genv *)
(* Basic_2A1: uses: snv *)
-inductive cnv (a) (h): relation3 genv lenv term ≝
-| cnv_sort: ∀G,L,s. cnv a h G L (⋆s)
-| cnv_zero: ∀I,G,K,V. cnv a h G K V → cnv a h G (K.ⓑ{I}V) (#0)
-| cnv_lref: ∀I,G,K,i. cnv a h G K (#i) → cnv a h G (K.ⓘ{I}) (#↑i)
-| cnv_bind: ∀p,I,G,L,V,T. cnv a h G L V → cnv a h G (L.ⓑ{I}V) T → cnv a h G L (ⓑ{p,I}V.T)
-| cnv_appl: ∀n,p,G,L,V,W0,T,U0. appl a n → cnv a h G L V → cnv a h G L T →
- ⦃G,L⦄ ⊢ V ➡*[1,h] W0 → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
-| cnv_cast: ∀G,L,U,T,U0. cnv a h G L U → cnv a h G L T →
- ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[1,h] U0 → cnv a h G L (ⓝU.T)
+inductive cnv (h) (a): relation3 genv lenv term ≝
+| cnv_sort: ∀G,L,s. cnv h a G L (⋆s)
+| cnv_zero: ∀I,G,K,V. cnv h a G K V → cnv h a G (K.ⓑ{I}V) (#0)
+| cnv_lref: ∀I,G,K,i. cnv h a G K (#i) → cnv h a G (K.ⓘ{I}) (#↑i)
+| cnv_bind: ∀p,I,G,L,V,T. cnv h a G L V → cnv h a G (L.ⓑ{I}V) T → cnv h a G L (ⓑ{p,I}V.T)
+| cnv_appl: ∀n,p,G,L,V,W0,T,U0. ad a n → cnv h a G L V → cnv h a G L T →
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0 → cnv h a G L (ⓐV.T)
+| cnv_cast: ∀G,L,U,T,U0. cnv h a G L U → cnv h a G L T →
+ ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[1,h] U0 → cnv h a G L (ⓝU.T)
.
interpretation "context-sensitive native validity (term)"
- 'Exclaim a h G L T = (cnv a h G L T).
-
-interpretation "context-sensitive restricted native validity (term)"
- 'Exclaim h G L T = (cnv (ac_eq (S O)) h G L T).
-
-interpretation "context-sensitive extended native validity (term)"
- 'ExclaimStar h G L T = (cnv ac_top h G L T).
+ 'Exclaim h a G L T = (cnv h a G L T).
(* Basic inversion lemmas ***************************************************)
-fact cnv_inv_zero_aux (a) (h):
- ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → X = #0 →
- ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V.
-#a #h #G #L #X * -G -L -X
+fact cnv_inv_zero_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → X = #0 →
+ ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![h,a] & L = K.ⓑ{I}V.
+#h #a #G #L #X * -G -L -X
[ #G #L #s #H destruct
| #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/
| #I #G #K #i #_ #H destruct
]
qed-.
-lemma cnv_inv_zero (a) (h):
- ∀G,L. ⦃G,L⦄ ⊢ #0 ![a,h] →
- ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V.
+lemma cnv_inv_zero (h) (a):
+ ∀G,L. ⦃G,L⦄ ⊢ #0 ![h,a] →
+ ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![h,a] & L = K.ⓑ{I}V.
/2 width=3 by cnv_inv_zero_aux/ qed-.
-fact cnv_inv_lref_aux (a) (h):
- ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀i. X = #(↑i) →
- ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}.
-#a #h #G #L #X * -G -L -X
+fact cnv_inv_lref_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → ∀i. X = #(↑i) →
+ ∃∃I,K. ⦃G,K⦄ ⊢ #i ![h,a] & L = K.ⓘ{I}.
+#h #a #G #L #X * -G -L -X
[ #G #L #s #j #H destruct
| #I #G #K #V #_ #j #H destruct
| #I #G #L #i #Hi #j #H destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma cnv_inv_lref (a) (h):
- ∀G,L,i. ⦃G,L⦄ ⊢ #↑i ![a,h] →
- ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}.
+lemma cnv_inv_lref (h) (a):
+ ∀G,L,i. ⦃G,L⦄ ⊢ #↑i ![h,a] →
+ ∃∃I,K. ⦃G,K⦄ ⊢ #i ![h,a] & L = K.ⓘ{I}.
/2 width=3 by cnv_inv_lref_aux/ qed-.
-fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀l. X = §l → ⊥.
-#a #h #G #L #X * -G -L -X
+fact cnv_inv_gref_aux (h) (a): ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → ∀l. X = §l → ⊥.
+#h #a #G #L #X * -G -L -X
[ #G #L #s #l #H destruct
| #I #G #K #V #_ #l #H destruct
| #I #G #K #i #_ #l #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_gref *)
-lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G,L⦄ ⊢ §l ![a,h] → ⊥.
+lemma cnv_inv_gref (h) (a): ∀G,L,l. ⦃G,L⦄ ⊢ §l ![h,a] → ⊥.
/2 width=8 by cnv_inv_gref_aux/ qed-.
-fact cnv_inv_bind_aux (a) (h):
- ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] →
+fact cnv_inv_bind_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] →
∀p,I,V,T. X = ⓑ{p,I}V.T →
- ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h].
-#a #h #G #L #X * -G -L -X
+ ∧∧ ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![h,a].
+#h #a #G #L #X * -G -L -X
[ #G #L #s #q #Z #X1 #X2 #H destruct
| #I #G #K #V #_ #q #Z #X1 #X2 #H destruct
| #I #G #K #i #_ #q #Z #X1 #X2 #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_bind *)
-lemma cnv_inv_bind (a) (h):
- ∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
- ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h].
+lemma cnv_inv_bind (h) (a):
+ ∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![h,a] →
+ ∧∧ ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![h,a].
/2 width=4 by cnv_inv_bind_aux/ qed-.
-fact cnv_inv_appl_aux (a) (h):
- ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀V,T. X = ⓐV.T →
- ∃∃n,p,W0,U0. appl a n & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+fact cnv_inv_appl_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → ∀V,T. X = ⓐV.T →
+ ∃∃n,p,W0,U0. ad a n & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
-#a #h #G #L #X * -L -X
+#h #a #G #L #X * -L -X
[ #G #L #s #X1 #X2 #H destruct
| #I #G #K #V #_ #X1 #X2 #H destruct
| #I #G #K #i #_ #X1 #X2 #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_appl *)
-lemma cnv_inv_appl (a) (h):
- ∀G,L,V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
- ∃∃n,p,W0,U0. appl a n & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+lemma cnv_inv_appl (h) (a):
+ ∀G,L,V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h,a] →
+ ∃∃n,p,W0,U0. ad a n & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
/2 width=3 by cnv_inv_appl_aux/ qed-.
-fact cnv_inv_cast_aux (a) (h):
- ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀U,T. X = ⓝU.T →
- ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+fact cnv_inv_cast_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → ∀U,T. X = ⓝU.T →
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
-#a #h #G #L #X * -G -L -X
+#h #a #G #L #X * -G -L -X
[ #G #L #s #X1 #X2 #H destruct
| #I #G #K #V #_ #X1 #X2 #H destruct
| #I #G #K #i #_ #X1 #X2 #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_cast *)
-lemma cnv_inv_cast (a) (h):
- ∀G,L,U,T. ⦃G,L⦄ ⊢ ⓝU.T ![a,h] →
- ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+lemma cnv_inv_cast (h) (a):
+ ∀G,L,U,T. ⦃G,L⦄ ⊢ ⓝU.T ![h,a] →
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
/2 width=3 by cnv_inv_cast_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma cnv_fwd_flat (a) (h) (I) (G) (L):
- ∀V,T. ⦃G,L⦄ ⊢ ⓕ{I}V.T ![a,h] →
- ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h].
-#a #h * #G #L #V #T #H
+lemma cnv_fwd_flat (h) (a) (I) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓕ{I}V.T ![h,a] →
+ ∧∧ ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ T ![h,a].
+#h #a * #G #L #V #T #H
[ elim (cnv_inv_appl … H) #n #p #W #U #_ #HV #HT #_ #_
| elim (cnv_inv_cast … H) #U #HV #HT #_ #_
] -H /2 width=1 by conj/
qed-.
-lemma cnv_fwd_pair_sn (a) (h) (I) (G) (L):
- ∀V,T. ⦃G,L⦄ ⊢ ②{I}V.T ![a,h] → ⦃G,L⦄ ⊢ V ![a,h].
-#a #h * [ #p ] #I #G #L #V #T #H
+lemma cnv_fwd_pair_sn (h) (a) (I) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ②{I}V.T ![h,a] → ⦃G,L⦄ ⊢ V ![h,a].
+#h #a * [ #p ] #I #G #L #V #T #H
[ elim (cnv_inv_bind … H) -H #HV #_
| elim (cnv_fwd_flat … H) -H #HV #_
] //
(* Forward lemmas on atomic arity assignment for terms **********************)
(* Basic_2A1: uses: snv_fwd_aaa *)
-lemma cnv_fwd_aaa (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ∃A. ⦃G,L⦄ ⊢ T ⁝ A.
-#a #h #G #L #T #H elim H -G -L -T
+lemma cnv_fwd_aaa (h) (a): ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ∃A. ⦃G,L⦄ ⊢ T ⁝ A.
+#h #a #G #L #T #H elim H -G -L -T
[ /2 width=2 by aaa_sort, ex_intro/
| #I #G #L #V #_ * /3 width=2 by aaa_zero, ex_intro/
| #I #G #L #K #_ * /3 width=2 by aaa_lref, ex_intro/
(* Forward lemmas with t_bound rt_transition for terms **********************)
-lemma cnv_fwd_cpm_SO (a) (h) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
-#a #h #G #L #T #H
+lemma cnv_fwd_cpm_SO (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
+#h #a #G #L #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by aaa_cpm_SO/
qed-.
(* Forward lemmas with t_bound rt_computation for terms *********************)
-lemma cnv_fwd_cpms_total (a) (h) (n) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
-#a #h #n #G #L #T #H
+lemma cnv_fwd_cpms_total (h) (a) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
+#h #a #n #G #L #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by cpms_total_aaa/
qed-.
-lemma cnv_fwd_cpms_abst_dx_le (a) (h) (G) (L) (W) (p):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] →
+lemma cnv_fwd_cpms_abst_dx_le (h) (a) (G) (L) (W) (p):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
∀n1,U1. ⦃G,L⦄ ⊢ T ➡*[n1,h] ⓛ{p}W.U1 → ∀n2. n1 ≤ n2 →
∃∃U2. ⦃G,L⦄ ⊢ T ➡*[n2,h] ⓛ{p}W.U2 & ⦃G,L.ⓛW⦄ ⊢ U1 ➡*[n2-n1,h] U2.
-#a #h #G #L #W #p #T #H
+#h #a #G #L #W #p #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by cpms_abst_dx_le_aaa/
qed-.
(* Advanced properties ******************************************************)
-lemma cnv_appl_ge (a) (h) (n1) (p) (G) (L):
- ∀n2. n1 ≤ n2 → appl a n2 →
- ∀V. ⦃G,L⦄ ⊢ V ![a,h] → ∀T. ⦃G,L⦄ ⊢ T ![a,h] →
+lemma cnv_appl_ge (h) (a) (n1) (p) (G) (L):
+ ∀n2. n1 ≤ n2 → ad a n2 →
+ ∀V. ⦃G,L⦄ ⊢ V ![h,a] → ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
∀X. ⦃G,L⦄ ⊢ V ➡*[1,h] X → ∀W. ⦃G,L⦄ ⊢ W ➡*[h] X →
- ∀U. ⦃G,L⦄ ⊢ T ➡*[n1,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![a,h].
-#a #h #n1 #p #G #L #n2 #Hn12 #Ha #V #HV #T #HT #X #HVX #W #HW #X #HTX
+ ∀U. ⦃G,L⦄ ⊢ T ➡*[n1,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![h,a].
+#h #a #n1 #p #G #L #n2 #Hn12 #Ha #V #HV #T #HT #X #HVX #W #HW #X #HTX
elim (cnv_fwd_cpms_abst_dx_le … HT … HTX … Hn12) #U #HTU #_ -n1
/4 width=11 by cnv_appl, cpms_bind, cpms_cprs_trans/
qed.
(* Properties with context-sensitive parallel eta-conversion for terms ******)
-lemma cpce_total_cnv (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∃T2. ⦃G,L⦄ ⊢ T1 ⬌η[h] T2.
-#a #h #G #L #T1 #HT1
+lemma cpce_total_cnv (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∃T2. ⦃G,L⦄ ⊢ T1 ⬌η[h] T2.
+#h #a #G #L #T1 #HT1
lapply (cnv_fwd_csx … HT1) #H
generalize in match HT1; -HT1
@(csx_ind_fpbg … H) -G -L -T1
(* Properties with r-equivalence ********************************************)
-lemma cnv_cpcs_dec (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∀T2. ⦃G,L⦄ ⊢ T2 ![a,h] →
+lemma cnv_cpcs_dec (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∀T2. ⦃G,L⦄ ⊢ T2 ![h,a] →
Decidable … (⦃G,L⦄ ⊢ T1 ⬌*[h] T2).
-#a #h #G #L #T1 #HT1 #T2 #HT2
+#h #a #G #L #T1 #HT1 #T2 #HT2
elim (cnv_fwd_aaa … HT1) -HT1 #A1 #HA1
elim (cnv_fwd_aaa … HT2) -HT2 #A2 #HA2
/3 width=2 by csx_cpcs_dec, aaa_csx/
(* Properties with t-bound rt-equivalence for terms *************************)
-lemma cnv_appl_cpes (a) (h) (G) (L):
- ∀n. appl a n →
- ∀V. ⦃G,L⦄ ⊢ V ![a,h] → ∀T. ⦃G,L⦄ ⊢ T ![a,h] →
+lemma cnv_appl_cpes (h) (a) (G) (L):
+ ∀n. ad a n →
+ ∀V. ⦃G,L⦄ ⊢ V ![h,a] → ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
∀W. ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W →
- ∀p,U. ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![a,h].
-#a #h #G #L #n #Hn #V #HV #T #HT #W *
+ ∀p,U. ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![h,a].
+#h #a #G #L #n #Hn #V #HV #T #HT #W *
/4 width=11 by cnv_appl, cpms_cprs_trans, cpms_bind/
qed.
-lemma cnv_cast_cpes (a) (h) (G) (L):
- ∀U. ⦃G,L⦄ ⊢ U ![a,h] →
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T → ⦃G,L⦄ ⊢ ⓝU.T ![a,h].
-#a #h #G #L #U #HU #T #HT * /2 width=3 by cnv_cast/
+lemma cnv_cast_cpes (h) (a) (G) (L):
+ ∀U. ⦃G,L⦄ ⊢ U ![h,a] →
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T → ⦃G,L⦄ ⊢ ⓝU.T ![h,a].
+#h #a #G #L #U #HU #T #HT * /2 width=3 by cnv_cast/
qed.
(* Inversion lemmas with t-bound rt-equivalence for terms *******************)
-lemma cnv_inv_appl_cpes (a) (h) (G) (L):
- ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
- ∃∃n,p,W,U. appl a n & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+lemma cnv_inv_appl_cpes (h) (a) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h,a] →
+ ∃∃n,p,W,U. ad a n & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
⦃G,L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U.
-#a #h #G #L #V #T #H
+#h #a #G #L #V #T #H
elim (cnv_inv_appl … H) -H #n #p #W #U #Hn #HV #HT #HVW #HTU
/3 width=7 by cpms_div, ex5_4_intro/
qed-.
-lemma cnv_inv_cast_cpes (a) (h) (G) (L):
- ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T ![a,h] →
- ∧∧ ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T.
-#a #h #G #L #U #T #H
+lemma cnv_inv_cast_cpes (h) (a) (G) (L):
+ ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T ![h,a] →
+ ∧∧ ⦃G,L⦄ ⊢ U ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] & ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T.
+#h #a #G #L #U #T #H
elim (cnv_inv_cast … H) -H
/3 width=3 by cpms_div, and3_intro/
qed-.
(* Eliminators with t-bound rt-equivalence for terms ************************)
-lemma cnv_ind_cpes (a) (h) (Q:relation3 genv lenv term):
+lemma cnv_ind_cpes (h) (a) (Q:relation3 genv lenv term):
(∀G,L,s. Q G L (⋆s)) →
- (∀I,G,K,V. ⦃G,K⦄ ⊢ V![a,h] → Q G K V → Q G (K.ⓑ{I}V) (#O)) →
- (∀I,G,K,i. ⦃G,K⦄ ⊢ #i![a,h] → Q G K (#i) → Q G (K.ⓘ{I}) (#(↑i))) →
- (∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ V![a,h] → ⦃G,L.ⓑ{I}V⦄⊢T![a,h] →
+ (∀I,G,K,V. ⦃G,K⦄ ⊢ V![h,a] → Q G K V → Q G (K.ⓑ{I}V) (#O)) →
+ (∀I,G,K,i. ⦃G,K⦄ ⊢ #i![h,a] → Q G K (#i) → Q G (K.ⓘ{I}) (#(↑i))) →
+ (∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ V![h,a] → ⦃G,L.ⓑ{I}V⦄⊢T![h,a] →
Q G L V →Q G (L.ⓑ{I}V) T →Q G L (ⓑ{p,I}V.T)
) →
- (∀n,p,G,L,V,W,T,U. appl a n → ⦃G,L⦄ ⊢ V![a,h] → ⦃G,L⦄ ⊢ T![a,h] →
+ (∀n,p,G,L,V,W,T,U. ad a n → ⦃G,L⦄ ⊢ V![h,a] → ⦃G,L⦄ ⊢ T![h,a] →
⦃G,L⦄ ⊢ V ⬌*[h,1,0]W → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U →
Q G L V → Q G L T → Q G L (ⓐV.T)
) →
- (∀G,L,U,T. ⦃G,L⦄⊢ U![a,h] → ⦃G,L⦄ ⊢ T![a,h] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T →
+ (∀G,L,U,T. ⦃G,L⦄⊢ U![h,a] → ⦃G,L⦄ ⊢ T![h,a] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T →
Q G L U → Q G L T → Q G L (ⓝU.T)
) →
- ∀G,L,T. ⦃G,L⦄⊢ T![a,h] → Q G L T.
