- first letter
+c: contex-sensitive for terms
f: context-freee for closures
-l: sn contex-sensitive for terms
-r: dx contex-sensitive for terms
+l: sn contex-sensitive for local environments
+r: dx contex-sensitive for local environments
t: context-free for terms
-second letter
- forth letter (if present)
p: non-reflexive transitive closure
+q: reflexive closure
s: reflexive transitive closure
non associative with precedence 45
for @{ 'SupTerm $L1 $T1 $L2 $T2 }.
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃⸮ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTermOpt $L1 $T1 $L2 $T2 }.
+
notation "hvbox( L ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )"
non associative with precedence 45
for @{ 'ICM $L $T $k }.
non associative with precedence 45
for @{ 'PRedSn $L1 $L2 }.
+notation "hvbox( ⦃ term 46 h, break term 46 L ⦄ ⊢ break term 46 T1 ➡ break [ term 46 g ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRed $h $g $L $T1 $T2 }.
+
+notation "hvbox( ⦃ term 46 h, break term 46 L ⦄ ⊢ break term 46 T1 ➡ ➡ break [ term 46 g ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRedAlt $h $g $L $T1 $T2 }.
+
(* Computation **************************************************************)
notation "hvbox( L ⊢ break term 46 T1 ➡ * break term 46 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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ssta.ma".
+include "basic_2/reduction/cpr.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+inductive cpx (h) (g): lenv → relation term ≝
+| cpx_cpr : ∀I,L,U2. L ⊢ ⓪{I} ➡ U2 → cpx h g L (⓪{I}) U2
+| cpx_ssta: ∀I,L,U2,l. ⦃h, L⦄ ⊢ ⓪{I} •[g] ⦃l+1, U2⦄ → cpx h g L (⓪{I}) U2
+| cpx_bind: ∀a,I,L,V1,V2,T1,T2,U2. cpx h g L V1 V2 → cpx h g (L.ⓑ{I}V1) T1 T2 →
+ L ⊢ ⓑ{a,I}V2.T2 ➡ U2 → cpx h g L (ⓑ{a,I}V1.T1) U2
+| cpx_flat: ∀I,L,V1,V2,T1,T2,U2. cpx h g L V1 V2 → cpx h g L T1 T2 →
+ L ⊢ ⓕ{I}V2.T2 ➡ U2 → cpx h g L (ⓕ{I}V1.T1) U2
+.
+
+interpretation
+ "context-sensitive extended parallel reduction (term)"
+ 'PRed h g L T1 T2 = (cpx h g L T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
+lemma cpx_refl: ∀h,g,T,L. ⦃h, L⦄ ⊢ T ➡[g] T.
+#h #g #T elim T -T /2 width=1/ * /2 width=5/
+qed.
+
+lemma cpr_cpx: ∀h,g,T1,L,T2. L ⊢ T1 ➡ T2 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+#h #g #T1 elim T1 -T1 /2 width=1/ * /2 width=5/
+qed.
+
+lemma ssta_cpx: ∀h,g,T1,L,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+#h #g #T1 elim T1 -T1 /2 width=2/ * [|*]
+[ #a #I #V1 #T1 #_ #IHT1 #L #X #l #H
+ elim (ssta_inv_bind1 … H) -H #T2 #HT12 #H destruct /3 width=5/
+| #V1 #T1 #_ #IHT1 #L #X #l #H
+ elim (ssta_inv_appl1 … H) -H #T2 #HT12 #H destruct /3 width=5/
+| #V1 #T1 #_ #IHT1 #L #X #l #H
+ lapply (ssta_inv_cast1 … H) -H /3 width=5/
+]
+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/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+include "basic_2/substitution/lpss_ldrop.ma".
+include "basic_2/substitution/fsups.ma".
+
+lemma fsup_cpss_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶* U2 →
+ ∃∃U1. L1 ⊢ T1 ▶* U1 & ⦃L1, U1⦄ ⊃ ⦃L2, U2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [2: * ] [1,2,3,4,5: /3 width=5/ ]
+[ * #L1
+ [ #V2 #U2 #HVU2
+ elim (lift_total U2 0 1) #W2 #HUW2
+ @(ex2_intro … W2) /2 width=7/
+ @(fsup_ldrop … L1 … HUW2) /2 width=1/
+ | #W #U2 #H
+ @(ex2_intro … (#0)) /2 width=7/
+
+| #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+ elim (IHT12 … HTU2) -IHT12 -HTU2 #T #HT1 #HT2
+ elim (lift_total T d e) #U #HTU
+ lapply (cpss_lift … HT1 … HLK1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
+]
+qed-.
+
+
+
+(*
+include "basic_2/relocation/lift_lift.ma".
+include "basic_2/substitution/fsups.ma".
+*)
+(*
+lemma fsup_ssta_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
+ ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
+ ∃∃L,U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃* ⦃L2, U2⦄.
+
+*)
+lemma fsup_ssta_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
+ ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
+ ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃ ⦃L2, U2⦄.
