"cic:/matita/arithmetics/nat/f_ind.con"
"cic:/matita/arithmetics/nat/f2_ind.con"
"cic:/matita/arithmetics/nat/f3_ind.con"
-
-####################################
-
-"cic:/matita/arithmetics/nat/lt_plus_Sn_r.con"
-"cic:/matita/arithmetics/nat/lt_plus_to_lt_l.con"
+#
"cic:/matita/arithmetics/nat/monotonic_lt_plus_l.con"
"cic:/matita/arithmetics/nat/monotonic_lt_plus_r.con"
-
-"cic:/matita/arithmetics/nat/le_pred_n.con"
+"cic:/matita/arithmetics/nat/lt_plus_Sn_r.con"
+"cic:/matita/arithmetics/nat/lt_plus_to_lt_l.con"
+#
"cic:/matita/arithmetics/nat/pred.con"
+#
"cic:/matita/arithmetics/nat/pred_Sn.con"
-"cic:/matita/arithmetics/nat/S_pred.con"
+#
+"cic:/matita/arithmetics/nat/le_pred_n.con"
"cic:/matita/arithmetics/nat/monotonic_pred.con"
+#
+"cic:/matita/arithmetics/nat/S_pred.con"
+
+####################################
"cic:/matita/arithmetics/nat/discr_minus_x_xy.con"
"cic:/matita/arithmetics/nat/discr_plus_xy_minus_xz.con"
+++ /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/arith/nat_succ.ma".
-include "ground/arith/nat_iter.ma".
-
-(* NON-NEGATIVE INTEGERS ****************************************************)
-
-(* Rewrites with nsucc ******************************************************)
-
-lemma niter_succ (A) (f) (n) (a): f (f^n a) = f^{A}(↑n) a.
-#A #f * //
-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/arith/nat_pred_succ.ma".
+include "ground/arith/nat_le.ma".
+
+(* NON-NEGATIVE INTEGERS ****************************************************)
+
+(* Basic constructions with pred ********************************************)
+
+(*** le_pred_n *)
+lemma nle_pred_sn_refl (m): ↓m ≤ m.
+#m elim m -m //
+qed.
+
+(*** monotonic_pred *)
+lemma nle_pred_bi (m) (n): m ≤ n → ↓m ≤ ↓n.
+#m #n #H elim H -n
+/2 width=3 by nle_trans/
+qed.
(* Basic constructions ******************************************************)
+lemma nlt_i (m) (n): ↑m ≤ n → m < n.
+// qed.
+
+lemma nlt_refl_succ (n): n < ↑n.
+// qed.
+
(*** lt_O_S *)
lemma nlt_zero_succ (m): 𝟎 < ↑m.
/2 width=1 by nle_succ_bi/ 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/arith/nat_le_plus.ma".
+include "ground/arith/nat_lt.ma".
+
+(* NON-NEGATIVE INTEGERS ****************************************************)
+
+(* Basic constructions with plus ********************************************)
+
+(*** monotonic_lt_plus_l *)
+lemma nlt_plus_bi_dx (m) (n1) (n2): n1 < n2 → n1 + m < n2 + m.
+#m #n1 #n2 #H
+@nlt_i >nplus_succ_sn /2 width=1 by nle_plus_bi_dx/
+qed.
+
+(*** monotonic_lt_plus_r *)
+lemma nlt_plus_bi_sn (m) (n1) (n2): n1 < n2 → m + n1 < m + n2.
+#m #n1 #n2 #H
+@nlt_i >nplus_succ_dx /2 width=1 by nle_plus_bi_sn/
+qed.
+
+(*** lt_plus_Sn_r *) (**)
+lemma lt_plus_Sn_r: ∀a,x,n. a < a + x + ↑n.
+/2 width=1/ qed-.
+
+(* Basic inversions with plus ***********************************************)
+
+(*** lt_plus_to_lt_l *)
+lemma nlt_inv_plus_bi_dx (m) (n1) (n2): n1 + m < n2 + m → n1 < n2.
+/2 width=2 by nle_inv_plus_bi_dx/ qed-.
+
+(*** lt_plus_to_lt_r *)
+lemma nlt_inv_plus_bi_sn (m) (n1) (n2): m + n1 < m + n2 → n1 < n2.
+/2 width=2 by nle_inv_plus_bi_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/arith/nat_pred_succ.ma".
+include "ground/arith/nat_lt.ma".
+
+(* NON-NEGATIVE INTEGERS ****************************************************)
+
+(* Basic constructions with pred ********************************************)
+
+lemma nlt_zero_sn (m): m = ↑↓m → 𝟎 < m.
+// qed.
+
+(* Basic inversions with pred ***********************************************)
+
+(*** S_pred *)
+lemma nlt_inv_zero_sn (m): 𝟎 < m → m = ↑↓m.