-#a #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #G #L #T #H
+ ∀G,L,T. ⦃G,L⦄⊢ T![h,a] → Q G L T.
+#h #a #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #G #L #T #H
elim H -G -L -T [5,6: /3 width=7 by cpms_div/ |*: /2 width=1 by/ ]
qed-.
∃∃T. ⦃G,L1⦄ ⊢ ⋆s ➡*[1,h] T & ⦃G,L2⦄ ⊢ ⋆(⫯[h]s) ➡*[h] T.
/3 width=3 by cpm_cpms, ex2_intro/ qed-.
-fact cnv_cpm_conf_lpr_atom_delta_aux (a) (h) (G) (L) (i):
- (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄⊢#i![a,h] →
+fact cnv_cpm_conf_lpr_atom_delta_aux (h) (a) (G) (L) (i):
+ (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄⊢#i![h,a] →
∀K,V. ⬇*[i]L ≘ K.ⓓV →
∀n,XV. ⦃G,K⦄ ⊢ V ➡[n,h] XV →
∀X. ⬆*[↑i]XV ≘ X →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
-#a #h #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #HL1 #L2 #HL2
+#h #a #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #HL1 #L2 #HL2
lapply (cnv_inv_lref_pair … HT … HLK) -HT #HV
elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #HV1 #H destruct
/3 width=6 by cpms_delta_drops, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_atom_ell_aux (a) (h) (G) (L) (i):
- (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄⊢#i![a,h] →
+fact cnv_cpm_conf_lpr_atom_ell_aux (h) (a) (G) (L) (i):
+ (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄⊢#i![h,a] →
∀K,W. ⬇*[i]L ≘ K.ⓛW →
∀n,XW. ⦃G,K⦄ ⊢ W ➡[n,h] XW →
∀X. ⬆*[↑i]XW ≘ X →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[↑n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
-#a #h #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #HL1 #L2 #HL2
+#h #a #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #HL1 #L2 #HL2
lapply (cnv_inv_lref_pair … HT … HLK) -HT #HW
elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK1 #HW1 #H destruct
/3 width=6 by cpms_ell_drops, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_delta_delta_aux (a) (h) (I) (G) (L) (i):
- (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄⊢#i![a,h] →
+fact cnv_cpm_conf_lpr_delta_delta_aux (h) (a) (I) (G) (L) (i):
+ (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄⊢#i![h,a] →
∀K1,V1. ⬇*[i]L ≘ K1.ⓑ{I}V1 → ∀K2,V2. ⬇*[i]L ≘ K2.ⓑ{I}V2 →
∀n1,XV1. ⦃G,K1⦄ ⊢ V1 ➡[n1,h] XV1 → ∀n2,XV2. ⦃G,K2⦄ ⊢ V2 ➡[n2,h] XV2 →
∀X1. ⬆*[↑i]XV1 ≘ X1 → ∀X2. ⬆*[↑i]XV2 ≘ X2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ X1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ X2 ➡*[n1-n2,h] T.
-#a #h #I #G #L #i #IH #HT
+#h #a #I #G #L #i #IH #HT
#K #V #HLK #Y #X #HLY #n1 #XV1 #HVX1 #n2 #XV2 #HVX2 #X1 #HXV1 #X2 #HXV2
#L1 #HL1 #L2 #HL2
lapply (drops_mono … HLY … HLK) -HLY #H destruct
lapply (drops_mono … HLK2 … HLK1) -L -i #H destruct
qed-.
-fact cnv_cpm_conf_lpr_bind_bind_aux (a) (h) (p) (I) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓑ{p,I}V.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
+fact cnv_cpm_conf_lpr_bind_bind_aux (h) (a) (p) (I) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓑ{p,I}V.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀n1,T1. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡*[n1-n2,h] T.
-#a #h #p #I #G0 #L0 #V0 #T0 #IH #H0
+#h #a #p #I #G0 #L0 #V0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #HV0 #HT0
/3 width=5 by cpms_bind_dx, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_bind_zeta_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
+fact cnv_cpm_conf_lpr_bind_zeta_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ +ⓓV.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢V ➡[h] V1 → ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 →
∀T2. ⬆*[1]T2 ≘ T → ∀n2,XT2. ⦃G,L⦄ ⊢ T2 ➡[n2,h] XT2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ +ⓓV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#V1 #HV01 #n1 #T1 #HT01 #T2 #HT20 #n2 #XT2 #HXT2
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #_ #HT0
/3 width=3 by cpms_zeta, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_zeta_zeta_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
+fact cnv_cpm_conf_lpr_zeta_zeta_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ +ⓓV.T ![h,a] →
∀T1. ⬆*[1]T1 ≘ T → ∀T2. ⬆*[1]T2 ≘ T →
∀n1,XT1. ⦃G,L⦄ ⊢ T1 ➡[n1,h] XT1 → ∀n2,XT2. ⦃G,L⦄ ⊢ T2 ➡[n2,h] XT2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ XT1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#T1 #HT10 #T2 #HT20 #n1 #XT1 #HXT1 #n2 #XT2 #HXT2
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #_ #HT0
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_appl_appl_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
+fact cnv_cpm_conf_lpr_appl_appl_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓐV2.T2 ➡*[n1-n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #HT0 #_ #_ -n0 -p0 -X01 -X02
/3 width=5 by cpms_appl_dx, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_appl_beta_aux (a) (h) (p) (G) (L) (V) (W) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
+fact cnv_cpm_conf_lpr_appl_beta_aux (h) (a) (p) (G) (L) (V) (W) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
∀n1,T1. ⦃G,L⦄ ⊢ ⓛ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
-#a #h #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#h #a #p #G0 #L0 #V0 #W0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
/4 width=5 by cpms_beta_dx, cpms_bind_dx, cpm_cast, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_appl_theta_aux (a) (h) (p) (G) (L) (V) (W) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
+fact cnv_cpm_conf_lpr_appl_theta_aux (h) (a) (p) (G) (L) (V) (W) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
∀n1,T1. ⦃G,L⦄ ⊢ ⓓ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
∀U2. ⬆*[1]V2 ≘ U2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
-#a #h #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#h #a #p #G0 #L0 #V0 #W0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02 #U2 #HVU2
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
]
qed-.
-fact cnv_cpm_conf_lpr_beta_beta_aux (a) (h) (p) (G) (L) (V) (W) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
+fact cnv_cpm_conf_lpr_beta_beta_aux (h) (a) (p) (G) (L) (V) (W) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
∀n1,T1. ⦃G,L.ⓛW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
-#a #h #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#h #a #p #G0 #L0 #V0 #W0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
/4 width=5 by cpms_bind_dx, cpm_eps, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_theta_theta_aux (a) (h) (p) (G) (L) (V) (W) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
+fact cnv_cpm_conf_lpr_theta_theta_aux (h) (a) (p) (G) (L) (V) (W) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
∀n1,T1. ⦃G,L.ⓓW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
∀U1. ⬆*[1]V1 ≘ U1 → ∀U2. ⬆*[1]V2 ≘ U2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
-#a #h #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#h #a #p #G0 #L0 #V0 #W0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02 #U1 #HVU1 #U2 #HVU2
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
/4 width=7 by cpms_appl_dx, cpms_bind_dx, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_cast_cast_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_cast_cast_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓝV2.T2 ➡*[n1-n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
/3 width=5 by cpms_cast, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_cast_epsilon_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_cast_epsilon_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 →
∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #HV01 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
/3 width=3 by cpms_eps, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_cast_ee_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_cast_ee_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr h a G0 L0 T0) →
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
+#h #a #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
#n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01
#L1 #HL01 #L2 #HL02 -HV01
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
/3 width=3 by cpms_eps, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_epsilon_epsilon_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_epsilon_epsilon_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #_ #HT0 #_ #_ -X0
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_epsilon_ee_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_epsilon_ee_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr h a G0 L0 T0) →
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
+#h #a #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
#n1 #T1 #HT01 #n2 #V2 #HV02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_ee_ee_aux (a) (h) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_ee_ee_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ V1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-n2,h] T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #HV01 #n2 #V2 #HV02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #_ #_ #_ -X0
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_aux (a) (h):
+fact cnv_cpm_conf_lpr_aux (h) (a):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
-#a #h #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr h a G1 L1 T1.
+#h #a #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
elim (cpm_inv_atom1_drops … HX1) -HX1 *
elim (cpm_inv_atom1_drops … HX2) -HX2 *
(* Inversion lemmas with restricted rt-transition for terms *****************)
-lemma cnv_cpr_tdeq_fwd_refl (a) (h) (G) (L):
- ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 → ⦃G,L⦄ ⊢ T1 ![a,h] → T1 = T2.
-#a #h #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
+lemma cnv_cpr_tdeq_fwd_refl (h) (a) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 → ⦃G,L⦄ ⊢ T1 ![h,a] → T1 = T2.
+#h #a #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ //
| #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
lapply (tdeq_inv_lref1 … H1) -H1 #H destruct //
]
qed-.
-lemma cpm_tdeq_inv_bind_sn (a) (h) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
+lemma cpm_tdeq_inv_bind_sn (h) (a) (n) (p) (I) (G) (L):
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![h,a] →
∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
-#a #h #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
+ ∃∃T2. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
+#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_bind1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
]
qed-.
-lemma cpm_tdeq_inv_appl_sn (a) (h) (n) (G) (L):
- ∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![a,h] →
+lemma cpm_tdeq_inv_appl_sn (h) (a) (n) (G) (L):
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![h,a] →
∀X. ⦃G,L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛ X →
- ∃∃m,q,W,U1,T2. appl a m & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ V ➡*[1,h] W & ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
- & ⦃G,L⦄⊢ T1 ![a,h] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
-#a #h #n #G #L #V #T1 #H0 #X #H1 #H2
+ ∃∃m,q,W,U1,T2. ad a m & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ V ➡*[1,h] W & ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
+ & ⦃G,L⦄⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
+#h #a #n #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_appl1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
]
qed-.
-lemma cpm_tdeq_inv_cast_sn (a) (h) (n) (G) (L):
- ∀U1,T1. ⦃G,L⦄ ⊢ ⓝU1.T1 ![a,h] →
+lemma cpm_tdeq_inv_cast_sn (h) (a) (n) (G) (L):
+ ∀U1,T1. ⦃G,L⦄ ⊢ ⓝU1.T1 ![h,a] →
∀X. ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
∃∃U0,U2,T2. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 & ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0
- & ⦃G,L⦄ ⊢ U1 ![a,h] & ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
- & ⦃G,L⦄ ⊢ T1 ![a,h] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
-#a #h #n #G #L #U1 #T1 #H0 #X #H1 #H2
+ & ⦃G,L⦄ ⊢ U1 ![h,a] & ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
+ & ⦃G,L⦄ ⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
+#h #a #n #G #L #U1 #T1 #H0 #X #H1 #H2
elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
[ #U2 #T2 #HU12 #HT12 #H destruct
elim (tdeq_inv_pair … H2) -H2 #_ #H2U12 #H2T12
]
qed-.
-lemma cpm_tdeq_inv_bind_dx (a) (h) (n) (p) (I) (G) (L):
- ∀X. ⦃G,L⦄ ⊢ X ![a,h] →
+lemma cpm_tdeq_inv_bind_dx (h) (a) (n) (p) (I) (G) (L):
+ ∀X. ⦃G,L⦄ ⊢ X ![h,a] →
∀V,T2. ⦃G,L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 →
- ∃∃T1. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
-#a #h #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
+ ∃∃T1. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
+#h #a #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
elim (tdeq_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
/2 width=5 by ex5_intro/
(* Eliminators with restricted rt-transition for terms **********************)
-lemma cpm_tdeq_ind (a) (h) (n) (G) (Q:relation3 …):
+lemma cpm_tdeq_ind (h) (a) (n) (G) (Q:relation3 …):
(∀I,L. n = 0 → Q L (⓪{I}) (⓪{I})) →
(∀L,s. n = 1 → Q L (⋆s) (⋆(⫯[h]s))) →
- (∀p,I,L,V,T1. ⦃G,L⦄⊢ V![a,h] → ⦃G,L.ⓑ{I}V⦄⊢T1![a,h] →
+ (∀p,I,L,V,T1. ⦃G,L⦄⊢ V![h,a] → ⦃G,L.ⓑ{I}V⦄⊢T1![h,a] →
∀T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
Q (L.ⓑ{I}V) T1 T2 → Q L (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
) →
- (∀m. appl a m →
- ∀L,V. ⦃G,L⦄ ⊢ V ![a,h] → ∀W. ⦃G,L⦄ ⊢ V ➡*[1,h] W →
- ∀p,T1,U1. ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![a,h] →
+ (∀m. ad a m →
+ ∀L,V. ⦃G,L⦄ ⊢ V ![h,a] → ∀W. ⦃G,L⦄ ⊢ V ➡*[1,h] W →
+ ∀p,T1,U1. ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![h,a] →
∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
) →
(∀L,U0,U1,T1. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 → ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 →
- ∀U2. ⦃G,L⦄ ⊢ U1 ![a,h] → ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
- ∀T2. ⦃G,L⦄ ⊢ T1 ![a,h] → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀U2. ⦃G,L⦄ ⊢ U1 ![h,a] → ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ![h,a] → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
) →
- ∀L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
+ ∀L,T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → Q L T1 T2.
-#a #h #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
+#h #a #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
@(insert_eq_0 … G) #F
@(fqup_wf_ind_eq (Ⓣ) … F L T1) -L -T1 -F
#G0 #L0 #T0 #IH #F #L * [| * [| * ]]
(* Advanced properties with restricted rt-transition for terms **************)
-lemma cpm_tdeq_free (a) (h) (n) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
+lemma cpm_tdeq_free (h) (a) (n) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
∀F,K. ⦃F,K⦄ ⊢ T1 ➡[n,h] T2.
-#a #h #n #G #L #T1 #H0 #T2 #H1 #H2
+#h #a #n #G #L #T1 #H0 #T2 #H1 #H2
@(cpm_tdeq_ind … H0 … H1 H2) -L -T1 -T2
[ #I #L #H #F #K destruct //
| #L #s #H #F #K destruct //
(* Advanced inversion lemmas with restricted rt-transition for terms ********)
-lemma cpm_tdeq_inv_bind_sn_void (a) (h) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
+lemma cpm_tdeq_inv_bind_sn_void (h) (a) (n) (p) (I) (G) (L):
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![h,a] →
∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
-#a #h #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
+ ∃∃T2. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
+#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
/3 width=5 by ex5_intro, cpm_tdeq_free/
qed-.
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-definition IH_cnv_cpm_tdeq_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
+definition IH_cnv_cpm_tdeq_conf_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
/3 width=5 by tdeq_sort, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_bind_bind_aux (a) (h) (p) (I) (G0) (L0) (V0) (T0):
- (∀G,L,T. ⦃G0,L0,ⓑ{p,I}V0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ ⓑ{p,I}V0.T0 ![a,h] →
+fact cnv_cpm_tdeq_conf_lpr_bind_bind_aux (h) (a) (p) (I) (G0) (L0) (V0) (T0):
+ (∀G,L,T. ⦃G0,L0,ⓑ{p,I}V0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓑ{p,I}V0.T0 ![h,a] →
∀n1,T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
∀n2,T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V0.T1 ➡[n2-n1,h] T & ⓑ{p,I}V0.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V0.T2 ➡[n1-n2,h] T & ⓑ{p,I}V0.T2 ≛ T.
-#a #h #p #I #G0 #L0 #V0 #T0 #IH #H0
+#h #a #p #I #G0 #L0 #V0 #T0 #IH #H0
#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #_ #HT0
/3 width=7 by cpm_bind, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_appl_appl_aux (a) (h) (G0) (L0) (V0) (T0):
- (∀G,L,T. ⦃G0,L0,ⓐV0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ ⓐV0.T0 ![a,h] →
+fact cnv_cpm_tdeq_conf_lpr_appl_appl_aux (h) (a) (G0) (L0) (V0) (T0):
+ (∀G,L,T. ⦃G0,L0,ⓐV0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓐV0.T0 ![h,a] →
∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ ⓐV0.T1 ➡[n2-n1,h] T & ⓐV0.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓐV0.T2 ➡[n1-n2,h] T & ⓐV0.T2 ≛ T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #_ #HT0 #_ #_ -n0 -p0 -X01 -X02
/3 width=7 by cpm_appl, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_cast_cast_aux (a) (h) (G0) (L0) (V0) (T0):
- (∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ ⓝV0.T0 ![a,h] →
+fact cnv_cpm_tdeq_conf_lpr_cast_cast_aux (h) (a) (G0) (L0) (V0) (T0):
+ (∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓝV0.T0 ![h,a] →
∀n1,V1. ⦃G0,L0⦄ ⊢ V0 ➡[n1,h] V1 → V0 ≛ V1 →
∀n2,V2. ⦃G0,L0⦄ ⊢ V0 ➡[n2,h] V2 → V0 ≛ V2 →
∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⓝV1.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓝV2.T2 ➡[n1-n2,h] T & ⓝV2.T2 ≛ T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #H1V01 #H2V01 #n2 #V2 #H1V02 #H2V02 #T1 #H1T01 #H2T01 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
/3 width=7 by cpm_cast, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_aux (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
- ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpm_tdeq_conf_lpr a h G L T.
-#a #h #G0 #L0 #T0 #IH1 #G #L * [| * [| * ]]
+fact cnv_cpm_tdeq_conf_lpr_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpm_tdeq_conf_lpr h a G L T.
+#h #a #G0 #L0 #T0 #IH1 #G #L * [| * [| * ]]
[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #H1X2 #H2X2 #L1 #HL1 #L2 #HL2 destruct
elim (cpm_tdeq_inv_atom_sn … H1X1 H2X1) -H1X1 -H2X1 *
elim (cpm_tdeq_inv_atom_sn … H1X2 H2X2) -H1X2 -H2X2 *
]
qed-.
-lemma cnv_cpm_tdeq_conf_lpr (a) (h) (G0) (L0) (T0):
- IH_cnv_cpm_tdeq_conf_lpr a h G0 L0 T0.
-#a #h #G0 #L0 #T0
+lemma cnv_cpm_tdeq_conf_lpr (h) (a) (G0) (L0) (T0):
+ IH_cnv_cpm_tdeq_conf_lpr h a G0 L0 T0.