+#h #g #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2
+[ * #L1 #V1 #U2 #l #H
+ elim (lift_total U2 0 1) #W2 #HUW2
+(*
+ [ @(ex2_intro … W2) [ @(cpx_ssta … l) @(ssta_ldef … H) //
+ | @(fsups_ldrop … L1 … HUW2) /2 width=1/
+*)
+|
+| #a #I #L1 #V1 #T1 #U2 #l #HT1
+ @(ex2_intro … (ⓑ{a,I}V1.U2)) /2 width=1/
+++ /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/static/ssta.ma".
-include "basic_2/reduction/cpr.ma".
-
-lemma arith1: ∀x,y,z,w. z < y → x + (y + w) - z = x + y - z + w.
-#x #y #z #w #H <le_plus_minus_comm // /3 width=1 by lt_to_le, le_plus_a/
-qed-.
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-
-notation "hvbox( ⦃term 46 h, break term 46 L⦄ ⊢ break term 46 T1 ➡ break [ term 46 g ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'XPRed $h $g $L $T1 $T2 }.
-
-inductive xpr (h) (g): lenv → relation term ≝
-| xpr_cpr : ∀I,L,U2. L ⊢ ⓪{I} ➡ U2 → xpr h g L (⓪{I}) U2
-| xpr_ssta: ∀I,L,U2,l. ⦃h, L⦄ ⊢ ⓪{I} •[g] ⦃l+1, U2⦄ → xpr h g L (⓪{I}) U2
-| xpr_bind: ∀a,I,L,V1,V2,T1,T2,U2. xpr h g L V1 V2 → xpr h g (L.ⓑ{I}V1) T1 T2 →
- L ⊢ ⓑ{a,I}V2.T2 ➡ U2 → xpr h g L (ⓑ{a,I}V1.T1) U2
-| xpr_flat: ∀I,L,V1,V2,T1,T2,U2. xpr h g L V1 V2 → xpr h g L T1 T2 →
- L ⊢ ⓕ{I}V2.T2 ➡ U2 → xpr h g L (ⓕ{I}V1.T1) U2
-.
-
-interpretation
- "context-sensitive extended parallel reduction (term)"
- 'XPRed h g L T1 T2 = (xpr h g L T1 T2).
-
-(* Basic properties *********************************************************)
-
-(* Note: this is "∀h,g,L. reflexive … (xpr h g L)" *)
-lemma xpr_refl: ∀h,g,T,L. ⦃h, L⦄ ⊢ T ➡[g] T.
-#h #g #T elim T -T /2 width=1/ * /2 width=5/
-qed.
-
-lemma cpr_xpr: ∀h,g,T1,L,T2. L ⊢ T1 ➡ T2 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
-#h #g #T1 elim T1 -T1 /2 width=1/ * /2 width=5/
-qed.
-
-lemma ssta_xpr: ∀h,g,T1,L,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
-#h #g #T1 elim T1 -T1 /2 width=2/ * [|*]
-[ #a #I #V1 #T1 #_ #IHT1 #L #X #l #H
- elim (ssta_inv_bind1 … H) -H #T2 #HT12 #H destruct /3 width=5/
-| #V1 #T1 #_ #IHT1 #L #X #l #H
- elim (ssta_inv_appl1 … H) -H #T2 #HT12 #H destruct /3 width=5/
-| #V1 #T1 #_ #IHT1 #L #X #l #H
- lapply (ssta_inv_cast1 … H) -H /3 width=5/
-]
-qed.
-
-include "basic_2/relocation/lift_lift.ma".
-include "basic_2/relocation/fsup.ma".
-include "basic_2/substitution/lpss_ldrop.ma".
-
--- /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/relocation/fsup.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+definition fsupq: bi_relation lenv term ≝ bi_RC … fsup.
+
+interpretation
+ "optional structural successor (closure)"
+ 'SupTermOpt L1 T1 L2 T2 = (fsup L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fsupq_refl: bi_reflexive … fsupq.
+/2 width=1/ qed.
+
+lemma fsup_fsupq: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄.
+/2 width=1/ qed.
+
definition bi_transitive: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
∀a1,a,b1,b. R a1 b1 a b →
∀a2,b2. R a b a2 b2 → R a1 b1 a2 b2.
+
+definition bi_RC: ∀A,B:Type[0]. bi_relation A B → bi_relation A B ≝
+ λA,B,R,x1,y1,x2,y2. R … x1 y1 x2 y2 ∨ (x1 = x2 ∧ y1 = y2).
+
+lemma bi_RC_reflexive: ∀A,B,R. bi_reflexive A B (bi_RC … R).
+/3 width=1/ qed.
@(star_ind_l_aux … H1 H2 … Ha12) //
qed.
-(* RC and star *)
+(* TC and star *)
lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
#R #A #a #b #TCH (elim TCH) /2/
let predefined_classes = [
["!"; "¡"; ];
+ ["?"; "¿"; "⸮"; ];
[":"; "⁝"; ];
["."; "•"; "◦"; ];
["#"; "♯"; "⌘"; ];
projects / helm.git / commitdiff