--- /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 "ground_2/notation/functions/append_2.ma".
+include "ground_2/ynat/ynat_plus.ma".
+include "basic_2/notation/functions/snbind2_3.ma".
+include "basic_2/notation/functions/snabbr_2.ma".
+include "basic_2/notation/functions/snabst_2.ma".
+include "basic_2/grammar/lenv_length.ma".
+
+(* LOCAL ENVIRONMENTS *******************************************************)
+
+let rec append L K on K ≝ match K with
+[ LAtom ⇒ L
+| LPair K I V ⇒ (append L K). ⓑ{I} V
+].
+
+interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
+
+interpretation "local environment tail binding construction (binary)"
+ 'SnBind2 I T L = (append (LPair LAtom I T) L).
+
+interpretation "tail abbreviation (local environment)"
+ 'SnAbbr T L = (append (LPair LAtom Abbr T) L).
+
+interpretation "tail abstraction (local environment)"
+ 'SnAbst L T = (append (LPair LAtom Abst T) L).
+
+definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
+ ∀K,T1,T2. R K T1 T2 → ∀L. R (L @@ K) T1 T2.
+
+(* Basic properties *********************************************************)
+
+lemma append_atom: ∀L. L @@ ⋆ = L.
+// qed.
+
+lemma append_pair: ∀I,L,K,V. L @@ (K.ⓑ{I}V) = (L @@ K).ⓑ{I}V.
+// qed.
+
+lemma append_atom_sn: ∀L. ⋆ @@ L = L.
+#L elim L -L //
+#L #I #V >append_pair //
+qed.
+
+lemma append_assoc: associative … append.
+#L1 #L2 #L3 elim L3 -L3 //
+qed.
+
+lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
+#L1 #L2 elim L2 -L2 //
+#L2 #I #V2 >append_pair >length_pair >length_pair //
+qed.
+
+lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
+#I #L #V >append_length //
+qed.
+
+(* Basic_1: was just: chead_ctail *)
+lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
+#L elim L -L /2 width=5 by ex2_3_intro/
+#L #Z #X #IHL #I #V elim (IHL Z X) -IHL
+#J #K #W #H #_ >H -H >ltail_length
+@(ex2_3_intro … J (K.ⓑ{I}V) W) //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
+ L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * /2 width=1 by conj/
+ #K2 #I2 #V2 #L1 #L2 #_ >length_atom >length_pair
+ #H elim (ysucc_inv_O_sn … H)
+| #K1 #I1 #V1 #IH *
+ [ #L1 #L2 #_ >length_atom >length_pair
+ #H elim (ysucc_inv_O_dx … H)
+ | #K2 #I2 #V2 #L1 #L2 #H1 #H2
+ elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+ elim (IH … H1) -IH -H1 /3 width=1 by ysucc_inv_inj, conj/
+ ]
+]
+qed-.
+
+(* Note: lemma 750 *)
+lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
+ L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * /2 width=1 by conj/
+ #K2 #I2 #V2 #L1 #L2 >append_atom >append_pair #H destruct
+ >length_pair >append_length <yplus_succ2 #H
+ elim (discr_yplus_xy_x … H) -H #H
+ [ elim (ylt_yle_false (|L2|) (∞)) // | elim (ysucc_inv_O_dx … H) ]
+| #K1 #I1 #V1 #IH *
+ [ #L1 #L2 >append_pair >append_atom #H destruct
+ >length_pair >append_length <yplus_succ2 #H
+ elim (discr_yplus_x_xy … H) -H #H
+ [ elim (ylt_yle_false (|L1|) (∞)) // | elim (ysucc_inv_O_dx … H) ]
+ | #K2 #I2 #V2 #L1 #L2 >append_pair >append_pair #H1 #H2
+ elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+ elim (IH … H1) -IH -H1 /2 width=1 by conj/
+ ]
+]
+qed-.
+
+lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆.
+#L #K #H elim (append_inj_dx … (⋆) … H) //
+qed-.
+
+lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
+#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
+qed-.
+
+lemma length_inv_pos_dx_ltail: ∀L,l. |L| = ⫯l →
+ ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_pos_dx … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
+qed-.
+
+lemma length_inv_pos_sn_ltail: ∀L,l. ⫯l = |L| →
+ ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_pos_sn … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
+qed-.
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_1: was: c_tail_ind *)
+lemma lenv_ind_alt: ∀R:predicate lenv.
+ R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
+ ∀L. R L.