+#h #a #G0 #L0 #T0
@(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH
/3 width=17 by cnv_cpm_tdeq_conf_lpr_aux/
qed-.
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-definition IH_cnv_cpm_tdeq_cpm_trans (a) (h): relation3 genv lenv term ≝
- λG,L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → T1 ≛ T →
+definition IH_cnv_cpm_tdeq_cpm_trans (h) (a): relation3 genv lenv term ≝
+ λG,L,T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → T1 ≛ T →
∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∃∃T0. ⦃G,L⦄ ⊢ T1 ➡[n2,h] T0 & ⦃G,L⦄ ⊢ T0 ➡[n1,h] T2 & T0 ≛ T2.
(* Transitive properties restricted rt-transition for terms *****************)
-fact cnv_cpm_tdeq_cpm_trans_sub (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_cpm_trans a h G L T) →
- ∀G,L,T1. G0 = G → L0 = L → T0 = T1 → IH_cnv_cpm_tdeq_cpm_trans a h G L T1.
-#a #h #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
+fact cnv_cpm_tdeq_cpm_trans_sub (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_cpm_trans h a G L T) →
+ ∀G,L,T1. G0 = G → L0 = L → T0 = T1 → IH_cnv_cpm_tdeq_cpm_trans h a G L T1.
+#h #a #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
[ #I #_ #_ #_ #_ #n1 #X1 #H1X #H2X #n2 #X2 #HX2 destruct -G0 -L0 -T0
elim (cpm_tdeq_inv_atom_sn … H1X H2X) -H1X -H2X *
[ #H1 #H2 destruct /2 width=4 by ex3_intro/
]
qed-.
-fact cnv_cpm_tdeq_cpm_trans_aux (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- IH_cnv_cpm_tdeq_cpm_trans a h G0 L0 T0.
-#a #h #G0 #L0 #T0
+fact cnv_cpm_tdeq_cpm_trans_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ IH_cnv_cpm_tdeq_cpm_trans h a G0 L0 T0.
+#h #a #G0 #L0 #T0
@(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #IH0
/5 width=10 by cnv_cpm_tdeq_cpm_trans_sub, fqup_fpbg_trans/
qed-.
(* Sub preservation propery with t-bound rt-transition for terms ************)
-fact cnv_cpm_trans_lpr_aux (a) (h):
+fact cnv_cpm_trans_lpr_aux (h) (a):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_trans_lpr a h G1 L1 T1.
-#a #h #G0 #L0 #T0 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_trans_lpr h a G1 L1 T1.
+#h #a #G0 #L0 #T0 #IH2 #IH1 #G1 #L1 * * [|||| * ]
[ #s #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #_ destruct -IH2 -IH1 -H1
elim (cpm_inv_sort1 … H2) -H2 #H #_ destruct //
| #i #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #HL12 destruct -IH2
(* Properties with t-bound evaluation on terms ******************************)
-lemma cnv_cpme_trans (a) (h) (n) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T2 ![a,h].
-#a #h #n #G #L #T1 #HT1 #T2 * #HT12 #_
+lemma cnv_cpme_trans (h) (a) (n) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T2 ![h,a].
+#h #a #n #G #L #T1 #HT1 #T2 * #HT12 #_
/2 width=4 by cnv_cpms_trans/
qed-.
-lemma cnv_cpme_cpms_conf (a) (h) (n) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∀T1. ⦃G,L⦄ ⊢ T ➡*[n,h] T1 →
+lemma cnv_cpme_cpms_conf (h) (a) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀T1. ⦃G,L⦄ ⊢ T ➡*[n,h] T1 →
∀T2. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
-#a #h #n #G #L #T0 #HT0 #T1 #HT01 #T2 * #HT02 #HT2
+#h #a #n #G #L #T0 #HT0 #T1 #HT01 #T2 * #HT02 #HT2
elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 <minus_n_n #T0 #HT10 #HT20
lapply (cprs_inv_cnr_sn … HT20 HT2) -HT20 #H destruct
/2 width=1 by cpme_intro/
(* Main properties with evaluation for t-bound rt-transition on terms *****)
-theorem cnv_cpme_mono (a) (h) (n) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∀T1. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍⦃T1⦄ →
+theorem cnv_cpme_mono (h) (a) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀T1. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍⦃T1⦄ →
∀T2. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍⦃T2⦄ → T1 = T2.
-#a #h #n #G #L #T0 #HT0 #T1 * #HT01 #HT1 #T2 * #HT02 #HT2
+#h #a #n #G #L #T0 #HT0 #T1 * #HT01 #HT1 #T2 * #HT02 #HT2
elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 <minus_n_n #T0 #HT10 #HT20
/3 width=7 by cprs_inv_cnr_sn, canc_dx_eq/
qed-.
(* Sub confluence propery with t-bound rt-computation for terms *************)
-fact cnv_cpms_conf_lpr_tdeq_tdeq_aux (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
+fact cnv_cpms_conf_lpr_tdeq_tdeq_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛ T1 →
∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-#a #h #G #L0 #T0 #IH2 #IH1 #HT0
+#h #a #G #L0 #T0 #IH2 #IH1 #HT0
#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_cpms_tdeq_conf_lpr_aux … IH2 IH1 … H1T01 … H1T02 … HL01 … HL02) -IH2 -IH1 -H1T01 -H1T02 -HL01 -HL02
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_refl_tdneq_sub (a) (h) (G0) (L0) (T0) (m21) (m22):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
+fact cnv_cpms_conf_lpr_refl_tdneq_sub (h) (a) (G0) (L0) (T0) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛ X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T0 ➡*[m21+m22,h] T& ⦃G0,L2⦄ ⊢ T2 ➡*[h] T.
-#a #h #G0 #L0 #T0 #m21 #m22 #IH2 #IH1 #H0
+#h #a #G0 #L0 #T0 #m21 #m22 #IH2 #IH1 #H0
#X2 #HX02 #HnX02 #T2 #HXT2
#L1 #HL01 #L2 #HL02
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_step_tdneq_sub (a) (h) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
+fact cnv_cpms_conf_lpr_step_tdneq_sub (h) (a) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → T0 ≛ X1 → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 → X1 ≛ T1 →
∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛ X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ((∀G,L,T. ⦃G0,L0,X1⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,X1⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
+ ((∀G,L,T. ⦃G0,L0,X1⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,X1⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
∀m21,m22.
∀X2. ⦃G0,L0⦄ ⊢ X1 ➡[m21,h] X2 → (X1 ≛ X2 → ⊥) →
∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-m12,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m12-(m21+m22),h]T
) →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
-#a #h #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #HT0
+#h #a #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #HT0
#X1 #H1X01 #H2X01 #T1 #H1XT1 #H2XT1 #X2 #H1X02 #H2X02 #T2 #HXT2
#L1 #HL01 #L2 #HL02 #IH
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X01 … L0 ?) // #HX1
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_tdeq_tdneq_aux (a) (h) (G0) (L0) (T0) (n1) (m21) (m22):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
+fact cnv_cpms_conf_lpr_tdeq_tdneq_aux (h) (a) (G0) (L0) (T0) (n1) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
∀T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛ T1 →
∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛ X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-n1,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-(m21+m22),h] T.
-#a #h #G0 #L0 #T0 #n1 #m21 #m22 #IH2 #IH1 #HT0
+#h #a #G0 #L0 #T0 #n1 #m21 #m22 #IH2 #IH1 #HT0
#T1 #H1T01 #H2T01
generalize in match m22; generalize in match m21; -m21 -m22
generalize in match IH1; generalize in match IH2;
]
qed-.
-fact cnv_cpms_conf_lpr_tdneq_tdneq_aux (a) (h) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
+fact cnv_cpms_conf_lpr_tdneq_tdneq_aux (h) (a) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → (T0 ≛ X1 → ⊥) → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 →
∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛ X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
-#a #h #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #H0
+#h #a #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #H0
#X1 #HX01 #HnX01 #T1 #HXT1 #X2 #HX02 #HnX02 #T2 #HXT2
#L1 #HL01 #L2 #HL02
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX01 … L0 ?) // #HX1
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_aux (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpms_conf_lpr a h G L T.
-#a #h #G #L #T #IH2 #IH1 #G0 #L0 #T0 #HG #HL #HT
+fact cnv_cpms_conf_lpr_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpms_conf_lpr h a G L T.
+#h #a #G #L #T #IH2 #IH1 #G0 #L0 #T0 #HG #HL #HT
#HT0 #n1 #T1 #HT01 #n2 #T2 #HT02 #L1 #HL01 #L2 #HL02 destruct
elim (tdeq_dec T0 T1) #H2T01
elim (tdeq_dec T0 T2) #H2T02
(* Properties with restricted rt-computation for terms **********************)
-fact cpms_tdneq_fwd_step_sn_aux (a) (h) (n) (G) (L) (T1):
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![a,h] → (T1 ≛ T2 → ⊥) →
- (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
+fact cpms_tdneq_fwd_step_sn_aux (h) (a) (n) (G) (L) (T1):
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![h,a] → (T1 ≛ T2 → ⊥) →
+ (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr h a G0 L0 T0) →
∃∃n1,n2,T0. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛ T0 → ⊥ & ⦃G,L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n.
-#a #h #n #G #L #T1 #T2 #H
+#h #a #n #G #L #T1 #T2 #H
@(cpms_ind_sn … H) -n -T1
[ #_ #H2T2 elim H2T2 -H2T2 //
| #n1 #n2 #T1 #T #H1T1 #H1T2 #IH #H0T1 #H2T12 #IH2 #IH1
]
qed-.
-fact cpms_tdeq_ind_sn (a) (h) (G) (L) (T2) (Q:relation2 …):
- (⦃G,L⦄ ⊢ T2 ![a,h] → Q 0 T2) →
- (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T1 ![a,h] → T1 ≛ T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T ![a,h] → T ≛ T2 → Q n2 T → Q (n1+n2) T1) →
- ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![a,h] → T1 ≛ T2 →
- (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
+fact cpms_tdeq_ind_sn (h) (a) (G) (L) (T2) (Q:relation2 …):
+ (⦃G,L⦄ ⊢ T2 ![h,a] → Q 0 T2) →
+ (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T1 ![h,a] → T1 ≛ T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T ![h,a] → T ≛ T2 → Q n2 T → Q (n1+n2) T1) →
+ ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![h,a] → T1 ≛ T2 →
+ (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr h a G0 L0 T0) →
Q n T1.
-#a #h #G #L #T2 #Q #IB1 #IB2 #n #T1 #H
+#h #a #G #L #T2 #Q #IB1 #IB2 #n #T1 #H
@(cpms_ind_sn … H) -n -T1
[ -IB2 #H0T2 #_ #_ #_ /2 width=1 by/
| #n1 #n2 #T1 #T #H1T1 #H1T2 #IH #H0T1 #H2T12 #IH2 #IH1 -IB1
(* Sub confluence propery with restricted rt-transition for terms ***********)
-fact cnv_cpms_tdeq_strip_lpr_aux (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛ T1 →
+fact cnv_cpms_tdeq_strip_lpr_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![h,a] → T0 ≛ T1 →
∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛ T.
-#a #h #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
+#h #a #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
[ #H0T1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
<minus_O_n <minus_n_O
]
qed-.
-fact cnv_cpms_tdeq_conf_lpr_aux (a) (h) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛ T1 →
+fact cnv_cpms_tdeq_conf_lpr_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![h,a] → T0 ≛ T1 →
∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛ T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛ T.
-#a #h #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
+#h #a #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
generalize in match IH1; generalize in match IH2;
@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
[ #H0T1 #IH2 #IH1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
(* Properties with t-unbound whd evaluation on terms ************************)
-lemma cnv_cpmuwe_trans (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀T2,n. ⦃G,L⦄ ⊢ T1 ➡*𝐍𝐖*[h,n] T2 → ⦃G,L⦄ ⊢ T2 ![a,h].
+lemma cnv_cpmuwe_trans (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*𝐍𝐖*[h,n] T2 → ⦃G,L⦄ ⊢ T2 ![h,a].
/3 width=4 by cpmuwe_fwd_cpms, cnv_cpms_trans/ qed-.
-lemma cnv_R_cpmuwe_total (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∃n. R_cpmuwe h G L T1 n.
+lemma cnv_R_cpmuwe_total (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∃n. R_cpmuwe h G L T1 n.
/4 width=2 by cnv_fwd_fsb, fsb_inv_csx, R_cpmuwe_total_csx/ qed-.
(* Main inversions with head evaluation for t-bound rt-transition on terms **)
-theorem cnv_cpmuwe_mono (a) (h) (G) (L):
- ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] → ∀T1,n1. ⦃G,L⦄ ⊢ T0 ➡*𝐍𝐖*[h,n1] T1 →
- ∀T2,n2. ⦃G,L⦄ ⊢ T0 ➡*𝐍𝐖*[h,n2] T2 →
+theorem cnv_cpmuwe_mono (h) (a) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![h,a] →
+ ∀n1,T1. ⦃G,L⦄ ⊢ T0 ➡*𝐍𝐖*[h,n1] T1 →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T0 ➡*𝐍𝐖*[h,n2] T2 →
∧∧ ⦃G,L⦄ ⊢ T1 ⬌*[h,n2-n1,n1-n2] T2 & T1 ≅ T2.
-#a #h #G #L #T0 #HT0 #T1 #n1 * #HT01 #HT1 #T2 #n2 * #HT02 #HT2
+#h #a #G #L #T0 #HT0 #n1 #T1 * #HT01 #HT1 #n2 #T2 * #HT02 #HT2
elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
/4 width=4 by cpms_div, tweq_canc_dx, conj/
qed-.
(* Advanced Properties with t-unbound whd evaluation on terms ***************)
-lemma cnv_R_cpmuwe_dec (a) (h) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∀n. Decidable (R_cpmuwe h G L T n).
-#a #h #G #L #T1 #HT1 #n
+lemma cnv_R_cpmuwe_dec (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀n. Decidable (R_cpmuwe h G L T n).
+#h #a #G #L #T1 #HT1 #n
elim (cnv_fwd_aaa … HT1) #A #HA
elim (cpme_total_aaa h n … HA) -HA #T2 #HT12
elim (cnuw_dec h G L T2) #HnT1
(* Advanced dproperties *****************************************************)
(* Basic_2A1: uses: snv_lref *)
-lemma cnv_lref_drops (a) (h) (G):
- ∀I,K,V,i,L. ⦃G,K⦄ ⊢ V ![a,h] →
- ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L⦄ ⊢ #i ![a,h].
-#a #h #G #I #K #V #i elim i -i
+lemma cnv_lref_drops (h) (a) (G):
+ ∀I,K,V,i,L. ⦃G,K⦄ ⊢ V ![h,a] →
+ ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L⦄ ⊢ #i ![h,a].
+#h #a #G #I #K #V #i elim i -i
[ #L #HV #H
lapply (drops_fwd_isid … H ?) -H // #H destruct
/2 width=1 by cnv_zero/
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: uses: snv_inv_lref *)
-lemma cnv_inv_lref_drops (a) (h) (G):
- ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] →
- ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ![a,h].
-#a #h #G #i elim i -i
+lemma cnv_inv_lref_drops (h) (a) (G):
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![h,a] →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ![h,a].
+#h #a #G #i elim i -i
[ #L #H
elim (cnv_inv_zero … H) -H #I #K #V #HV #H destruct
/3 width=5 by drops_refl, ex2_3_intro/
]
qed-.
-lemma cnv_inv_lref_pair (a) (h) (G):
- ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] →
- ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ![a,h].
-#a #h #G #i #L #H #I #K #V #HLK
+lemma cnv_inv_lref_pair (h) (a) (G):
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![h,a] →
+ ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ![h,a].
+#h #a #G #i #L #H #I #K #V #HLK
elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #HX
lapply (drops_mono … HLY … HLK) -L #H destruct //
qed-.
-lemma cnv_inv_lref_atom (a) (h) (b) (G):
- ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] → ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⊥.
-#a #h #b #G #i #L #H #Hi
+lemma cnv_inv_lref_atom (h) (a) (b) (G):
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![h,a] → ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⊥.
+#h #a #b #G #i #L #H #Hi
elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
lapply (drops_gen b … HLY) -HLY #HLY
lapply (drops_mono … HLY … Hi) -L #H destruct
qed-.
-lemma cnv_inv_lref_unit (a) (h) (G):
- ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] →
+lemma cnv_inv_lref_unit (h) (a) (G):
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![h,a] →
∀I,K. ⬇*[i] L ≘ K.ⓤ{I} → ⊥.
-#a #h #G #i #L #H #I #K #HLK
+#h #a #G #i #L #H #I #K #HLK
elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
lapply (drops_mono … HLY … HLK) -L #H destruct
qed-.
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: uses: snv_lift *)
-lemma cnv_lifts (a) (h): ∀G. d_liftable1 (cnv a h G).
-#a #h #G #K #T
+lemma cnv_lifts (h) (a): ∀G. d_liftable1 (cnv h a G).
+#h #a #G #K #T
@(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * * [|||| * ]
[ #s #HG #HK #HT #_ #b #f #L #_ #X #H2 destruct
>(lifts_inv_sort1 … H2) -X -K -f //
(* Inversion lemmas with generic slicing for local environments *************)
(* Basic_2A1: uses: snv_inv_lift *)
-lemma cnv_inv_lifts (a) (h): ∀G. d_deliftable1 (cnv a h G).
-#a #h #G #L #U
+lemma cnv_inv_lifts (h) (a): ∀G. d_deliftable1 (cnv h a G).
+#h #a #G #L #U
@(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * * [|||| * ]
[ #s #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
>(lifts_inv_sort2 … H2) -X -L -f //
(* main properties with evaluations for rt-transition on terms **************)
-theorem cnv_dec (a) (h) (G) (L) (T): ac_props a →
- Decidable (⦃G,L⦄ ⊢ T ![a,h]).
-#a #h #G #L #T #Ha
+theorem cnv_dec (h) (a) (G) (L) (T): ac_props a →
+ Decidable (⦃G,L⦄ ⊢ T ![h,a]).
+#h #a #G #L #T #Ha
@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
[ #s #HG #HL #HT destruct -Ha -IH
/2 width=1 by cnv_sort, or_introl/
(* Properties with supclosure ***********************************************)
(* Basic_2A1: uses: snv_fqu_conf *)
-lemma cnv_fqu_conf (a) (h):
+lemma cnv_fqu_conf (h) (a):
∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂ ⦃G2,L2,T2⦄ →
- ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h].