+#m @(nat_ind … m) -m //
+#H elim (nlt_inv_refl … H)
+qed-.
(* *)
(**************************************************************************)
-include "ground/arith/nat_iter_succ.ma".
+include "ground/arith/nat_succ_iter.ma".
(* NON-NEGATIVE INTEGERS ****************************************************)
--- /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/notation/functions/downarrow_1.ma".
+include "ground/arith/pnat_dis.ma".
+include "ground/arith/nat.ma".
+
+(* NON-NEGATIVE INTEGERS ****************************************************)
+
+(*** pred *)
+definition npred (m): nat ≝ match m with
+[ nzero ⇒ 𝟎
+| ninj p ⇒ pdis … (𝟎) ninj p
+].
+
+interpretation
+ "predecessor (non-negative integers"
+ 'DownArrow m = (npred m).
+
+(* Basic rewrites ***********************************************************)
+
+lemma npred_zero: 𝟎 = ↓𝟎.
+// 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/arith/nat_succ.ma".
+include "ground/arith/nat_pred.ma".
+
+(* NON-NEGATIVE INTEGERS ****************************************************)
+
+(* Basic rewrites with succ *************************************************)
+
+(*** pred_Sn *)
+lemma npred_succ (n): n = ↓↑n.
+* //
+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/arith/nat_iter.ma".
+include "ground/arith/nat_succ.ma".
+
+(* NON-NEGATIVE INTEGERS ****************************************************)
+
+(* Rewrites with niter ******************************************************)
+
+lemma niter_succ (A) (f) (n) (a): f (f^n a) = f^{A}(↑n) a.
+#A #f * //
+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/arith/pnat.ma".
+
+(* POSITIVE INTEGERS ********************************************************)
+
+definition pdis (A:Type[0]) (a) (f) (p): A ≝
+match p with
+[ punit ⇒ a
+| psucc q ⇒ f q
+].
name "ground_src"
table {
- class "gray"
- [ { "component" * } {
- [ { "plane" * } {
- [ "files" * ]
- }
- ]
- }
- ]
- class "water"
- [ { "generic rt-transition counter" * } {
- [ { "" * } {
- [ "rtc_ist ( 𝐓❪?,?❫ )" "rtc_ist_shift" "rtc_ist_plus" "rtc_ist_max" * ]
- [ "rtc_isrc ( 𝐑𝐓❪?,?❫ )" "rtc_isrt_shift" "rtc_isrt_plus" "rtc_isrt_max" "rtc_isrt_max_shift" * ]
- [ "rtc ( 〈?,?,?,?〉 ) ( 𝟘𝟘 ) ( 𝟙𝟘 ) ( 𝟘𝟙 )" "rtc_shift ( ↕*? )" "rtc_plus ( ? + ? )" "rtc_max ( ? ∨ ? )" "rtc_max_shift" * ]
- }
- ]
- }
- ]
- class "green"
- [ { "multiple relocation" * } {
- [ { "" * } {
- [ "rtmap" "rtmap_eq ( ? ≡ ? )" "rtmap_pushs ( ⫯*[?]? )" "rtmap_nexts ( ↑*[?]? )"
- "rtmap_tl ( ⫱? )" "rtmap_tls ( ⫱*[?]? )" "rtmap_isid ( 𝐈❪?❫ )" "rtmap_id" "rtmap_isdiv ( 𝛀❪?❫ )"
- "rtmap_fcla ( 𝐂❪?❫ ≘ ? )" "rtmap_isfin ( 𝐅❪?❫ )" "rtmap_isuni ( 𝐔❪?❫ )" "rtmap_uni ( 𝐔❨?❩ )"