+#R #IH1 #IH2 #L @(ynat_f_ind … length … L) -L #x #IHx * // -IH1
+#L #I #V #H destruct elim (lpair_ltail L I V)
+/4 width=1 by monotonic_ylt_plus_sn/
+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 "ground_2/notation/functions/append_2.ma".
-include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/notation/functions/snbind2_3.ma".
-include "basic_2/notation/functions/snabbr_2.ma".
-include "basic_2/notation/functions/snabst_2.ma".
-include "basic_2/grammar/lenv_length.ma".
-
-(* LOCAL ENVIRONMENTS *******************************************************)
-
-let rec append L K on K ≝ match K with
-[ LAtom ⇒ L
-| LPair K I V ⇒ (append L K). ⓑ{I} V
-].
-
-interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
-
-interpretation "local environment tail binding construction (binary)"
- 'SnBind2 I T L = (append (LPair LAtom I T) L).
-
-interpretation "tail abbreviation (local environment)"
- 'SnAbbr T L = (append (LPair LAtom Abbr T) L).
-
-interpretation "tail abstraction (local environment)"
- 'SnAbst L T = (append (LPair LAtom Abst T) L).
-
-definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
- ∀K,T1,T2. R K T1 T2 → ∀L. R (L @@ K) T1 T2.
-
-(* Basic properties *********************************************************)
-
-lemma append_atom: ∀L. L @@ ⋆ = L.
-// qed.
-
-lemma append_pair: ∀I,L,K,V. L @@ (K.ⓑ{I}V) = (L @@ K).ⓑ{I}V.
-// qed.
-
-lemma append_atom_sn: ∀L. ⋆ @@ L = L.
-#L elim L -L //
-#L #I #V >append_pair //
-qed.
-
-lemma append_assoc: associative … append.
-#L1 #L2 #L3 elim L3 -L3 //
-qed.
-
-lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
-#L1 #L2 elim L2 -L2 //
-#L2 #I #V2 >append_pair >length_pair >length_pair //
-qed.
-
-lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
-#I #L #V >append_length //
-qed.
-
-(* Basic_1: was just: chead_ctail *)
-lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
-#L elim L -L /2 width=5 by ex2_3_intro/
-#L #Z #X #IHL #I #V elim (IHL Z X) -IHL
-#J #K #W #H #_ >H -H >ltail_length
-@(ex2_3_intro … J (K.ⓑ{I}V) W) //
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
- L1 = L2 ∧ K1 = K2.
-#K1 elim K1 -K1
-[ * /2 width=1 by conj/
- #K2 #I2 #V2 #L1 #L2 #_ >length_atom >length_pair
- #H elim (ysucc_inv_O_sn … H)
-| #K1 #I1 #V1 #IH *
- [ #L1 #L2 #_ >length_atom >length_pair
- #H elim (ysucc_inv_O_dx … H)
- | #K2 #I2 #V2 #L1 #L2 #H1 #H2
- elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
- elim (IH … H1) -IH -H1 /3 width=1 by ysucc_inv_inj, conj/
- ]
-]
-qed-.
-
-(* Note: lemma 750 *)
-lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
- L1 = L2 ∧ K1 = K2.
-#K1 elim K1 -K1
-[ * /2 width=1 by conj/
- #K2 #I2 #V2 #L1 #L2 >append_atom >append_pair #H destruct
- >length_pair >append_length <yplus_succ2 #H
- elim (discr_yplus_xy_x … H) -H #H
- [ elim (ylt_yle_false (|L2|) (∞)) // | elim (ysucc_inv_O_dx … H) ]
-| #K1 #I1 #V1 #IH *
- [ #L1 #L2 >append_pair >append_atom #H destruct
- >length_pair >append_length <yplus_succ2 #H
- elim (discr_yplus_x_xy … H) -H #H
- [ elim (ylt_yle_false (|L1|) (∞)) // | elim (ysucc_inv_O_dx … H) ]
- | #K2 #I2 #V2 #L1 #L2 >append_pair >append_pair #H1 #H2
- elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
- elim (IH … H1) -IH -H1 /2 width=1 by conj/
- ]
-]
-qed-.
-
-lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆.
-#L #K #H elim (append_inj_dx … (⋆) … H) //
-qed-.
-
-lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
-#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
-qed-.
-
-lemma length_inv_pos_dx_ltail: ∀L,l. |L| = ⫯l →
- ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
-#Y #l #H elim (length_inv_pos_dx … H) -H #I #L #V #Hl #HLK destruct
-elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
-qed-.
-
-lemma length_inv_pos_sn_ltail: ∀L,l. ⫯l = |L| →
- ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
-#Y #l #H elim (length_inv_pos_sn … H) -H #I #L #V #Hl #HLK destruct
-elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
-qed-.