-#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+ ⦃G1,L1⦄ ⊢ T1 ![h,a] → ⦃G2,L2⦄ ⊢ T2 ![h,a].
+#h #a #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I1 #G1 #L1 #V1 #H
elim (cnv_inv_zero … H) -H #I2 #L2 #V2 #HV2 #H destruct //
| * [ #p #I1 | * ] #G1 #L1 #V1 #T1 #H
qed-.
(* Basic_2A1: uses: snv_fquq_conf *)
-lemma cnv_fquq_conf (a) (h):
+lemma cnv_fquq_conf (h) (a):
∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮ ⦃G2,L2,T2⦄ →
- ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h].
-#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [|*]
+ ⦃G1,L1⦄ ⊢ T1 ![h,a] → ⦃G2,L2⦄ ⊢ T2 ![h,a].
+#h #a #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [|*]
/2 width=5 by cnv_fqu_conf/
qed-.
(* Basic_2A1: uses: snv_fqup_conf *)
-lemma cnv_fqup_conf (a) (h):
+lemma cnv_fqup_conf (h) (a):
∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+ ⦃G2,L2,T2⦄ →
- ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h].
-#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+ ⦃G1,L1⦄ ⊢ T1 ![h,a] → ⦃G2,L2⦄ ⊢ T2 ![h,a].
+#h #a #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=5 by fqup_strap1, cnv_fqu_conf/
qed-.
(* Basic_2A1: uses: snv_fqus_conf *)
-lemma cnv_fqus_conf (a) (h):
+lemma cnv_fqus_conf (h) (a):
∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂* ⦃G2,L2,T2⦄ →
- ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h].
-#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H [|*]
+ ⦃G1,L1⦄ ⊢ T1 ![h,a] → ⦃G2,L2⦄ ⊢ T2 ![h,a].
+#h #a #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H [|*]
/2 width=5 by cnv_fqup_conf/
qed-.
(* Note: this is the "big tree" theorem *)
(* Basic_2A1: uses: snv_fwd_fsb *)
-lemma cnv_fwd_fsb (a) (h):
- ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ≥[h] 𝐒⦃G,L,T⦄.
-#a #h #G #L #T #H elim (cnv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
+lemma cnv_fwd_fsb (h) (a):
+ ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ≥[h] 𝐒⦃G,L,T⦄.
+#h #a #G #L #T #H elim (cnv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
qed-.
(* Forward lemmas with strongly rt-normalizing terms ************************)
-lemma cnv_fwd_csx (a) (h):
- ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
-#a #h #G #L #T #H
+lemma cnv_fwd_csx (h) (a):
+ ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #a #G #L #T #H
/3 width=2 by cnv_fwd_fsb, fsb_inv_csx/
qed-.
(* Inversion lemmas with proper parallel rst-computation for closures *******)
-lemma cnv_fpbg_refl_false (a) (h):
- ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥.
+lemma cnv_fpbg_refl_false (h) (a):
+ ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥.
/3 width=7 by cnv_fwd_fsb, fsb_fpbg_refl_false/ qed-.
(* Main preservation properties *********************************************)
(* Basic_2A1: uses: snv_preserve *)
-lemma cnv_preserve (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] →
- ∧∧ IH_cnv_cpms_conf_lpr a h G L T
- & IH_cnv_cpm_trans_lpr a h G L T.
-#a #h #G #L #T #HT
+lemma cnv_preserve (h) (a): ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∧∧ IH_cnv_cpms_conf_lpr h a G L T
+ & IH_cnv_cpm_trans_lpr h a G L T.
+#h #a #G #L #T #HT
lapply (cnv_fwd_fsb … HT) -HT #H
@(fsb_ind_fpbg … H) -G -L -T #G #L #T #_ #IH
@conj [ letin aux ≝ cnv_cpms_conf_lpr_aux | letin aux ≝ cnv_cpm_trans_lpr_aux ]
elim (IH … H) -IH -H //
qed-.
-theorem cnv_cpms_conf_lpr (a) (h) (G) (L) (T): IH_cnv_cpms_conf_lpr a h G L T.
-#a #h #G #L #T #HT elim (cnv_preserve … HT) /2 width=1 by/
+theorem cnv_cpms_conf_lpr (h) (a) (G) (L) (T): IH_cnv_cpms_conf_lpr h a G L T.
+#h #a #G #L #T #HT elim (cnv_preserve … HT) /2 width=1 by/
qed-.
(* Basic_2A1: uses: snv_cpr_lpr *)
-theorem cnv_cpm_trans_lpr (a) (h) (G) (L) (T): IH_cnv_cpm_trans_lpr a h G L T.
-#a #h #G #L #T #HT elim (cnv_preserve … HT) /2 width=2 by/
+theorem cnv_cpm_trans_lpr (h) (a) (G) (L) (T): IH_cnv_cpm_trans_lpr h a G L T.
+#h #a #G #L #T #HT elim (cnv_preserve … HT) /2 width=2 by/
qed-.
(* Advanced preservation properties *****************************************)
-lemma cnv_cpms_conf (a) (h) (G) (L):
- ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] →
+lemma cnv_cpms_conf (h) (a) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![h,a] →
∀n1,T1. ⦃G,L⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T0 ➡*[n2,h] T2 →
∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L⦄ ⊢ T2 ➡*[n1-n2,h] T.
/2 width=8 by cnv_cpms_conf_lpr/ qed-.
(* Basic_2A1: uses: snv_cprs_lpr *)
-lemma cnv_cpms_trans_lpr (a) (h) (G) (L) (T): IH_cnv_cpms_trans_lpr a h G L T.
-#a #h #G #L1 #T1 #HT1 #n #T2 #H
+lemma cnv_cpms_trans_lpr (h) (a) (G) (L) (T): IH_cnv_cpms_trans_lpr h a G L T.
+#h #a #G #L1 #T1 #HT1 #n #T2 #H
@(cpms_ind_dx … H) -n -T2 /3 width=6 by cnv_cpm_trans_lpr/
qed-.
-lemma cnv_cpm_trans (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T2 ![a,h].
+lemma cnv_cpm_trans (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T2 ![h,a].
/2 width=6 by cnv_cpm_trans_lpr/ qed-.
(* Note: this is the preservation property *)
-lemma cnv_cpms_trans (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T2 ![a,h].
+lemma cnv_cpms_trans (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T2 ![h,a].
/2 width=6 by cnv_cpms_trans_lpr/ qed-.
-lemma cnv_lpr_trans (a) (h) (G):
- ∀L1,T. ⦃G,L1⦄ ⊢ T ![a,h] → ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T ![a,h].
+lemma cnv_lpr_trans (h) (a) (G):
+ ∀L1,T. ⦃G,L1⦄ ⊢ T ![h,a] → ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T ![h,a].
/2 width=6 by cnv_cpm_trans_lpr/ qed-.
-lemma cnv_lprs_trans (a) (h) (G):
- ∀L1,T. ⦃G,L1⦄ ⊢ T ![a,h] → ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T ![a,h].
-#a #h #G #L1 #T #HT #L2 #H
+lemma cnv_lprs_trans (h) (a) (G):
+ ∀L1,T. ⦃G,L1⦄ ⊢ T ![h,a] → ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T ![h,a].
+#h #a #G #L1 #T #HT #L2 #H
@(lprs_ind_dx … H) -L2 /2 width=3 by cnv_lpr_trans/
qed-.
(* Forward lemmas with r-equivalence ****************************************)
-lemma cnv_cpms_conf_eq (a) (h) (n) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] →
+lemma cnv_cpms_conf_eq (h) (a) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
∀T1. ⦃G,L⦄ ⊢ T ➡*[n,h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
-#a #h #n #G #L #T #HT #T1 #HT1 #T2 #HT2
+#h #a #n #G #L #T #HT #T1 #HT1 #T2 #HT2
elim (cnv_cpms_conf … HT … HT1 … HT2) -T <minus_n_n #T #HT1 #HT2
/2 width=3 by cprs_div/
qed-.
-lemma cnv_cpms_fwd_appl_sn_decompose (a) (h) (G) (L):
- ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] → ∀n,X. ⦃G,L⦄ ⊢ ⓐV.T ➡*[n,h] X →
- ∃∃U. ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ T ➡*[n,h] U & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X.
-#a #h #G #L #V #T #H0 #n #X #HX
+lemma cnv_cpms_fwd_appl_sn_decompose (h) (a) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h,a] → ∀n,X. ⦃G,L⦄ ⊢ ⓐV.T ➡*[n,h] X →
+ ∃∃U. ⦃G,L⦄ ⊢ T ![h,a] & ⦃G,L⦄ ⊢ T ➡*[n,h] U & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X.
+#h #a #G #L #V #T #H0 #n #X #HX
elim (cnv_inv_appl … H0) #m #p #W #U #_ #_ #HT #_ #_ -m -p -W -U
-elim (cnv_fwd_cpms_total … h n … HT) #U #HTU
+elim (cnv_fwd_cpms_total h … n … HT) #U #HTU
lapply (cpms_appl_dx … V V … HTU) [ // ] #H
/3 width=8 by cnv_cpms_conf_eq, ex3_intro/
qed-.
(* Properties with t-bound rt-equivalence for terms *************************)
-lemma cnv_cpes_dec (a) (h) (n1) (n2) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∀T2. ⦃G,L⦄ ⊢ T2 ![a,h] →
+lemma cnv_cpes_dec (h) (a) (n1) (n2) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∀T2. ⦃G,L⦄ ⊢ T2 ![h,a] →
Decidable … (⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2).
-#a #h #n1 #n2 #G #L #T1 #HT1 #T2 #HT2
+#h #a #n1 #n2 #G #L #T1 #HT1 #T2 #HT2
elim (cnv_fwd_aaa … HT1) #A1 #HA1
elim (cnv_fwd_aaa … HT2) #A2 #HA2
elim (cpme_total_aaa h n1 … HA1) -HA1 #U1 #HTU1
(* Inductive premises for the preservation results **************************)
-definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝
- λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![a,h] →
+definition IH_cnv_cpm_trans_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![h,a] →
∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡[n,h] T2 →
- ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![a,h].
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![h,a].
-definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝
- λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![a,h] →
+definition IH_cnv_cpms_trans_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![h,a] →
∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2 →
- ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![a,h].
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![h,a].
-definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
+definition IH_cnv_cpm_conf_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 →
∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-definition IH_cnv_cpms_strip_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
+definition IH_cnv_cpms_strip_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 →
∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-definition IH_cnv_cpms_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] →
+definition IH_cnv_cpms_conf_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡*[n2,h] T2 →
∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
(* Auxiliary properties for preservation ************************************)
-fact cnv_cpms_trans_lpr_sub (a) (h):
+fact cnv_cpms_trans_lpr_sub (h) (a):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1.
-#a #h #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_trans_lpr h a G1 L1 T1.
+#h #a #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
@(cpms_ind_dx … H) -n -T2
/3 width=7 by fpbg_cpms_trans/
qed-.
-fact cnv_cpm_conf_lpr_sub (a) (h):
+fact cnv_cpm_conf_lpr_sub (h) (a):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_conf_lpr h a G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.
-fact cnv_cpms_strip_lpr_sub (a) (h):
+fact cnv_cpms_strip_lpr_sub (h) (a):
∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_strip_lpr h a G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.
(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE VALIDITY *************************)
-inductive lsubv (a) (h) (G): relation lenv ≝
-| lsubv_atom: lsubv a h G (⋆) (⋆)
-| lsubv_bind: ∀I,L1,L2. lsubv a h G L1 L2 → lsubv a h G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubv_beta: ∀L1,L2,W,V. ⦃G,L1⦄ ⊢ ⓝW.V ![a,h] →
- lsubv a h G L1 L2 → lsubv a h G (L1.ⓓⓝW.V) (L2.ⓛW)
+inductive lsubv (h) (a) (G): relation lenv ≝
+| lsubv_atom: lsubv h a G (⋆) (⋆)
+| lsubv_bind: ∀I,L1,L2. lsubv h a G L1 L2 → lsubv h a G (L1.ⓘ{I}) (L2.ⓘ{I})
+| lsubv_beta: ∀L1,L2,W,V. ⦃G,L1⦄ ⊢ ⓝW.V ![h,a] →
+ lsubv h a G L1 L2 → lsubv h a G (L1.ⓓⓝW.V) (L2.ⓛW)
.
interpretation
"local environment refinement (native validity)"
- 'LRSubEqV a h G L1 L2 = (lsubv a h G L1 L2).
+ 'LRSubEqV h a G L1 L2 = (lsubv h a G L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact lsubv_inv_atom_sn_aux (a) (h) (G):
- ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L1 = ⋆ → L2 = ⋆.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_atom_sn_aux (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → L1 = ⋆ → L2 = ⋆.
+#h #a #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #_ #H destruct
| #L1 #L2 #W #V #_ #_ #H destruct
qed-.
(* Basic_2A1: uses: lsubsv_inv_atom1 *)
-lemma lsubv_inv_atom_sn (a) (h) (G):
- ∀L2. G ⊢ ⋆ ⫃![a,h] L2 → L2 = ⋆.
+lemma lsubv_inv_atom_sn (h) (a) (G):
+ ∀L2. G ⊢ ⋆ ⫃![h,a] L2 → L2 = ⋆.
/2 width=6 by lsubv_inv_atom_sn_aux/ qed-.
-fact lsubv_inv_bind_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
+fact lsubv_inv_bind_sn_aux (h) (a) (G): ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
∀I,K1. L1 = K1.ⓘ{I} →
- ∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] & G ⊢ K1 ⫃![a,h] K2
+ ∨∨ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![h,a] & G ⊢ K1 ⫃![h,a] K2
& I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
-#a #h #G #L1 #L2 * -L1 -L2
+#h #a #G #L1 #L2 * -L1 -L2
[ #J #K1 #H destruct
| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
| #L1 #L2 #W #V #HWV #HL12 #J #K1 #H destruct /3 width=7 by ex4_3_intro, or_intror/
qed-.
(* Basic_2A1: uses: lsubsv_inv_pair1 *)
-lemma lsubv_inv_bind_sn (a) (h) (G):
- ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⫃![a,h] L2 →
- ∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] & G ⊢ K1 ⫃![a,h] K2
+lemma lsubv_inv_bind_sn (h) (a) (G):
+ ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⫃![h,a] L2 →
+ ∨∨ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![h,a] & G ⊢ K1 ⫃![h,a] K2
& I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
/2 width=3 by lsubv_inv_bind_sn_aux/ qed-.
-fact lsubv_inv_atom_dx_aux (a) (h) (G):
- ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L2 = ⋆ → L1 = ⋆.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_atom_dx_aux (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → L2 = ⋆ → L1 = ⋆.
+#h #a #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #_ #H destruct
| #L1 #L2 #W #V #_ #_ #H destruct
qed-.
(* Basic_2A1: uses: lsubsv_inv_atom2 *)
-lemma lsubv_inv_atom2 (a) (h) (G):
- ∀L1. G ⊢ L1 ⫃![a,h] ⋆ → L1 = ⋆.
+lemma lsubv_inv_atom2 (h) (a) (G):
+ ∀L1. G ⊢ L1 ⫃![h,a] ⋆ → L1 = ⋆.
/2 width=6 by lsubv_inv_atom_dx_aux/ qed-.
-fact lsubv_inv_bind_dx_aux (a) (h) (G):
- ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
+fact lsubv_inv_bind_dx_aux (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
∀I,K2. L2 = K2.ⓘ{I} →
- ∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
- G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
-#a #h #G #L1 #L2 * -L1 -L2
+ ∨∨ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![h,a] &
+ G ⊢ K1 ⫃![h,a] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+#h #a #G #L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
| #L1 #L2 #W #V #HWV #HL12 #J #K2 #H destruct /3 width=7 by ex4_3_intro, or_intror/
qed-.
(* Basic_2A1: uses: lsubsv_inv_pair2 *)
-lemma lsubv_inv_bind_dx (a) (h) (G):
- ∀I,L1,K2. G ⊢ L1 ⫃![a,h] K2.ⓘ{I} →
- ∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] &
- G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+lemma lsubv_inv_bind_dx (h) (a) (G):
+ ∀I,L1,K2. G ⊢ L1 ⫃![h,a] K2.ⓘ{I} →
+ ∨∨ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![h,a] &
+ G ⊢ K1 ⫃![h,a] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
/2 width=3 by lsubv_inv_bind_dx_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsubv_inv_abst_sn (a) (h) (G):
- ∀K1,L2,W. G ⊢ K1.ⓛW ⫃![a,h] L2 →
- ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓛW.
-#a #h #G #K1 #L2 #W #H
+lemma lsubv_inv_abst_sn (h) (a) (G):
+ ∀K1,L2,W. G ⊢ K1.ⓛW ⫃![h,a] L2 →
+ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓛW.
+#h #a #G #K1 #L2 #W #H
elim (lsubv_inv_bind_sn … H) -H // *
#K2 #XW #XV #_ #_ #H1 #H2 destruct
qed-.
(* Basic properties *********************************************************)
(* Basic_2A1: uses: lsubsv_refl *)
-lemma lsubv_refl (a) (h) (G): reflexive … (lsubv a h G).
-#a #h #G #L elim L -L /2 width=1 by lsubv_atom, lsubv_bind/
+lemma lsubv_refl (h) (a) (G): reflexive … (lsubv h a G).
+#h #a #G #L elim L -L /2 width=1 by lsubv_atom, lsubv_bind/
qed.
(* Basic_2A1: removed theorems 3:
(* Forward lemmas with native validity **************************************)
(* Basic_2A1: uses: lsubsv_snv_trans *)
-lemma lsubv_cnv_trans (a) (h) (G):
- ∀L2,T. ⦃G,L2⦄ ⊢ T ![a,h] →
- ∀L1. G ⊢ L1 ⫃![a,h] L2 → ⦃G,L1⦄ ⊢ T ![a,h].
-#a #h #G #L2 #T #H elim H -G -L2 -T //
+lemma lsubv_cnv_trans (h) (a) (G):
+ ∀L2,T. ⦃G,L2⦄ ⊢ T ![h,a] →
+ ∀L1. G ⊢ L1 ⫃![h,a] L2 → ⦃G,L1⦄ ⊢ T ![h,a].