- "rtmap_sle ( ? ⊆ ? )" "rtmap_sdj ( ? ∥ ? )" "rtmap_sand ( ? ⋒ ? ≘ ? )" "rtmap_sor ( ? ⋓ ? ≘ ? )"
- "rtmap_at ( @❪?,?❫ ≘ ? )" "rtmap_istot ( 𝐓❪?❫ )" "rtmap_after ( ? ⊚ ? ≘ ? )" "rtmap_coafter ( ? ~⊚ ? ≘ ? )"
- "rtmap_basic ( 𝐁❨?,?❩ )" "rtmap_basic_after"
- * ]
- [ "nstream ( ⫯? ) ( ↑? )" "nstream_eq" "" ""
- "" "" "nstream_isid" "nstream_id ( 𝐈𝐝 )" ""
- "" "" "" ""
- "" "" "" "nstream_sor"
- "" "nstream_istot ( ?@❨?❩ )" "nstream_after ( ? ∘ ? )" "nstream_coafter ( ? ~∘ ? )"
- "nstream_basic" ""
- * ]
+ class "gray"
+ [ { "component" * } {
+ [ { "plane" * } {
+ [ "files" * ]
+ }
+ ]
+ }
+ ]
+ class "water"
+ [ { "generic rt-transition counter" * } {
+ [ { "" * } {
+ [ "rtc_ist ( 𝐓❪?,?❫ )" "rtc_ist_shift" "rtc_ist_plus" "rtc_ist_max" * ]
+ [ "rtc_isrc ( 𝐑𝐓❪?,?❫ )" "rtc_isrt_shift" "rtc_isrt_plus" "rtc_isrt_max" "rtc_isrt_max_shift" * ]
+ [ "rtc ( 〈?,?,?,?〉 ) ( 𝟘𝟘 ) ( 𝟙𝟘 ) ( 𝟘𝟙 )" "rtc_shift ( ↕*? )" "rtc_plus ( ? + ? )" "rtc_max ( ? ∨ ? )" "rtc_max_shift" * ]
+ }
+ ]
+ }
+ ]
+ class "green"
+ [ { "multiple relocation" * } {
+ [ { "" * } {
+ [ "rtmap" "rtmap_eq ( ? ≡ ? )" "rtmap_pushs ( ⫯*[?]? )" "rtmap_nexts ( ↑*[?]? )"
+ "rtmap_tl ( ⫱? )" "rtmap_tls ( ⫱*[?]? )" "rtmap_isid ( 𝐈❪?❫ )" "rtmap_id" "rtmap_isdiv ( 𝛀❪?❫ )"
+ "rtmap_fcla ( 𝐂❪?❫ ≘ ? )" "rtmap_isfin ( 𝐅❪?❫ )" "rtmap_isuni ( 𝐔❪?❫ )" "rtmap_uni ( 𝐔❨?❩ )"
+ "rtmap_sle ( ? ⊆ ? )" "rtmap_sdj ( ? ∥ ? )" "rtmap_sand ( ? ⋒ ? ≘ ? )" "rtmap_sor ( ? ⋓ ? ≘ ? )"
+ "rtmap_at ( @❪?,?❫ ≘ ? )" "rtmap_istot ( 𝐓❪?❫ )" "rtmap_after ( ? ⊚ ? ≘ ? )" "rtmap_coafter ( ? ~⊚ ? ≘ ? )"
+ "rtmap_basic ( 𝐁❨?,?❩ )" "rtmap_basic_after"
+ * ]
+ [ "nstream ( ⫯? ) ( ↑? )" "nstream_eq" "" ""
+ "" "" "nstream_isid" "nstream_id ( 𝐈𝐝 )" ""
+ "" "" "" ""
+ "" "" "" "nstream_sor"
+ "" "nstream_istot ( ?@❨?❩ )" "nstream_after ( ? ∘ ? )" "nstream_coafter ( ? ~∘ ? )"
+ "nstream_basic" ""
+ * ]
(*
- [ "trace ( ∥?∥ )" "trace_at ( @❪?,?❫ ≘ ? )" "trace_after ( ? ⊚ ? ≘ ? )" "trace_isid ( 𝐈❪?❫ )" "trace_isun ( 𝐔❪?❫ )"
- "trace_sle ( ? ⊆ ? )" "trace_sor ( ? ⋓ ? ≘ ? )" "trace_snot ( ∁ ? )" * ]
+ [ "trace ( ∥?∥ )" "trace_at ( @❪?,?❫ ≘ ? )" "trace_after ( ? ⊚ ? ≘ ? )" "trace_isid ( 𝐈❪?❫ )" "trace_isun ( 𝐔❪?❫ )"
+ "trace_sle ( ? ⊆ ? )" "trace_sor ( ? ⋓ ? ≘ ? )" "trace_snot ( ∁ ? )" * ]
*)
- [ "mr2 ( ◊ ) ( ❨?,?❩;? )" "mr2_append ( ? @@? )" "mr2_at ( @❪?,?❫ ≘ ? )" "mr2_plus ( ? + ? )" "mr2_minus ( ? ▭ ? ≘ ? )" * ]
- }
- ]
- }
- ]
- class "grass"
- [ { "natural numbers with infinity" * } {
- [ { "" * } {
- [ "ynat ( ∞ )" "ynat_pred ( ↓? )" "ynat_succ ( ↑? )"
- "ynat_le ( ? ≤ ? )" "ynat_lt ( ? < ? )"