-
-(* Basic eliminators ********************************************************)
-
-(* Basic_1: was: c_tail_ind *)
-lemma lenv_ind_alt: ∀R:predicate lenv.
- R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
- ∀L. R L.
-#R #IH1 #IH2 #L @(ynat_f_ind … length … L) -L #x #IHx * // -IH1
-#L #I #V #H destruct elim (lpair_ltail L I V)
-/4 width=1 by monotonic_ylt_plus_sn/
-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 "ground_2/ynat/ynat_lt.ma".
+include "basic_2/grammar/lenv.ma".
+
+(* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
+
+let rec length L ≝ match L with
+[ LAtom ⇒ 0
+| LPair L _ _ ⇒ ⫯(length L)
+].
+
+interpretation "length (local environment)" 'card L = (length L).
+
+(* Basic properties *********************************************************)
+
+lemma length_atom: |⋆| = 0.
+// qed.
+
+lemma length_pair: ∀I,L,V. |L.ⓑ{I}V| = ⫯|L|.
+// qed.
+
+lemma length_inj: ∀L. |L| < ∞.
+#L elim L -L /2 width=1 by ylt_succ_Y/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆.
+* // #L #I #V >length_pair
+#H elim (ysucc_inv_O_dx … H)
+qed-.
+
+lemma length_inv_zero_sn: ∀L. yinj 0 = |L| → L = ⋆.
+/2 width=1 by length_inv_zero_dx/ qed-.
+
+lemma length_inv_pos_dx: ∀l,L. |L| = ⫯l →
+ ∃∃I,K,V. |K| = l & L = K. ⓑ{I}V.
+#l * /3 width=5 by ysucc_inj, ex2_3_intro/
+>length_atom #H elim (ysucc_inv_O_sn … H)
+qed-.
+
+lemma length_inv_pos_sn: ∀l,L. ⫯l = |L| →
+ ∃∃I,K,V. l = |K| & L = K. ⓑ{I}V.
+#l #L #H lapply (sym_eq ??? H) -H
+#H elim (length_inv_pos_dx … H) -H /2 width=5 by ex2_3_intro/
+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 "ground_2/ynat/ynat_lt.ma".
-include "basic_2/grammar/lenv.ma".
-
-(* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
-
-let rec length L ≝ match L with
-[ LAtom ⇒ 0
-| LPair L _ _ ⇒ ⫯(length L)
-].
-
-interpretation "length (local environment)" 'card L = (length L).
-
-(* Basic properties *********************************************************)
-
-lemma length_atom: |⋆| = 0.
-// qed.
-
-lemma length_pair: ∀I,L,V. |L.ⓑ{I}V| = ⫯|L|.
-// qed.
-
-lemma length_inj: ∀L. |L| < ∞.
-#L elim L -L /2 width=1 by ylt_succ_Y/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆.
-* // #L #I #V >length_pair
-#H elim (ysucc_inv_O_dx … H)
-qed-.
-
-lemma length_inv_zero_sn: ∀L. yinj 0 = |L| → L = ⋆.
-/2 width=1 by length_inv_zero_dx/ qed-.
-
-lemma length_inv_pos_dx: ∀l,L. |L| = ⫯l →
- ∃∃I,K,V. |K| = l & L = K. ⓑ{I}V.
-#l * /3 width=5 by ysucc_inj, ex2_3_intro/
->length_atom #H elim (ysucc_inv_O_sn … H)
-qed-.
-
-lemma length_inv_pos_sn: ∀l,L. ⫯l = |L| →
- ∃∃I,K,V. l = |K| & L = K. ⓑ{I}V.
-#l #L #H lapply (sym_eq ??? H) -H
-#H elim (length_inv_pos_dx … H) -H /2 width=5 by ex2_3_intro/
-qed-.
interpretation "application to vector (term)"
'SnApplVector Vs T = (applv Vs T).
+(* Basic properties *********************************************************)
+
+lemma applv_nil: ∀T. Ⓐ ◊.T = T.
+// qed.
+
+lemma applv_cons: ∀V,Vs,T. Ⓐ V@Vs.T = ⓐV.ⒶVs.T.
+// qed.
+
(* properties concerning simple terms ***************************************)
lemma applv_simple: ∀T,Vs. 𝐒⦃T⦄ → 𝐒⦃ⒶVs.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 l , break term 46 m ] break term 46 T1 ≡ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'RLift $l $m $T1 $T2 }.
projects / helm.git / commitdiff