+#h #a #G #L2 #T #H elim H -G -L2 -T //
[ #I #G #K2 #V #HV #IH #L1 #H
elim (lsubv_inv_bind_dx … H) -H * /3 width=1 by cnv_zero/
| #I #G #K2 #i #_ #IH #L1 #H
(* Forward lemmas with context-sensitive r-equivalence for terms ************)
(* Basic_2A1: uses: lsubsv_cprs_trans *)
-lemma lsubv_cpcs_trans (a) (h) (G): lsub_trans … (cpcs h G) (lsubv a h G).
+lemma lsubv_cpcs_trans (h) (a) (G): lsub_trans … (cpcs h G) (lsubv h a G).
/3 width=6 by lsubv_fwd_lsubr, lsubr_cpcs_trans/
qed-.
(* Forward lemmas with t-bound contex-sensitive rt-computation for terms ****)
(* Basic_2A1: uses: lsubsv_cprs_trans lsubsv_scpds_trans *)
-lemma lsubv_cpms_trans (a) (n) (h) (G):
- lsub_trans … (λL. cpms h G L n) (lsubv a h G).
+lemma lsubv_cpms_trans (h) (a) (n) (G):
+ lsub_trans … (λL. cpms h G L n) (lsubv h a G).
/3 width=6 by lsubv_fwd_lsubr, lsubr_cpms_trans/
qed-.
(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
(* Basic_2A1: includes: lsubsv_drop_O1_conf *)
-lemma lsubv_drops_conf_isuni (a) (h) (G):
- ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
+lemma lsubv_drops_conf_isuni (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b,f] L1 ≘ K1 →
- ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & ⬇*[b,f] L2 ≘ K2.
-#a #h #G #L1 #L2 #H elim H -L1 -L2
+ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & ⬇*[b,f] L2 ≘ K2.
+#h #a #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K1 #Hf #H
elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
(* Basic_2A1: includes: lsubsv_drop_O1_trans *)
-lemma lsubv_drops_trans_isuni (a) (h) (G):
- ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
+lemma lsubv_drops_trans_isuni (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2 →
- ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & ⬇*[b,f] L1 ≘ K1.
-#a #h #G #L1 #L2 #H elim H -L1 -L2
+ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & ⬇*[b,f] L1 ≘ K1.
+#h #a #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
(* Forward lemmas with lenv refinement for atomic arity assignment **********)
(* Basic_2A1: uses: lsubsv_fwd_lsuba *)
-lemma lsubsv_fwd_lsuba (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → G ⊢ L1 ⫃⁝ L2.
-#a #h #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsuba_bind/
+lemma lsubsv_fwd_lsuba (h) (a) (G): ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → G ⊢ L1 ⫃⁝ L2.
+#h #a #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsuba_bind/
#L1 #L2 #W #V #H #_ #IH
elim (cnv_inv_cast … H) -H #W0 #HW #HV #HW0 #HVW0
elim (cnv_fwd_aaa … HW) -HW #B #HW
(* Forward lemmas with restricted refinement for local environments *********)
(* Basic_2A1: uses: lsubsv_fwd_lsubr *)
-lemma lsubv_fwd_lsubr (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L1 ⫃ L2.
-#a #h #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_beta/
+lemma lsubv_fwd_lsubr (h) (a) (G): ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → L1 ⫃ L2.
+#h #a #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_beta/
qed-.
(* Main properties **********************************************************)
(* Note: not valid in Basic_2A1 *)
-theorem lsubv_trans (a) (h) (G): Transitive … (lsubv a h G).
-#a #h #G #L1 #L #H elim H -L1 -L //
+theorem lsubv_trans (h) (a) (G): Transitive … (lsubv h a G).
+#h #a #G #L1 #L #H elim H -L1 -L //
[ #I #K1 #K #HK1 #IH #Y #H
elim (lsubv_inv_bind_sn … H) -H *
[ #K2 #HK2 #H destruct /3 width=1 by lsubv_bind/
(**************************************************************************)
include "basic_2/notation/relations/colon_6.ma".
-include "basic_2/notation/relations/colon_5.ma".
-include "basic_2/notation/relations/colonstar_5.ma".
include "basic_2/dynamic/cnv.ma".
(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
-definition nta (a) (h): relation4 genv lenv term term ≝
- λG,L,T,U. ⦃G,L⦄ ⊢ ⓝU.T ![a,h].
+definition nta (h) (a): relation4 genv lenv term term ≝
+ λG,L,T,U. ⦃G,L⦄ ⊢ ⓝU.T ![h,a].
interpretation "native type assignment (term)"
- 'Colon a h G L T U = (nta a h G L T U).
-
-interpretation "restricted native type assignment (term)"
- 'Colon h G L T U = (nta (ac_eq (S O)) h G L T U).
-
-interpretation "extended native type assignment (term)"
- 'ColonStar h G L T U = (nta ac_top h G L T U).
+ 'Colon h a G L T U = (nta h a G L T U).
(* Basic properties *********************************************************)
(* Basic_1: was by definition: ty3_sort *)
(* Basic_2A1: was by definition: nta_sort ntaa_sort *)
-lemma nta_sort (a) (h) (G) (L) (s): ⦃G,L⦄ ⊢ ⋆s :[a,h] ⋆(⫯[h]s).
-#a #h #G #L #s /2 width=3 by cnv_sort, cnv_cast, cpms_sort/
+lemma nta_sort (h) (a) (G) (L): ∀s. ⦃G,L⦄ ⊢ ⋆s :[h,a] ⋆(⫯[h]s).
+#h #a #G #L #s /2 width=3 by cnv_sort, cnv_cast, cpms_sort/
qed.
-lemma nta_bind_cnv (a) (h) (G) (K):
- ∀V. ⦃G,K⦄ ⊢ V ![a,h] →
- ∀I,T,U. ⦃G,K.ⓑ{I}V⦄ ⊢ T :[a,h] U →
- ∀p. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[a,h] ⓑ{p,I}V.U.
-#a #h #G #K #V #HV #I #T #U #H #p
+lemma nta_bind_cnv (h) (a) (G) (K):
+ ∀V. ⦃G,K⦄ ⊢ V ![h,a] →
+ ∀I,T,U. ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,a] U →
+ ∀p. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[h,a] ⓑ{p,I}V.U.
+#h #a #G #K #V #HV #I #T #U #H #p
elim (cnv_inv_cast … H) -H #X #HU #HT #HUX #HTX
/3 width=5 by cnv_bind, cnv_cast, cpms_bind_dx/
qed.
(* Basic_2A1: was by definition: nta_cast *)
-lemma nta_cast (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ ⓝU.T :[a,h] U.
-#a #h #G #L #T #U #H
+lemma nta_cast (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ ⓝU.T :[h,a] U.
+#h #a #G #L #T #U #H
elim (cnv_inv_cast … H) #X #HU #HT #HUX #HTX
/3 width=3 by cnv_cast, cpms_eps/
qed.
(* Basic_1: was by definition: ty3_cast *)
-lemma nta_cast_old (a) (h) (G) (L):
- ∀T0,T1. ⦃G,L⦄ ⊢ T0 :[a,h] T1 →
- ∀T2. ⦃G,L⦄ ⊢ T1 :[a,h] T2 → ⦃G,L⦄ ⊢ ⓝT1.T0 :[a,h] ⓝT2.T1.
-#a #h #G #L #T0 #T1 #H1 #T2 #H2
+lemma nta_cast_old (h) (a) (G) (L):
+ ∀T0,T1. ⦃G,L⦄ ⊢ T0 :[h,a] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T1 :[h,a] T2 → ⦃G,L⦄ ⊢ ⓝT1.T0 :[h,a] ⓝT2.T1.
+#h #a #G #L #T0 #T1 #H1 #T2 #H2
elim (cnv_inv_cast … H1) #X1 #_ #_ #HTX1 #HTX01
elim (cnv_inv_cast … H2) #X2 #_ #_ #HTX2 #HTX12
/3 width=3 by cnv_cast, cpms_eps/
(* Basic inversion lemmas ***************************************************)
-lemma nta_inv_gref_sn (a) (h) (G) (L):
- ∀X2,l. ⦃G,L⦄ ⊢ §l :[a,h] X2 → ⊥.
-#a #h #G #L #X2 #l #H
+lemma nta_inv_gref_sn (h) (a) (G) (L):
+ ∀X2,l. ⦃G,L⦄ ⊢ §l :[h,a] X2 → ⊥.
+#h #a #G #L #X2 #l #H
elim (cnv_inv_cast … H) -H #X #_ #H #_ #_
elim (cnv_inv_gref … H)
qed-.
(* Basic_forward lemmas *****************************************************)
-lemma nta_fwd_cnv_sn (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ T ![a,h].
-#a #h #G #L #T #U #H
+lemma nta_fwd_cnv_sn (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ T ![h,a].
+#h #a #G #L #T #U #H
elim (cnv_inv_cast … H) -H #X #_ #HT #_ #_ //
qed-.
(* Note: this is nta_fwd_correct_cnv *)
-lemma nta_fwd_cnv_dx (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ U ![a,h].
-#a #h #G #L #T #U #H
+lemma nta_fwd_cnv_dx (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ U ![h,a].
+#h #a #G #L #T #U #H
elim (cnv_inv_cast … H) -H #X #HU #_ #_ #_ //
qed-.
(* Forward lemmas with atomic arity assignment for terms ********************)
(* Note: this means that no type is a universe *)
-lemma nta_fwd_aaa (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃∃A. ⦃G,L⦄ ⊢ T ⁝ A & ⦃G,L⦄ ⊢ U ⁝ A.
-#a #h #G #L #T #U #H
+lemma nta_fwd_aaa (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ∃∃A. ⦃G,L⦄ ⊢ T ⁝ A & ⦃G,L⦄ ⊢ U ⁝ A.
+#h #a #G #L #T #U #H
elim (cnv_fwd_aaa … H) -H #A #H
elim (aaa_inv_cast … H) -H #HU #HT
/2 width=3 by ex2_intro/
(* Advanced inversion lemmas ************************************************)
(* Basic_1: uses: ty3_predicative *)
-lemma nta_abst_predicative (a) (h) (p) (G) (L):
- ∀W,T. ⦃G,L⦄ ⊢ ⓛ{p}W.T :[a,h] W → ⊥.
-#a #h #p #G #L #W #T #H
+lemma nta_abst_predicative (h) (a) (p) (G) (L):
+ ∀W,T. ⦃G,L⦄ ⊢ ⓛ{p}W.T :[h,a] W → ⊥.
+#h #a #p #G #L #W #T #H
elim (nta_fwd_aaa … H) -a -h #X #H #H1W
elim (aaa_inv_abst … H) -p #B #A #H2W #_ #H destruct -T
lapply (aaa_mono … H1W … H2W) -G -L -W #H
qed-.
(* Basic_1: uses: ty3_repellent *)
-theorem nta_abst_repellent (a) (h) (p) (G) (K):
- ∀W,T,U1. ⦃G,K⦄ ⊢ ⓛ{p}W.T :[a,h] U1 →
- ∀U2. ⦃G,K.ⓛW⦄ ⊢ T :[a,h] U2 → ⬆*[1] U1 ≘ U2 → ⊥.
-#a #h #p #G #K #W #T #U1 #H1 #U2 #H2 #HU12
+theorem nta_abst_repellent (h) (a) (p) (G) (K):
+ ∀W,T,U1. ⦃G,K⦄ ⊢ ⓛ{p}W.T :[h,a] U1 →
+ ∀U2. ⦃G,K.ⓛW⦄ ⊢ T :[h,a] U2 → ⬆*[1] U1 ≘ U2 → ⊥.
+#h #a #p #G #K #W #T #U1 #H1 #U2 #H2 #HU12
elim (nta_fwd_aaa … H2) -H2 #A2 #H2T #H2U2
elim (nta_fwd_aaa … H1) -H1 #X1 #H1 #HU1
elim (aaa_inv_abst … H1) -a -h -p #B #A1 #_ #H1T #H destruct
(* Properties with r-equivalence for terms **********************************)
-lemma nta_conv_cnv (a) (h) (G) (L) (T):
- ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 →
- ∀U2. ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![a,h] → ⦃G,L⦄ ⊢ T :[a,h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #HU12 #HU2
+lemma nta_conv_cnv (h) (a) (G) (L) (T):
+ ∀U1. ⦃G,L⦄ ⊢ T :[h,a] U1 →
+ ∀U2. ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,a] → ⦃G,L⦄ ⊢ T :[h,a] U2.
+#h #a #G #L #T #U1 #H1 #U2 #HU12 #HU2
elim (cnv_inv_cast … H1) -H1 #X1 #HU1 #HT #HUX1 #HTX1
lapply (cpcs_cprs_conf … HUX1 … HU12) -U1 #H
elim (cpcs_inv_cprs … H) -H #X2 #HX12 #HU12
(* Basic_1: was by definition: ty3_conv *)
(* Basic_2A1: was by definition: nta_conv ntaa_conv *)
-lemma nta_conv (a) (h) (G) (L) (T):
- ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 →
+lemma nta_conv (h) (a) (G) (L) (T):
+ ∀U1. ⦃G,L⦄ ⊢ T :[h,a] U1 →
∀U2. ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 →
- ∀W2. ⦃G,L⦄ ⊢ U2 :[a,h] W2 → ⦃G,L⦄ ⊢ T :[a,h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #HU12 #W2 #H2
+ ∀W2. ⦃G,L⦄ ⊢ U2 :[h,a] W2 → ⦃G,L⦄ ⊢ T :[h,a] U2.
+#h #a #G #L #T #U1 #H1 #U2 #HU12 #W2 #H2
/3 width=3 by nta_conv_cnv, nta_fwd_cnv_sn/
qed-.
(* Basic_1: was: ty3_gen_sort *)
(* Basic_2A1: was: nta_inv_sort1 *)
-lemma nta_inv_sort_sn (a) (h) (G) (L) (X2):
- ∀s. ⦃G,L⦄ ⊢ ⋆s :[a,h] X2 →
- ∧∧ ⦃G,L⦄ ⊢ ⋆(⫯[h]s) ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #s #H
+lemma nta_inv_sort_sn (h) (a) (G) (L) (X2):
+ ∀s. ⦃G,L⦄ ⊢ ⋆s :[h,a] X2 →
+ ∧∧ ⦃G,L⦄ ⊢ ⋆(⫯[h]s) ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #s #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #_ #HX21 #H
lapply (cpms_inv_sort1 … H) -H #H destruct
/3 width=1 by cpcs_cprs_sn, conj/
qed-.
-lemma nta_inv_ldec_sn_cnv (a) (h) (G) (K) (V):
- ∀X2. ⦃G,K.ⓛV⦄ ⊢ #0 :[a,h] X2 →
- ∃∃U. ⦃G,K⦄ ⊢ V ![a,h] & ⬆*[1] V ≘ U & ⦃G,K.ⓛV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓛV⦄ ⊢ X2 ![a,h].
-#a #h #G #Y #X #X2 #H
+lemma nta_inv_ldec_sn_cnv (h) (a) (G) (K) (V):
+ ∀X2. ⦃G,K.ⓛV⦄ ⊢ #0 :[h,a] X2 →
+ ∃∃U. ⦃G,K⦄ ⊢ V ![h,a] & ⬆*[1] V ≘ U & ⦃G,K.ⓛV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓛV⦄ ⊢ X2 ![h,a].
+#h #a #G #Y #X #X2 #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
elim (cpms_inv_ell_sn … H2) -H2 *
(* Properties with advanced rt_computation for terms ************************)
(* Basic_2A1: uses by definition nta_appl ntaa_appl *)
-lemma nta_appl_abst (a) (h) (p) (G) (L):
- ∀n. appl a n →
- ∀V,W. ⦃G,L⦄ ⊢ V :[a,h] W →
- ∀T,U. ⦃G,L.ⓛW⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T :[a,h] ⓐV.ⓛ{p}W.U.
-#a #h #p #G #L #n #Ha #V #W #H1 #T #U #H2
+lemma nta_appl_abst (h) (a) (p) (G) (L):
+ ∀n. ad a n →
+ ∀V,W. ⦃G,L⦄ ⊢ V :[h,a] W →
+ ∀T,U. ⦃G,L.ⓛW⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T :[h,a] ⓐV.ⓛ{p}W.U.
+#h #a #p #G #L #n #Ha #V #W #H1 #T #U #H2
elim (cnv_inv_cast … H1) -H1 #X1 #HW #HV #HWX1 #HVX1
elim (cnv_inv_cast … H2) -H2 #X2 #HU #HT #HUX2 #HTX2
/4 width=11 by cnv_appl_ge, cnv_cast, cnv_bind, cpms_appl_dx, cpms_bind_dx/
(* Basic_1: was by definition: ty3_appl *)
(* Basic_2A1: was nta_appl_old *)
-lemma nta_appl (a) (h) (p) (G) (L):
- ∀n. 1 ≤ n → appl a n →
- ∀V,W. ⦃G,L⦄ ⊢ V :[a,h] W →
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T :[a,h] ⓐV.ⓛ{p}W.U.
-#a #h #p #G #L #n #Hn #Ha #V #W #H1 #T #U #H2
+lemma nta_appl (h) (a) (p) (G) (L):
+ ∀n. 1 ≤ n → ad a n →
+ ∀V,W. ⦃G,L⦄ ⊢ V :[h,a] W →
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T :[h,a] ⓐV.ⓛ{p}W.U.
+#h #a #p #G #L #n #Hn #Ha #V #W #H1 #T #U #H2
elim (cnv_inv_cast … H1) -H1 #X1 #HW #HV #HWX1 #HVX1
elim (cnv_inv_cast … H2) -H2 #X2 #HU #HT #HUX2 #HTX2
elim (cpms_inv_abst_sn … HUX2) #W0 #U0 #HW0 #HU0 #H destruct
(* Inversion lemmas with advanced rt_computation for terms ******************)
-lemma nta_inv_abst_bi_cnv (a) (h) (p) (G) (K) (W):
- ∀T,U. ⦃G,K⦄ ⊢ ⓛ{p}W.T :[a,h] ⓛ{p}W.U →
- ∧∧ ⦃G,K⦄ ⊢ W ![a,h] & ⦃G,K.ⓛW⦄ ⊢ T :[a,h] U.
-#a #h #p #G #K #W #T #U #H
+lemma nta_inv_abst_bi_cnv (h) (a) (p) (G) (K) (W):
+ ∀T,U. ⦃G,K⦄ ⊢ ⓛ{p}W.T :[h,a] ⓛ{p}W.U →
+ ∧∧ ⦃G,K⦄ ⊢ W ![h,a] & ⦃G,K.ⓛW⦄ ⊢ T :[h,a] U.