- "ynat_plus ( ? + ? )" "ynat_minus_sn ( ? - ? )" *
- ]
- }
- ]
- }
- ]
- class "yellow"
- [ { "extensions to the library" * } {
- [ { "" * } {
- [ "stream ( ? ⨮{?} ? )" "stream_eq ( ? ≗{?} ? )" "stream_hdtl ( ⫰{?}? )" "stream_tls ( ⫰*{?}[?]? )" * ]
- [ "list ( Ⓔ{?} ) ( ? ⨮{?} ? )" "list_length ( |?| )" * ]
- [ "bool ( Ⓕ ) ( Ⓣ )" "arith ( ?^? ) ( ↑? ) ( ↓? ) ( ? ∨ ? ) ( ? ∧ ? )" "arith_2a" "arith_2b" * ]
- [ "ltc" "ltc_ctc" * ]
- [ "logic ( ⊥ ) ( ⊤ )" "relations ( ? ⊆ ? )" "functions" "exteq ( ? ≐{?,?} ? )" "star" "lstar_2a" * ]
- }
- ]
- }
- ]
- class "orange"
- [ { "generated library" * } {
- [ { "generalization with equality" * } {
- [ "insert_eq" * ]
- }
- ]
- [ { "permutation of quantifiers" * } {
- [ "pull" * ]
- }
- ]
- [ { "logical decomposables" * } {
- [ "xoa ( ∃∃ ) ( ∨∨ ) ( ∧∧ )" * ]
- }
- ]
- }
- ]
- class "red"
- [ { "" * } {
- [ { "" * } {
- [ * ]
- }
- ]
- }
- ]
+ [ "mr2 ( ◊ ) ( ❨?,?❩;? )" "mr2_append ( ? @@? )" "mr2_at ( @❪?,?❫ ≘ ? )" "mr2_plus ( ? + ? )" "mr2_minus ( ? ▭ ? ≘ ? )" * ]
+ }
+ ]
+ }
+ ]
+ class "grass"
+ [ { "natural numbers with infinity" * } {
+ [ { "" * } {
+ [ "ynat ( ∞ )" "ynat_pred ( ↓? )" "ynat_succ ( ↑? )"
+ "ynat_le ( ? ≤ ? )" "ynat_lt ( ? < ? )"
+ "ynat_plus ( ? + ? )" "ynat_minus_sn ( ? - ? )" *
+ ]
+ }
+ ]
+ }
+ ]
+ class "grass"
+ [ { "arithmetics" * } {
+ [ { "non-negative integers" * } {
+ [ "nat_lt ( ?<? )" "nat_lt_pred" "nat_lt_plus" * ]
+ [ "nat_le ( ?≤? )" "nat_le_pred" "nat_le_plus" * ]
+ [ "nat_plus ( ?+? )" * ]
+ [ "nat_pred ( ↓? )" "nat_pred_succ" * ]
+ [ "nat_succ ( ↑? )" "nat_succ_iter" * ]
+ [ "nat ( 𝟎 )" "nat_iter ( ?^{?}? )" * ]
+ }
+ ]
+ [ { "positive integers" * } {
+ [ "pnat_plus ( ?+? )" * ]
+ [ "pnat ( 𝟏 ) ( ↑? )" "pnat_dis" "pnat_iter ( ?^{?}? )" * ]
+ }
+ ]
+ }
+ ]
+ class "yellow"
+ [ { "extensions to the library" * } {
+ [ { "" * } {
+ [ "stream ( ? ⨮{?} ? )" "stream_eq ( ? ≗{?} ? )" "stream_hdtl ( ⫰{?}? )" "stream_tls ( ⫰*{?}[?]? )" * ]
+ [ "list ( Ⓔ{?} ) ( ? ⨮{?} ? )" "list_length ( |?| )" * ]
+ [ "bool ( Ⓕ ) ( Ⓣ )" "arith ( ?^? ) ( ↑? ) ( ↓? ) ( ? ∨ ? ) ( ? ∧ ? )" "arith_2a" "arith_2b" * ]
+ [ "ltc" "ltc_ctc" * ]
+ [ "logic ( ⊥ ) ( ⊤ )" "relations ( ? ⊆ ? )" "functions" "exteq ( ? ≐{?,?} ? )" "star" "lstar_2a" * ]
+ }
+ ]
+ }
+ ]
+ class "orange"
+ [ { "generated library" * } {
+ [ { "generalization with equality" * } {
+ [ "insert_eq" * ]
+ }
+ ]
+ [ { "permutation of quantifiers" * } {
+ [ "pull" * ]
+ }
+ ]
+ [ { "logical decomposables" * } {
+ [ "xoa ( ∃∃ ) ( ∨∨ ) ( ∧∧ )" * ]
+ }
+ ]
+ }
+ ]
+ class "red"
+ [ { "" * } {
+ [ { "" * } {
+ [ * ]
+ }
+ ]
+ }
+ ]
}
class "top" { * }
projects / helm.git / commitdiff