+#h #a #p #G #K #W #T #U #H
elim (cnv_inv_cast … H) -H #X #HWU #HWT #HUX #HTX
elim (cnv_inv_bind … HWU) -HWU #HW #HU
elim (cnv_inv_bind … HWT) -HWT #_ #HT
(* Advanced properties ******************************************************)
-lemma nta_ldef (a) (h) (G) (K):
- ∀V,W. ⦃G,K⦄ ⊢ V :[a,h] W →
- ∀U. ⬆*[1] W ≘ U → ⦃G,K.ⓓV⦄ ⊢ #0 :[a,h] U.
-#a #h #G #K #V #W #H #U #HWU
+lemma nta_ldef (h) (a) (G) (K):
+ ∀V,W. ⦃G,K⦄ ⊢ V :[h,a] W →
+ ∀U. ⬆*[1] W ≘ U → ⦃G,K.ⓓV⦄ ⊢ #0 :[h,a] U.
+#h #a #G #K #V #W #H #U #HWU
elim (cnv_inv_cast … H) -H #X #HW #HV #HWX #HVX
lapply (cnv_lifts … HW (Ⓣ) … (K.ⓓV) … HWU) -HW
[ /3 width=3 by drops_refl, drops_drop/ ] #HU
/3 width=5 by cnv_zero, cnv_cast, cpms_delta/
qed.
-lemma nta_ldec_cnv (a) (h) (G) (K):
- ∀W. ⦃G,K⦄ ⊢ W ![a,h] →
- ∀U. ⬆*[1] W ≘ U → ⦃G,K.ⓛW⦄ ⊢ #0 :[a,h] U.
-#a #h #G #K #W #HW #U #HWU
+lemma nta_ldec_cnv (h) (a) (G) (K):
+ ∀W. ⦃G,K⦄ ⊢ W ![h,a] →
+ ∀U. ⬆*[1] W ≘ U → ⦃G,K.ⓛW⦄ ⊢ #0 :[h,a] U.
+#h #a #G #K #W #HW #U #HWU
lapply (cnv_lifts … HW (Ⓣ) … (K.ⓛW) … HWU)
/3 width=5 by cnv_zero, cnv_cast, cpms_ell, drops_refl, drops_drop/
qed.
-lemma nta_lref (a) (h) (I) (G) (K):
- ∀T,i. ⦃G,K⦄ ⊢ #i :[a,h] T →
- ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #(↑i) :[a,h] U.
-#a #h #I #G #K #T #i #H #U #HTU
+lemma nta_lref (h) (a) (I) (G) (K):
+ ∀T,i. ⦃G,K⦄ ⊢ #i :[h,a] T →
+ ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #(↑i) :[h,a] U.
+#h #a #I #G #K #T #i #H #U #HTU
elim (cnv_inv_cast … H) -H #X #HT #Hi #HTX #H2
lapply (cnv_lifts … HT (Ⓣ) … (K.ⓘ{I}) … HTU) -HT
[ /3 width=3 by drops_refl, drops_drop/ ] #HU
(* Properties with generic slicing for local environments *******************)
-lemma nta_lifts_sn (a) (h) (G): d_liftable2_sn … lifts (nta a h G).
-#a #h #G #K #T1 #T2 #H #b #f #L #HLK #U1 #HTU1
+lemma nta_lifts_sn (h) (a) (G): d_liftable2_sn … lifts (nta a h G).
+#h #a #G #K #T1 #T2 #H #b #f #L #HLK #U1 #HTU1
elim (cnv_inv_cast … H) -H #X #HT2 #HT1 #HT2X #HT1X
elim (lifts_total T2 f) #U2 #HTU2
lapply (cnv_lifts … HT2 … HLK … HTU2) -HT2 #HU2
(* Basic_1: was: ty3_lift *)
(* Basic_2A1: was: nta_lift ntaa_lift *)
-lemma nta_lifts_bi (a) (h) (G): d_liftable2_bi … lifts (nta a h G).
+lemma nta_lifts_bi (h) (a) (G): d_liftable2_bi … lifts (nta a h G).
/3 width=12 by nta_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
(* Basic_1: was by definition: ty3_abbr *)
(* Basic_2A1: was by definition: nta_ldef ntaa_ldef *)
-lemma nta_ldef_drops (a) (h) (G) (K) (L) (i):
- ∀V,W. ⦃G,K⦄ ⊢ V :[a,h] W →
- ∀U. ⬆*[↑i] W ≘ U → ⬇*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ #i :[a,h] U.
-#a #h #G #K #L #i #V #W #HVW #U #HWU #HLK
+lemma nta_ldef_drops (h) (a) (G) (K) (L) (i):
+ ∀V,W. ⦃G,K⦄ ⊢ V :[h,a] W →
+ ∀U. ⬆*[↑i] W ≘ U → ⬇*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ #i :[h,a] U.
+#h #a #G #K #L #i #V #W #HVW #U #HWU #HLK
elim (lifts_split_trans … HWU (𝐔❴1❵) (𝐔❴i❵)) [| // ] #X #HWX #HXU
/3 width=9 by nta_lifts_bi, nta_ldef/
qed.
-lemma nta_ldec_drops_cnv (a) (h) (G) (K) (L) (i):
- ∀W. ⦃G,K⦄ ⊢ W ![a,h] →
- ∀U. ⬆*[↑i] W ≘ U → ⬇*[i] L ≘ K.ⓛW → ⦃G,L⦄ ⊢ #i :[a,h] U.
-#a #h #G #K #L #i #W #HW #U #HWU #HLK
+lemma nta_ldec_drops_cnv (h) (a) (G) (K) (L) (i):
+ ∀W. ⦃G,K⦄ ⊢ W ![h,a] →
+ ∀U. ⬆*[↑i] W ≘ U → ⬇*[i] L ≘ K.ⓛW → ⦃G,L⦄ ⊢ #i :[h,a] U.
+#h #a #G #K #L #i #W #HW #U #HWU #HLK
elim (lifts_split_trans … HWU (𝐔❴1❵) (𝐔❴i❵)) [| // ] #X #HWX #HXU
/3 width=9 by nta_lifts_bi, nta_ldec_cnv/
qed.
(* Properties with evaluations for rt-transition on terms *******************)
-lemma nta_typecheck_dec (a) (h) (G) (L): ac_props a →
- ∀T,U. Decidable … (⦃G,L⦄ ⊢ T :[a,h] U).
+lemma nta_typecheck_dec (h) (a) (G) (L): ac_props a →
+ ∀T,U. Decidable … (⦃G,L⦄ ⊢ T :[h,a] U).
/2 width=1 by cnv_dec/ qed-.
(* Basic_1: uses: ty3_inference *)
-lemma nta_inference_dec (a) (h) (G) (L) (T): ac_props a →
- ∨∨ ∃U. ⦃G,L⦄ ⊢ T :[a,h] U
- | ∀U. (⦃G,L⦄ ⊢ T :[a,h] U → ⊥).
-#a #h #G #L #T #Ha
-elim (cnv_dec … h G L T Ha) -Ha
+lemma nta_inference_dec (h) (a) (G) (L) (T): ac_props a →
+ Decidable (∃U. ⦃G,L⦄ ⊢ T :[h,a] U).
+#h #a #G #L #T #Ha
+elim (cnv_dec h … G L T Ha) -Ha #HT
[ /3 width=1 by cnv_nta_sn, or_introl/
-| /4 width=2 by nta_fwd_cnv_sn, or_intror/
+| @or_intror * #U #HTU
+ /3 width=2 by nta_fwd_cnv_sn/
]
qed-.
(* Note: this might use fsb_inv_cast (still to be proved) *)
(* Basic_1: uses: ty3_sn3 *)
(* Basic_2A1: uses: nta_fwd_csn *)
-theorem nta_fwd_fsb (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U →
+theorem nta_fwd_fsb (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U →
∧∧ ≥[h] 𝐒⦃G,L,T⦄ & ≥[h] 𝐒⦃G,L,U⦄.
-#a #h #G #L #T #U #H
+#h #a #G #L #T #U #H
elim (cnv_inv_cast … H) #X #HU #HT #_ #_ -X
/3 width=2 by cnv_fwd_fsb, conj/
qed-.
lemma nta_ind_rest_cnv (h) (Q:relation4 …):
(∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
(∀G,K,V,W,U.
- ⦃G,K⦄ ⊢ V :[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ V :[h,𝟐] W → ⬆*[1] W ≘ U →
Q G K V W → Q G (K.ⓓV) (#0) U
) →
- (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+ (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h,𝟐] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
(∀I,G,K,W,U,i.
- ⦃G,K⦄ ⊢ #i :[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ #i :[h,𝟐] W → ⬆*[1] W ≘ U →
Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
) →
(∀p,I,G,K,V,T,U.
- ⦃G,K⦄ ⊢ V ![h] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h] U →
+ ⦃G,K⦄ ⊢ V ![h,𝟐] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,𝟐] U →
Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
) →
(∀p,G,L,V,W,T,U.
- ⦃G,L⦄ ⊢ V :[h] W → ⦃G,L⦄ ⊢ T :[h] ⓛ{p}W.U →
+ ⦃G,L⦄ ⊢ V :[h,𝟐] W → ⦃G,L⦄ ⊢ T :[h,𝟐] ⓛ{p}W.U →
Q G L V W → Q G L T (ⓛ{p}W.U) → Q G L (ⓐV.T) (ⓐV.ⓛ{p}W.U)
) →
- (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h] U → Q G L T U → Q G L (ⓝU.T) U
+ (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝟐] U → Q G L T U → Q G L (ⓝU.T) U
) →
(∀G,L,T,U1,U2.
- ⦃G,L⦄ ⊢ T :[h] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h] →
+ ⦃G,L⦄ ⊢ T :[h,𝟐] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,𝟐] →
Q G L T U1 → Q G L T U2
) →
- ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h] U → Q G L T U.
+ ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝟐] U → Q G L T U.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #G #L #T
@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
[ #s #HG #HL #HT #X #H destruct -IH
lemma nta_ind_ext_cnv_mixed (h) (Q:relation4 …):
(∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
(∀G,K,V,W,U.
- ⦃G,K⦄ ⊢ V :*[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ V :[h,𝛚] W → ⬆*[1] W ≘ U →
Q G K V W → Q G (K.ⓓV) (#0) U
) →
- (∀G,K,W,U. ⦃G,K⦄ ⊢ W !*[h] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+ (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h,𝛚] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
(∀I,G,K,W,U,i.
- ⦃G,K⦄ ⊢ #i :*[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ #i :[h,𝛚] W → ⬆*[1] W ≘ U →
Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
) →
(∀p,I,G,K,V,T,U.
- ⦃G,K⦄ ⊢ V !*[h] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :*[h] U →
+ ⦃G,K⦄ ⊢ V ![h,𝛚] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,𝛚] U →
Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
) →
(∀p,G,L,V,W,T,U.
- ⦃G,L⦄ ⊢ V :*[h] W → ⦃G,L⦄ ⊢ T :*[h] ⓛ{p}W.U →
+ ⦃G,L⦄ ⊢ V :[h,𝛚] W → ⦃G,L⦄ ⊢ T :[h,𝛚] ⓛ{p}W.U →
Q G L V W → Q G L T (ⓛ{p}W.U) → Q G L (ⓐV.T) (ⓐV.ⓛ{p}W.U)
) →
(∀G,L,V,T,U.
- ⦃G,L⦄ ⊢ T :*[h] U → ⦃G,L⦄ ⊢ ⓐV.U !*[h] →
+ ⦃G,L⦄ ⊢ T :[h,𝛚] U → ⦃G,L⦄ ⊢ ⓐV.U ![h,𝛚] →
Q G L T U → Q G L (ⓐV.T) (ⓐV.U)
) →
- (∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U → Q G L (ⓝU.T) U
+ (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U → Q G L T U → Q G L (ⓝU.T) U
) →
(∀G,L,T,U1,U2.
- ⦃G,L⦄ ⊢ T :*[h] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 !*[h] →
+ ⦃G,L⦄ ⊢ T :[h,𝛚] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,𝛚] →
Q G L T U1 → Q G L T U2
) →
- ∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U.
+ ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U → Q G L T U.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T
@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
[ #s #HG #HL #HT #X #H destruct -IH
lemma nta_ind_ext_cnv (h) (Q:relation4 …):
(∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
(∀G,K,V,W,U.
- ⦃G,K⦄ ⊢ V :*[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ V :[h,𝛚] W → ⬆*[1] W ≘ U →
Q G K V W → Q G (K.ⓓV) (#0) U
) →
- (∀G,K,W,U. ⦃G,K⦄ ⊢ W !*[h] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+ (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h,𝛚] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
(∀I,G,K,W,U,i.
- ⦃G,K⦄ ⊢ #i :*[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ #i :[h,𝛚] W → ⬆*[1] W ≘ U →
Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
) →
(∀p,I,G,K,V,T,U.
- ⦃G,K⦄ ⊢ V !*[h] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :*[h] U →
+ ⦃G,K⦄ ⊢ V ![h,𝛚] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,𝛚] U →
Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
) →
(∀p,G,K,V,W,T,U.
- ⦃G,K⦄ ⊢ V :*[h] W → ⦃G,K.ⓛW⦄ ⊢ T :*[h] U →
+ ⦃G,K⦄ ⊢ V :[h,𝛚] W → ⦃G,K.ⓛW⦄ ⊢ T :[h,𝛚] U →
Q G K V W → Q G (K.ⓛW) T U → Q G K (ⓐV.ⓛ{p}W.T) (ⓐV.ⓛ{p}W.U)
) →
(∀G,L,V,T,U.
- ⦃G,L⦄ ⊢ T :*[h] U → ⦃G,L⦄ ⊢ ⓐV.U !*[h] →
+ ⦃G,L⦄ ⊢ T :[h,𝛚] U → ⦃G,L⦄ ⊢ ⓐV.U ![h,𝛚] →
Q G L T U → Q G L (ⓐV.T) (ⓐV.U)
) →
- (∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U → Q G L (ⓝU.T) U
+ (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U → Q G L T U → Q G L (ⓝU.T) U
) →
(∀G,L,T,U1,U2.
- ⦃G,L⦄ ⊢ T :*[h] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 !*[h] →
+ ⦃G,L⦄ ⊢ T :[h,𝛚] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,𝛚] →
Q G L T U1 → Q G L T U2
) →
- ∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U.
+ ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U → Q G L T U.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T #U #H
@(nta_ind_ext_cnv_mixed … IH1 IH2 IH3 IH4 IH5 … IH7 IH8 IH9 … H) -G -L -T -U -IH1 -IH2 -IH3 -IH4 -IH5 -IH6 -IH8 -IH9
#p #G #L #V #W #T #U #HVW #HTU #_ #IHTU
(* Properties based on preservation *****************************************)
-lemma cnv_cpms_nta (a) (h) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∀U.⦃G,L⦄ ⊢ T ➡*[1,h] U → ⦃G,L⦄ ⊢ T :[a,h] U.
+lemma cnv_cpms_nta (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀U.⦃G,L⦄ ⊢ T ➡*[1,h] U → ⦃G,L⦄ ⊢ T :[h,a] U.
/3 width=4 by cnv_cast, cnv_cpms_trans/ qed.
-lemma cnv_nta_sn (a) (h) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T :[a,h] U.
-#a #h #G #L #T #HT
+lemma cnv_nta_sn (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∃U. ⦃G,L⦄ ⊢ T :[h,a] U.
+#h #a #G #L #T #HT
elim (cnv_fwd_cpm_SO … HT) #U #HTU
/4 width=2 by cnv_cpms_nta, cpm_cpms, ex_intro/
qed-.
(* Basic_1: was: ty3_typecheck *)
-lemma nta_typecheck (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃T0. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] T0.
+lemma nta_typecheck (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ∃T0. ⦃G,L⦄ ⊢ ⓝU.T :[h,a] T0.
/3 width=1 by cnv_cast, cnv_nta_sn/ qed-.
(* Basic_1: was: ty3_correct *)
(* Basic_2A1: was: ntaa_fwd_correct *)
-lemma nta_fwd_correct (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃T0. ⦃G,L⦄ ⊢ U :[a,h] T0.
+lemma nta_fwd_correct (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ∃T0. ⦃G,L⦄ ⊢ U :[h,a] T0.
/3 width=2 by nta_fwd_cnv_dx, cnv_nta_sn/ qed-.
lemma nta_pure_cnv (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :*[h] U →
- ∀V. ⦃G,L⦄ ⊢ ⓐV.U !*[h] → ⦃G,L⦄ ⊢ ⓐV.T :*[h] ⓐV.U.
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U →
+ ∀V. ⦃G,L⦄ ⊢ ⓐV.U ![h,𝛚] → ⦃G,L⦄ ⊢ ⓐV.T :[h,𝛚] ⓐV.U.
#h #G #L #T #U #H1 #V #H2
elim (cnv_inv_cast … H1) -H1 #X0 #HU #HT #HUX0 #HTX0
elim (cnv_inv_appl … H2) #n #p #X1 #X2 #_ #HV #_ #HVX1 #HUX2
qed.
(* Basic_1: uses: ty3_sred_wcpr0_pr0 *)
-lemma nta_cpr_conf_lpr (a) (h) (G):
- ∀L1,T1,U. ⦃G,L1⦄ ⊢ T1 :[a,h] U → ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
- ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
+lemma nta_cpr_conf_lpr (h) (a) (G):
+ ∀L1,T1,U. ⦃G,L1⦄ ⊢ T1 :[h,a] U → ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
/3 width=6 by cnv_cpm_trans_lpr, cpm_cast/
qed-.
(* Basic_1: uses: ty3_sred_pr2 ty3_sred_pr0 *)
-lemma nta_cpr_conf (a) (h) (G) (L):
- ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #H #T2 #HT12
+lemma nta_cpr_conf (h) (a) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[h,a] U →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #H #T2 #HT12
/3 width=6 by cnv_cpm_trans, cpm_cast/
qed-.
(* Note: this is the preservation property *)
(* Basic_1: uses: ty3_sred_pr3 ty3_sred_pr1 *)
-lemma nta_cprs_conf (a) (h) (G) (L):
- ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #H #T2 #HT12
+lemma nta_cprs_conf (h) (a) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[h,a] U →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #H #T2 #HT12
/3 width=6 by cnv_cpms_trans, cpms_cast/
qed-.
(* Basic_1: uses: ty3_cred_pr2 *)
-lemma nta_lpr_conf (a) (h) (G):
- ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[a,h] U →
- ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T :[a,h] U.
-#a #h #G #L1 #T #U #HTU #L2 #HL12
+lemma nta_lpr_conf (h) (a) (G):
+ ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[h,a] U →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T :[h,a] U.
+#h #a #G #L1 #T #U #HTU #L2 #HL12
/2 width=3 by cnv_lpr_trans/
qed-.
(* Basic_1: uses: ty3_cred_pr3 *)
-lemma nta_lprs_conf (a) (h) (G):
- ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[a,h] U →
- ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T :[a,h] U.
-#a #h #G #L1 #T #U #HTU #L2 #HL12
+lemma nta_lprs_conf (h) (a) (G):
+ ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[h,a] U →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T :[h,a] U.
+#h #a #G #L1 #T #U #HTU #L2 #HL12
/2 width=3 by cnv_lprs_trans/
qed-.
(* Inversion lemmas based on preservation ***********************************)
-lemma nta_inv_ldef_sn (a) (h) (G) (K) (V):
- ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 :[a,h] X2 →
- ∃∃W,U. ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[1] W ≘ U & ⦃G,K.ⓓV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓓV⦄ ⊢ X2 ![a,h].
-#a #h #G #Y #X #X2 #H
+lemma nta_inv_ldef_sn (h) (a) (G) (K) (V):
+ ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 :[h,a] X2 →
+ ∃∃W,U. ⦃G,K⦄ ⊢ V :[h,a] W & ⬆*[1] W ≘ U & ⦃G,K.ⓓV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓓV⦄ ⊢ X2 ![h,a].
+#h #a #G #Y #X #X2 #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
elim (cpms_inv_delta_sn … H2) -H2 *
]
qed-.
-lemma nta_inv_lref_sn (a) (h) (G) (L):
- ∀X2,i. ⦃G,L⦄ ⊢ #↑i :[a,h] X2 →
- ∃∃I,K,T2,U2. ⦃G,K⦄ ⊢ #i :[a,h] T2 & ⬆*[1] T2 ≘ U2 & ⦃G,K.ⓘ{I}⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,K.ⓘ{I}⦄ ⊢ X2 ![a,h] & L = K.ⓘ{I}.
-#a #h #G #L #X2 #i #H
+lemma nta_inv_lref_sn (h) (a) (G) (L):
+ ∀X2,i. ⦃G,L⦄ ⊢ #↑i :[h,a] X2 →
+ ∃∃I,K,T2,U2. ⦃G,K⦄ ⊢ #i :[h,a] T2 & ⬆*[1] T2 ≘ U2 & ⦃G,K.ⓘ{I}⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,K.ⓘ{I}⦄ ⊢ X2 ![h,a] & L = K.ⓘ{I}.
+#h #a #G #L #X2 #i #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_lref … H1) -H1 #I #K #Hi #H destruct
elim (cpms_inv_lref_sn … H2) -H2 *
]
qed-.
-lemma nta_inv_lref_sn_drops_cnv (a) (h) (G) (L):
- ∀X2,i. ⦃G,L⦄ ⊢ #i :[a,h] X2 →
- ∨∨ ∃∃K,V,W,U. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h]
- | ∃∃K,W,U. ⬇*[i] L ≘ K. ⓛW & ⦃G,K⦄ ⊢ W ![a,h] & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #i #H
+lemma nta_inv_lref_sn_drops_cnv (h) (a) (G) (L):
+ ∀X2,i. ⦃G,L⦄ ⊢ #i :[h,a] X2 →
+ ∨∨ ∃∃K,V,W,U. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V :[h,a] W & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a]
+ | ∃∃K,W,U. ⬇*[i] L ≘ K. ⓛW & ⦃G,K⦄ ⊢ W ![h,a] & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #i #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_lref_drops … H1) -H1 #I #K #V #HLK #HV
elim (cpms_inv_lref1_drops … H2) -H2 *
]
qed-.
-lemma nta_inv_bind_sn_cnv (a) (h) (p) (I) (G) (K) (X2):
- ∀V,T. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[a,h] X2 →
- ∃∃U. ⦃G,K⦄ ⊢ V ![a,h] & ⦃G,K.ⓑ{I}V⦄ ⊢ T :[a,h] U & ⦃G,K⦄ ⊢ ⓑ{p,I}V.U ⬌*[h] X2 & ⦃G,K⦄ ⊢ X2 ![a,h].
-#a #h #p * #G #K #X2 #V #T #H
+lemma nta_inv_bind_sn_cnv (h) (a) (p) (I) (G) (K) (X2):
+ ∀V,T. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[h,a] X2 →
+ ∃∃U. ⦃G,K⦄ ⊢ V ![h,a] & ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,a] U & ⦃G,K⦄ ⊢ ⓑ{p,I}V.U ⬌*[h] X2 & ⦃G,K⦄ ⊢ X2 ![h,a].
+#h #a #p * #G #K #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_bind … H1) -H1 #HV #HT
[ elim (cpms_inv_abbr_sn_dx … H2) -H2 *
(* Basic_1: uses: ty3_gen_appl *)
lemma nta_inv_appl_sn (h) (G) (L) (X2):
- ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :[h] X2 →
- ∃∃p,W,U. ⦃G,L⦄ ⊢ V :[h] W & ⦃G,L⦄ ⊢ T :[h] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h].
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :[h,𝟐] X2 →
+ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :[h,𝟐] W & ⦃G,L⦄ ⊢ T :[h,𝟐] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,𝟐].
#h #G #L #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X #HX2 #H1 #HX2 #H2
elim (cnv_inv_appl … H1) #n #p #W #U #H <H -n #HV #HT #HVW #HTU
(* Basic_2A1: uses: nta_fwd_pure1 *)
lemma nta_inv_pure_sn_cnv (h) (G) (L) (X2):
- ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :*[h] X2 →
- ∨∨ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :*[h] W & ⦃G,L⦄ ⊢ T :*[h] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h]
- | ∃∃U. ⦃G,L⦄ ⊢ T :*[h] U & ⦃G,L⦄ ⊢ ⓐV.U !*[h] & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h].
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :[h,𝛚] X2 →
+ ∨∨ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :[h,𝛚] W & ⦃G,L⦄ ⊢ T :[h,𝛚] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,𝛚]
+ | ∃∃U. ⦃G,L⦄ ⊢ T :[h,𝛚] U & ⦃G,L⦄ ⊢ ⓐV.U ![h,𝛚] & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,𝛚].
#h #G #L #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H
elim (cnv_inv_appl … H1) * [| #n ] #p #W0 #T0 #Hn #HV #HT #HW0 #HT0
qed-.
(* Basic_2A1: uses: nta_inv_cast1 *)
-lemma nta_inv_cast_sn (a) (h) (G) (L) (X2):
- ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] X2 →
- ∧∧ ⦃G,L⦄ ⊢ T :[a,h] U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #U #T #H
+lemma nta_inv_cast_sn (h) (a) (G) (L) (X2):
+ ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T :[h,a] X2 →
+ ∧∧ ⦃G,L⦄ ⊢ T :[h,a] U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #U #T #H
elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
elim (cnv_inv_cast … H1) #X #HU #HT #HUX #HTX
elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
qed-.
(* Basic_1: uses: ty3_gen_cast *)
-lemma nta_inv_cast_sn_old (a) (h) (G) (L) (X2):
- ∀T0,T1. ⦃G,L⦄ ⊢ ⓝT1.T0 :[a,h] X2 →
- ∃∃T2. ⦃G,L⦄ ⊢ T0 :[a,h] T1 & ⦃G,L⦄ ⊢ T1 :[a,h] T2 & ⦃G,L⦄ ⊢ ⓝT2.T1 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #T0 #T1 #H
+lemma nta_inv_cast_sn_old (h) (a) (G) (L) (X2):
+ ∀T0,T1. ⦃G,L⦄ ⊢ ⓝT1.T0 :[h,a] X2 →
+ ∃∃T2. ⦃G,L⦄ ⊢ T0 :[h,a] T1 & ⦃G,L⦄ ⊢ T1 :[h,a] T2 & ⦃G,L⦄ ⊢ ⓝT2.T1 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #T0 #T1 #H
elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
elim (cnv_inv_cast … H1) #X #HT1 #HT0 #HT1X #HT0X
elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
(* Basic_1: uses: ty3_gen_lift *)
(* Note: "⦃G, L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G, L⦄ ⊢ X2 ➡*[h] U2" *)
-lemma nta_inv_lifts_sn (a) (h) (G):
- ∀L,T2,X2. ⦃G,L⦄ ⊢ T2 :[a,h] X2 →
+lemma nta_inv_lifts_sn (h) (a) (G):
+ ∀L,T2,X2. ⦃G,L⦄ ⊢ T2 :[h,a] X2 →
∀b,f,K. ⬇*[b,f] L ≘ K → ∀T1. ⬆*[f] T1 ≘ T2 →
- ∃∃U1,U2. ⦃G,K⦄ ⊢ T1 :[a,h] U1 & ⬆*[f] U1 ≘ U2 & ⦃G,L⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
+ ∃∃U1,U2. ⦃G,K⦄ ⊢ T1 :[h,a] U1 & ⬆*[f] U1 ≘ U2 & ⦃G,L⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
elim (cnv_inv_cast … H) -H #U2 #HX2 #HT2 #HXU2 #HTU2
lapply (cnv_inv_lifts … HT2 … HLK … HT12) -HT2 #HT1
elim (cpms_inv_lifts_sn … HTU2 … HLK … HT12) -T2 -HLK #U1 #HU12 #HTU1
(* Forward lemmas based on preservation *************************************)
(* Basic_1: was: ty3_unique *)
-theorem nta_mono (a) (h) (G) (L) (T):
- ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 → ∀U2. ⦃G,L⦄ ⊢ T :[a,h] U2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #H2
+theorem nta_mono (h) (a) (G) (L) (T):
+ ∀U1. ⦃G,L⦄ ⊢ T :[h,a] U1 → ∀U2. ⦃G,L⦄ ⊢ T :[h,a] U2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
+#h #a #G #L #T #U1 #H1 #U2 #H2
elim (cnv_inv_cast … H1) -H1 #X1 #_ #_ #HUX1 #HTX1
elim (cnv_inv_cast … H2) -H2 #X2 #_ #HT #HUX2 #HTX2
lapply (cnv_cpms_conf_eq … HT … HTX1 … HTX2) -T #HX12
(* Advanced properties ******************************************************)
(* Basic_1: uses: ty3_sconv_pc3 *)
-lemma nta_cpcs_bi (a) (h) (G) (L):
- ∀T1,U1. ⦃G,L⦄ ⊢ T1 :[a,h] U1 → ∀T2,U2. ⦃G,L⦄ ⊢ T2 :[a,h] U2 →
+lemma nta_cpcs_bi (h) (a) (G) (L):
+ ∀T1,U1. ⦃G,L⦄ ⊢ T1 :[h,a] U1 → ∀T2,U2. ⦃G,L⦄ ⊢ T2 :[h,a] U2 →
⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
-#a #h #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+#h #a #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
elim (cpcs_inv_cprs … HT12) -HT12 #T0 #HT10 #HT02
/3 width=6 by nta_mono, nta_cprs_conf/
qed-.
(* Properties based on type equivalence and preservation *******************)
(* Basic_1: uses: ty3_tred *)
-lemma nta_cprs_trans (a) (h) (G) (L):
- ∀T,U1. ⦃G,L⦄ ⊢ T :[a,h] U1 → ∀U2. ⦃G,L⦄ ⊢ U1 ➡*[h] U2 → ⦃G,L⦄ ⊢ T :[a,h] U2.
-#a #h #G #L #T #U1 #H #U2 #HU12
+lemma nta_cprs_trans (h) (a) (G) (L):
+ ∀T,U1. ⦃G,L⦄ ⊢ T :[h,a] U1 → ∀U2. ⦃G,L⦄ ⊢ U1 ➡*[h] U2 → ⦃G,L⦄ ⊢ T :[h,a] U2.
+#h #a #G #L #T #U1 #H #U2 #HU12
/4 width=4 by nta_conv_cnv, nta_fwd_cnv_dx, cnv_cpms_trans, cpcs_cprs_dx/
qed-.
(* Basic_1: uses: ty3_sred_back *)
-lemma cprs_nta_trans (a) (h) (G) (L):
- ∀T1,U0. ⦃G,L⦄ ⊢ T1 :[a,h] U0 → ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
- ∀U. ⦃G,L⦄ ⊢ T2 :[a,h] U → ⦃G,L⦄ ⊢ T1 :[a,h] U.
-#a #h #G #L #T1 #U0 #HT1 #T2 #HT12 #U #H
+lemma cprs_nta_trans (h) (a) (G) (L):
+ ∀T1,U0. ⦃G,L⦄ ⊢ T1 :[h,a] U0 → ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ∀U. ⦃G,L⦄ ⊢ T2 :[h,a] U → ⦃G,L⦄ ⊢ T1 :[h,a] U.
+#h #a #G #L #T1 #U0 #HT1 #T2 #HT12 #U #H
lapply (nta_cprs_conf … HT1 … HT12) -HT12 #HT2
/4 width=6 by nta_mono, nta_conv_cnv, nta_fwd_cnv_dx/
qed-.
-lemma cprs_nta_trans_cnv (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
- ∀U. ⦃G,L⦄ ⊢ T2 :[a,h] U → ⦃G,L⦄ ⊢ T1 :[a,h] U.
-#a #h #G #L #T1 #HT1 #T2 #HT12 #U #H
+lemma cprs_nta_trans_cnv (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ∀U. ⦃G,L⦄ ⊢ T2 :[h,a] U → ⦃G,L⦄ ⊢ T1 :[h,a] U.
+#h #a #G #L #T1 #HT1 #T2 #HT12 #U #H
elim (cnv_nta_sn … HT1) -HT1 #U0 #HT1
/2 width=3 by cprs_nta_trans/
qed-.
(* Basic_1: uses: ty3_sconv *)
-lemma nta_cpcs_conf (a) (h) (G) (L):
- ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U → ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
- ∀U0. ⦃G,L⦄ ⊢ T2 :[a,h] U0 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #HT1 #T2 #HT12 #U0 #HT2
+lemma nta_cpcs_conf (h) (a) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[h,a] U → ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀U0. ⦃G,L⦄ ⊢ T2 :[h,a] U0 → ⦃G,L⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #HT1 #T2 #HT12 #U0 #HT2
elim (cpcs_inv_cprs … HT12) -HT12 #T0 #HT10 #HT02
/3 width=5 by cprs_nta_trans, nta_cprs_conf/
qed-.
(* Note: type preservation by valid r-equivalence *)
-lemma nta_cpcs_conf_cnv (a) (h) (G) (L):
- ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
- ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ T2 ![a,h] → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #HT1 #T2 #HT12 #HT2
+lemma nta_cpcs_conf_cnv (h) (a) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[h,a] U →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ T2 ![h,a] → ⦃G,L⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #HT1 #T2 #HT12 #HT2
elim (cnv_nta_sn … HT2) -HT2 #U0 #HT2
/2 width=3 by nta_cpcs_conf/
qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/colon_7.ma".
-include "basic_2/notation/relations/colon_6.ma".
-include "basic_2/notation/relations/colonstar_6.ma".
+include "basic_2/notation/relations/colonstar_7.ma".
include "basic_2/dynamic/cnv.ma".
(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
-definition ntas (a) (h) (n) (G) (L): relation term ≝ λT,U.
- ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[n,h] U0.
+definition ntas (h) (a) (n) (G) (L): relation term ≝ λT,U.
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] & ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[n,h] U0.
interpretation "iterated native type assignment (term)"
- 'Colon a h n G L T U = (ntas a h n G L T U).
-
-interpretation "restricted iterated native type assignment (term)"
- 'Colon h n G L T U = (ntas true h n G L T U).
-
-interpretation "extended iterated native type assignment (term)"
- 'ColonStar h n G L T U = (ntas false h n G L T U).
+ 'ColonStar h a n G L T U = (ntas h a n G L T U).
(* Basic properties *********************************************************)
-lemma ntas_refl (a) (h) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L⦄ ⊢ T :[a,h,0] T.
+lemma ntas_intro (h) (a) (n) (G) (L):
+ ∀U. ⦃G,L⦄ ⊢ U ![h,a] → ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀U0. ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[n,h] U0 → ⦃G,L⦄ ⊢ T :*[h,a,n] U.
/2 width=3 by ex4_intro/ qed.
+
+lemma ntas_refl (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L⦄ ⊢ T :*[h,a,0] T.
+/2 width=3 by ntas_intro/ qed.
+
+lemma ntas_sort (h) (a) (n) (G) (L):
+ ∀s. ⦃G,L⦄ ⊢ ⋆s :*[h,a,n] ⋆((next h)^n s).
+#h #a #n #G #L #s
+/2 width=3 by ntas_intro, cnv_sort, cpms_sort/
+qed.
+
+lemma ntas_bind_cnv (h) (a) (n) (G) (K):
+ ∀V. ⦃G,K⦄ ⊢ V ![h,a] →
+ ∀I,T,U. ⦃G,K.ⓑ{I}V⦄ ⊢ T :*[h,a,n] U →
+ ∀p. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :*[h,a,n] ⓑ{p,I}V.U.
+#h #a #n #G #K #V #HV #I #T #U
+* #X #HU #HT #HUX #HTX #p
+/3 width=5 by ntas_intro, cnv_bind, cpms_bind_dx/
+qed.
+
+(* Basic_forward lemmas *****************************************************)
+
+lemma ntas_fwd_cnv_sn (h) (a) (n) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :*[h,a,n] U → ⦃G,L⦄ ⊢ T ![h,a].
+#h #a #n #G #L #T #U
+* #X #_ #HT #_ #_ //
+qed-.
+
+(* Note: this is ntas_fwd_correct_cnv *)
+lemma ntas_fwd_cnv_dx (h) (a) (n) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :*[h,a,n] U → ⦃G,L⦄ ⊢ U ![h,a].
+#h #a #n #G #L #T #U
+* #X #HU #_ #_ #_ //
+qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-(*
-include "basic_2/dynamic/nta_lift.ma".
-include "basic_2/hod/ntas.ma".
-
-(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************)
-
-(* Advanced properties on native type assignment for terms ******************)
-
-lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h,L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h,L⦄ ⊢ T : U →
- ∀V. ⦃h,L⦄ ⊢ V : W → ⦃h,L⦄ ⊢ ⓐV.T : ⓐV.U.
-#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/
-qed.
-
-axiom pippo: ∀h,L,T,W,Y. ⦃h,L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h,L⦄ ⊢ T : U →
- ∃Z. ⦃h,L⦄ ⊢ U :* ⓛW.Z.
-(* REQUIRES SUBJECT CONVERSION
-#h #L #T #W #Y #H @(ntas_ind_dx … H) -T
-[ #U #HYU
- elim (nta_fwd_correct … HYU) #U0 #HU0
- elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U
-*)
-
-(* Advanced inversion lemmas on native type assignment for terms ************)
-
-fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h,L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X →
- ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
- L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
-#h #L #Z #U #H elim H -L -Z -U
-[ #L #k #X #Y #H destruct
-| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct
-| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct
-| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct
-| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/
-| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct
- elim (IHUW U Y ?) -IHUW // /3 width=9/
-| #L #Z #U #_ #_ #X #Y #H destruct
-| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
- elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT
- lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/
-]
-qed.
-
-(* Basic_1: was only: ty3_gen_appl *)
-lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h,L⦄ ⊢ ⓐY.X : U →
- ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
- L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
-/2 width=3/ qed-.
-
-axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h,L⦄ ⊢ ⓐZ.ⓛY.X : U →
- ∃∃W. ⦃h,L⦄ ⊢ Z : Y & ⦃h,L⦄ ⊢ ⓛY.X : ⓛY.W &
- L ⊢ ⓐZ.ⓛY.W ⬌* U.
-(* REQUIRES SUBJECT REDUCTION
-#h #L #Z #Y #X #U #H
-elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT
-elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V
-lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U
-@(ex3_1_intro … HX0U) /2 width=2/
-*)
-*)
(* *)
(**************************************************************************)
-include "basic_2/dynamic/nta.ma".
-include "basic_2/i_dynamic/ntas.ma".
+include "basic_2/dynamic/nta_preserve.ma".
+include "basic_2/i_dynamic/ntas_preserve.ma".
(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
+(* Properties with native type assignment for terms *************************)
+
+lemma nta_ntas (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ T :*[h,a,1] U.
+#h #a #G #L #T #U #H
+elim (cnv_inv_cast … H) -H /2 width=3 by ntas_intro/
+qed-.
+
+(* Inversion lemmas with native type assignment for terms *******************)
+
+lemma ntas_inv_nta (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :*[h,a,1] U → ⦃G,L⦄ ⊢ T :[h,a] U.
+#h #a #G #L #T #U
+* /2 width=3 by cnv_cast/
+qed-.
+
+(* Note: this follows from ntas_inv_appl_sn *)
+lemma nta_inv_appl_sn_ntas (h) (a) (G) (L) (V) (T):
+ ∀X. ⦃G,L⦄ ⊢ ⓐV.T :[h,a] X →
+ ∨∨ ∃∃p,W,U,U0. ad a 0 & ⦃G,L⦄ ⊢ V :[h,a] W & ⦃G,L⦄ ⊢ T :*[h,a,0] ⓛ{p}W.U0 & ⦃G,L.ⓛW⦄ ⊢ U0 :[h,a] U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a]
+ | ∃∃n,p,W,U,U0. ad a (↑n) & ⦃G,L⦄ ⊢ V :[h,a] W & ⦃G,L⦄ ⊢ T :[h,a] U & ⦃G,L⦄ ⊢ U :*[h,a,n] ⓛ{p}W.U0 & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a].
+#h #a #G #L #V #T #X #H
+(*
+lapply (nta_ntas … H) -H #H
+elim (ntas_inv_appl_sn … H) -H * #n #p #W #U #U0 #Hn #Ha #HVW #HTU #HU #HUX #HX
+[ elim (eq_or_gt n) #H destruct
+ [ <minus_n_O in HU; #HU -Hn
+ /4 width=8 by ntas_inv_nta, ex6_4_intro, or_introl/
+ | lapply (le_to_le_to_eq … Hn H) -Hn -H #H destruct
+ <minus_n_n in HU; #HU
+ @or_intror
+ @(ex6_5_intro … Ha … HUX HX) -Ha -X
+ [2,4: /2 width=2 by ntas_inv_nta/
+ |6: @ntas_refl
+*)
+elim (cnv_inv_cast … H) -H #X0 #HX #HVT #HX0 #HTX0
+elim (cnv_inv_appl … HVT) #n #p #W #U0 #Ha #HV #HT #HVW #HTU0
+elim (eq_or_gt n) #Hn destruct
+[ elim (cnv_fwd_cpms_abst_dx_le … HT … HTU0 1) [| // ] <minus_n_O #U #H #HU0
+ lapply (cpms_appl_dx … V V … H) [ // ] -H #H
+ elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+ lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+ lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+ lapply (cnv_cpms_trans … HT … HTU0) #H
+ elim (cnv_inv_bind … H) -H #_ #HU0
+ /4 width=8 by cnv_cpms_ntas, cnv_cpms_nta, ex6_4_intro, or_introl/
+| >(plus_minus_m_m_commutative … Hn) in HTU0; #H
+ elim (cpms_inv_plus … H) -H #U #HTU #HU0
+ lapply (cpms_appl_dx … V V … HTU) [ // ] #H
+ elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+ lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+ lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+ <(S_pred … Hn) in Ha; -Hn #Ha
+ /5 width=10 by cnv_cpms_ntas, cnv_cpms_nta, cnv_cpms_trans, ex6_5_intro, or_intror/
+]
+qed-.
+
(*
definition ntas: sh → lenv → relation term ≝
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_equivalence/cpcs_cprs.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+include "basic_2/i_dynamic/ntas.ma".
+
+(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
+
+(* Properties based on preservation *****************************************)
+
+lemma cnv_cpms_ntas (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀n,U.⦃G,L⦄ ⊢ T ➡*[n,h] U → ⦃G,L⦄ ⊢ T :*[h,a,n] U.
+/3 width=4 by ntas_intro, cnv_cpms_trans/ qed.
+
+(* Inversion lemmas based on preservation ***********************************)
+
+lemma ntas_inv_appl_sn (h) (a) (m) (G) (L) (V) (T):
+ ∀X. ⦃G,L⦄ ⊢ ⓐV.T :*[h,a,m] X →
+ ∨∨ ∃∃n,p,W,U,U0. n ≤ m & ad a n & ⦃G,L⦄ ⊢ V :*[h,a,1] W & ⦃G,L⦄ ⊢ T :*[h,a,n] ⓛ{p}W.U0 & ⦃G,L.ⓛW⦄ ⊢ U0 :*[h,a,m-n] U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a]
+ | ∃∃n,p,W,U,U0. m ≤ n & ad a n & ⦃G,L⦄ ⊢ V :*[h,a,1] W & ⦃G,L⦄ ⊢ T :*[h,a,m] U & ⦃G,L⦄ ⊢ U :*[h,a,n-m] ⓛ{p}W.U0 & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a].
+#h #a #m #G #L #V #T #X
+* #X0 #HX #HVT #HX0 #HTX0
+elim (cnv_inv_appl … HVT) #n #p #W #U0 #Ha #HV #HT #HVW #HTU0
+elim (le_or_ge n m) #Hnm
+[ elim (cnv_fwd_cpms_abst_dx_le … HT … HTU0 … Hnm) #U #H #HU0
+ lapply (cpms_appl_dx … V V … H) [ // ] -H #H
+ elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+ lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+ lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+ lapply (cnv_cpms_trans … HT … HTU0) #H
+ elim (cnv_inv_bind … H) -H #_ #HU0
+ /4 width=11 by cnv_cpms_ntas, ex7_5_intro, or_introl/
+| >(plus_minus_m_m_commutative … Hnm) in HTU0; #H
+ elim (cpms_inv_plus … H) -H #U #HTU #HU0
+ lapply (cpms_appl_dx … V V … HTU) [ // ] #H
+ elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+ lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+ lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+ /5 width=11 by cnv_cpms_ntas, cnv_cpms_trans, ex7_5_intro, or_intror/
+]
+qed-.
+
+(*
+(* Advanced properties on native type assignment for terms ******************)
+
+lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h,L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h,L⦄ ⊢ T : U →
+ ∀V. ⦃h,L⦄ ⊢ V : W → ⦃h,L⦄ ⊢ ⓐV.T : ⓐV.U.
+#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/
+qed.
+
+axiom pippo: ∀h,L,T,W,Y. ⦃h,L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h,L⦄ ⊢ T : U →
+ ∃Z. ⦃h,L⦄ ⊢ U :* ⓛW.Z.
+(* REQUIRES SUBJECT CONVERSION
+#h #L #T #W #Y #H @(ntas_ind_dx … H) -T
+[ #U #HYU
+ elim (nta_fwd_correct … HYU) #U0 #HU0
+ elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U
+*)
+
+(* Advanced inversion lemmas on native type assignment for terms ************)
+
+fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h,L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X →
+ ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
+ L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
+#h #L #Z #U #H elim H -L -Z -U
+[ #L #k #X #Y #H destruct
+| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct
+| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct
+| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct
+| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/
+| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct
+ elim (IHUW U Y ?) -IHUW // /3 width=9/
+| #L #Z #U #_ #_ #X #Y #H destruct
+| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
+ elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT
+ lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/
+]
+qed.
+
+(* Basic_1: was only: ty3_gen_appl *)
+lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h,L⦄ ⊢ ⓐY.X : U →
+ ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
+ L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
+/2 width=3/ qed-.
+
+axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h,L⦄ ⊢ ⓐZ.ⓛY.X : U →
+ ∃∃W. ⦃h,L⦄ ⊢ Z : Y & ⦃h,L⦄ ⊢ ⓛY.X : ⓛY.W &
+ L ⊢ ⓐZ.ⓛY.W ⬌* U.
+(* REQUIRES SUBJECT REDUCTION
+#h #L #Z #Y #X #U #H
+elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT
+elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V
+lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U
+@(ex3_1_intro … HX0U) /2 width=2/
+*)
+*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :[ break term 46 h ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'Colon $h $G $L $T1 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :[ break term 46 a, break term 46 h ] break term 46 T2 )"
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :[ break term 46 h, break term 46 a ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'Colon $a $h $G $L $T1 $T2 }.
+ for @{ 'Colon $h $a $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :[ break term 46 a, break term 46 h, break term 46 n ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'Colon $a $h $n $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :*[ break term 46 h ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'ColonStar $h $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :*[ break term 46 h, break term 46 n ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'ColonStar $h $n $G $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :*[ break term 46 h, break term 46 a, break term 46 n ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'ColonStar $h $a $n $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T ![ break term 46 h ] )"
- non associative with precedence 45
- for @{ 'Exclaim $h $G $L $T }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T ![ break term 46 a, break term 46 h ] )"
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T ![ break term 46 h, break term 46 a ] )"
non associative with precedence 45
- for @{ 'Exclaim $a $h $G $L $T }.
+ for @{ 'Exclaim $h $a $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T !*[ break term 46 h ] )"
- non associative with precedence 45
- for @{ 'ExclaimStar $h $G $L $T }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( G ⊢ break term 46 L1 ⫃![ break term 46 a, break term 46 h ] break term 46 L2 )"
+notation "hvbox( G ⊢ break term 46 L1 ⫃![ break term 46 h, break term 46 a ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'LRSubEqV $a $h $G $L1 $L2 }.
+ for @{ 'LRSubEqV $h $a $G $L1 $L2 }.
class "wine"
[ { "iterated dynamic typing" * } {
[ { "context-sensitive iterated native type assignment" * } {
- [ [ "for terms" ] "ntas" + "( ⦃?,?⦄ ⊢ ? :[?,?,?] ? )" + "( ⦃?,?⦄ ⊢ ? :[?,?] ? )" + "( ⦃?,?⦄ ⊢ ? :*[?,?] ? )" * ]
+ [ [ "for terms" ] "ntas" + "( ⦃?,?⦄ ⊢ ? :*[?,?,?] ? )" "ntas_nta" + "ntas_preserve" * ]
}
]
}
class "magenta"
[ { "dynamic typing" * } {
[ { "context-sensitive native type assignment" * } {
- [ [ "for terms" ] "nta" + "( ⦃?,?⦄ ⊢ ? :[?,?] ? )" + "( ⦃?,?⦄ ⊢ ? :[?] ? )" + "( ⦃?,?⦄ ⊢ ? :*[?] ? )" "nta_drops" + "nta_aaa" + "nta_fsb" + "nta_cpms" + "nta_cpcs" + "nta_preserve" + "nta_preserve_cpcs" + "nta_ind" + "nta_eval" * ]
+ [ [ "for terms" ] "nta" + "( ⦃?,?⦄ ⊢ ? :[?,?] ? )" "nta_drops" + "nta_aaa" + "nta_fsb" + "nta_cpms" + "nta_cpcs" + "nta_preserve" + "nta_preserve_cpcs" + "nta_ind" + "nta_eval" * ]
}
]
[ { "context-sensitive native validity" * } {
[ [ "restricted refinement for lenvs" ] "lsubv ( ? ⊢ ? ⫃![?,?] ? )" "lsubv_drops" + "lsubv_lsubr" + "lsubv_lsuba" + "lsubv_cpms" + "lsubv_cpcs" + "lsubv_cnv" + "lsubv_lsubv" * ]
- [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" + "( ⦃?,?⦄ ⊢ ? ![?] )" + "( ⦃?,?⦄ ⊢ ? !*[?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_cpme" + "cnv_cpmuwe" + "cnv_cpmuwe_cpme" + "cnv_eval" + "cnv_cpce" + "cnv_cpes" + "cnv_cpcs" + "cnv_preserve_sub" + "cnv_preserve" + "cnv_preserve_cpes" + "cnv_preserve_cpcs" * ]
+ [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_cpme" + "cnv_cpmuwe" + "cnv_cpmuwe_cpme" + "cnv_eval" + "cnv_cpce" + "cnv_cpes" + "cnv_cpcs" + "cnv_preserve_sub" + "cnv_preserve" + "cnv_preserve_cpes" + "cnv_preserve_cpcs" * ]
}
]
}
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "𝛚"
+ non associative with precedence 46
+ for @{ 'Omega }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "𝟏"
+ non associative with precedence 46
+ for @{ 'One }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "𝟐"
+ non associative with precedence 46
+ for @{ 'Two }.
(**************************************************************************)
include "ground_2/lib/arith.ma".
+include "static_2/notation/functions/one_0.ma".
+include "static_2/notation/functions/two_0.ma".
+include "static_2/notation/functions/omega_0.ma".
(* APPLICABILITY CONDITION *************************************************)
(* applicability condition specification *)
record ac: Type[0] ≝ {
-(* degree of the sort *)
- appl: predicate nat
+(* applicability domain *)
+ ad: predicate nat
}.
(* applicability condition postulates *)
record ac_props (a): Prop ≝ {
- ac_dec: ∀m. Decidable (∃∃n. m ≤ n & appl a n)
+ ac_dec: ∀m. Decidable (∃∃n. m ≤ n & ad a n)
}.
(* Notable specifications ***************************************************)
definition ac_top: ac ≝ mk_ac apply_top.
+interpretation "any number (applicability domain)"
+ 'Omega = (ac_top).
+
lemma ac_top_props: ac_props ac_top ≝ mk_ac_props ….
/3 width=3 by or_introl, ex2_intro/
qed.
definition ac_eq (k): ac ≝ mk_ac (eq … k).
+interpretation "one (applicability domain)"
+ 'Two = (ac_eq (S O)).
+
+interpretation "zero (applicability domain)"
+ 'One = (ac_eq O).
+
lemma ac_eq_props (k): ac_props (ac_eq k) ≝ mk_ac_props ….
#m elim (le_dec m k) #Hm
[ /3 width=3 by or_introl, ex2_intro/
class "red"
[ { "syntax" * } {
[ { "applicability condition" * } {
- [ [ "properties" ] "ac" * ]
+ [ [ "properties" ] "ac" + "( 𝟏 )" + "( 𝟐 )" + "( 𝛚 )" * ]
}
]
[ { "equivalence up to exclusion binders" * } {
["Y"; "ϒ"; "𝕐"; "𝐘"; "𝚼"; "Ⓨ"; ] ;
["z"; "ζ"; "𝕫"; "𝐳"; "𝛇"; "ⓩ"; ] ;
["Z"; "ℨ"; "ℤ"; "𝐙"; "Ⓩ"; ] ;
- ["0"; "𝟘"; "⓪"; ] ;
- ["1"; "𝟙"; "①"; "⓵"; ] ;
- ["2"; "𝟚"; "②"; "⓶"; ] ;
- ["3"; "𝟛"; "③"; "⓷"; ] ;
- ["4"; "𝟜"; "④"; "⓸"; ] ;
- ["5"; "𝟝"; "⑤"; "⓹"; ] ;
- ["6"; "𝟞"; "⑥"; "⓺"; ] ;
- ["7"; "𝟟"; "⑦"; "⓻"; ] ;
- ["8"; "𝟠"; "⑧"; "⓼"; "∞"; ] ;
- ["9"; "𝟡"; "⑨"; "⓽"; ] ;
+ ["0"; "𝟘"; "⓪"; "𝟎"; ] ;
+ ["1"; "𝟙"; "①"; "⓵"; "𝟏"; ] ;
+ ["2"; "𝟚"; "②"; "⓶"; "𝟐"; ] ;
+ ["3"; "𝟛"; "③"; "⓷"; "𝟑"; ] ;
+ ["4"; "𝟜"; "④"; "⓸"; "𝟒"; ] ;
+ ["5"; "𝟝"; "⑤"; "⓹"; "𝟓"; ] ;
+ ["6"; "𝟞"; "⑥"; "⓺"; "𝟔"; ] ;
+ ["7"; "𝟟"; "⑦"; "⓻"; "𝟕"; ] ;
+ ["8"; "𝟠"; "⑧"; "⓼"; "𝟖"; "∞"; ] ;
+ ["9"; "𝟡"; "⑨"; "⓽"; "𝟗"; ] ;
]
;;
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