set "baseuri" "cic:/matita/classical_pointwise/ordered_sets/".
-include "ordered_set.ma".
+include "excedence.ma".
-record is_dedekind_sigma_complete (O:ordered_set) : Type ≝
+record is_dedekind_sigma_complete (O:excedence) : Type ≝
{ dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? a m → ex ? (λs:O.is_inf O a s);
dsc_inf_proof_irrelevant:
∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? a m.∀p':is_lower_bound ? a m'.
}.
record dedekind_sigma_complete_ordered_set : Type ≝
- { dscos_ordered_set:> ordered_set;
+ { dscos_ordered_set:> excedence;
dscos_dedekind_sigma_complete_properties:>
is_dedekind_sigma_complete dscos_ordered_set
}.
qed.
definition is_sequentially_monotone ≝
- λO:ordered_set.λf:O→O.
+ λO:excedence.λf:O→O.
∀a:nat→O.∀p:is_increasing ? a.
is_increasing ? (λi.f (a i)).
--- /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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/divisible_group/".
+
+include "nat/orders.ma".
+include "group.ma".
+
+let rec pow (G : abelian_group) (x : G) (n : nat) on n : G ≝
+ match n with [ O ⇒ 0 | S m ⇒ x + pow ? x m].
+
+interpretation "abelian group pow" 'exp x n =
+ (cic:/matita/divisible_group/pow.con _ x n).
+
+record dgroup : Type ≝ {
+ dg_carr:> abelian_group;
+ dg_prop: ∀x:dg_carr.∀n:nat.∃y.pow ? y (S n) ≈ x
+}.
+
+lemma divide: ∀G:dgroup.G → nat → G.
+intros (G x n); cases (dg_prop G x n); apply w;
+qed.
+
+interpretation "divisible group divide" 'divide x n =
+ (cic:/matita/divisible_group/divide.con _ x n).
+
+lemma divide_divides:
+ ∀G:dgroup.∀x:G.∀n. pow ? (x / n) (S n) ≈ x.
+intro G; cases G; unfold divide; intros (x n); simplify;
+cases (f x n); simplify; exact H;
+qed.
+
+lemma feq_pow: ∀G:dgroup.∀x,y:G.∀n.x≈y → pow ? x n ≈ pow ? y n.
+intros (G x y n H); elim n; [apply eq_reflexive]
+simplify; apply (Eq≈ (x + (pow ? y n1)) H1);
+apply (Eq≈ (y+pow ? y n1) H (eq_reflexive ??));
+qed.
+
+lemma div1: ∀G:dgroup.∀x:G.x/O ≈ x.
+intro G; cases G; unfold divide; intros; simplify;
+cases (f x O); simplify; simplify in H; intro; apply H;
+apply (ap_rewl ???? (plus_comm ???));
+apply (ap_rewl ???w (zero_neutral ??)); assumption;
+qed.
apply ap_symmetric; assumption;
qed.
-lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x := eq_sym_.
+lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
coercion cic:/matita/excedence/eq_sym.con.
cases (exc_cotransitive ??? x EXx) (EXz EXz); [right; assumption]
cases LE; assumption;
qed.
-
lemma le_le_eq: ∀E:excedence.∀a,b:E. a ≤ b → b ≤ a → a ≈ b.
intros (E x y L1 L2); intro H; cases H; [apply L1|apply L2] assumption;
qed.
+lemma eq_le_le: ∀E:excedence.∀a,b:E. a ≈ b → a ≤ b ∧ b ≤ a.
+intros (E x y H); unfold apart_of_excedence in H; unfold apart in H;
+simplify in H; split; intro; apply H; [left|right] assumption.
+qed.
+
lemma ap_le_to_lt: ∀E:excedence.∀a,c:E.c # a → c ≤ a → c < a.
intros; split; assumption;
-qed.
\ No newline at end of file
+qed.
+
+definition total_order_property : ∀E:excedence. Type ≝
+ λE:excedence. ∀a,b:E. a ≰ b → b < a.
+
include "metric_space.ma".
include "lattice.ma".
-record mlattice_ (R : ogroup) : Type ≝ {
+record mlattice_ (R : todgroup) : Type ≝ {
ml_mspace_: metric_space R;
ml_lattice:> lattice;
ml_with_: ms_carr ? ml_mspace_ = ap_carr (l_carr ml_lattice)
coercion cic:/matita/metric_lattice/ml_mspace.con.
-record is_mlattice (R : ogroup) (ml: mlattice_ R) : Type ≝ {
+record is_mlattice (R : todgroup) (ml: mlattice_ R) : Type ≝ {
ml_prop1: ∀a,b:ml. 0 < δ a b → a # b;
ml_prop2: ∀a,b,c:ml. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ δ b c
}.
-record mlattice (R : ogroup) : Type ≝ {
+record mlattice (R : todgroup) : Type ≝ {
ml_carr :> mlattice_ R;
ml_props:> is_mlattice R ml_carr
}.
include "sequence.ma".
include "nat/plus.ma".
-definition d2s ≝
- λR.λml:mlattice R.λs:sequence (pordered_set_of_excedence ml).λk.λn. δ (s n) k.
-(*
-notation "s ⇝ 'Zero'" non associative with precedence 50 for @{'tends0 $s }.
-
-interpretation "tends to" 'tends s x =
- (cic:/matita/sequence/tends0.con _ s).
-*)
-
-lemma lew_opp : ∀O:ogroup.∀a,b,c:O.0 ≤ b → a ≤ c → a + -b ≤ c.
-intros (O a b c L0 L);
-apply (le_transitive ????? L);
-apply (plus_cancl_le ??? (-a));
-apply (le_rewr ??? 0 (opp_inverse ??));
-apply (le_rewl ??? (-a+a+-b) (plus_assoc ????));
-apply (le_rewl ??? (0+-b) (opp_inverse ??));
-apply (le_rewl ??? (-b) (zero_neutral ?(-b)));
-apply le_zero_x_to_le_opp_x_zero;
-assumption;
-qed.
-
-
-lemma ltw_opp : ∀O:ogroup.∀a,b,c:O.0 < b → a < c → a + -b < c.
-intros (O a b c P L);
-apply (lt_transitive ????? L);
-apply (plus_cancl_lt ??? (-a));
-apply (lt_rewr ??? 0 (opp_inverse ??));
-apply (lt_rewl ??? (-a+a+-b) (plus_assoc ????));
-apply (lt_rewl ??? (0+-b) (opp_inverse ??));
-apply (lt_rewl ??? ? (zero_neutral ??));
-apply lt_zero_x_to_lt_opp_x_zero;
-assumption;
-qed.
-
lemma ltwl: ∀a,b,c:nat. b + a < c → a < c.
intros 3 (x y z); elim y (H z IH H); [apply H]
apply IH; apply lt_S_to_lt; apply H;
intros 3 (x y z); rewrite > sym_plus; apply ltwl;
qed.
+
+definition d2s ≝
+ λR.λml:mlattice R.λs:sequence ml.λk.λn. δ (s n) k.
+(*
+notation "s ⇝ 'Zero'" non associative with precedence 50 for @{'tends0 $s }.
+
+interpretation "tends to" 'tends s x =
+ (cic:/matita/sequence/tends0.con _ s).
+*)
+
+axiom core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
+
alias symbol "leq" = "ordered sets less or equal than".
alias symbol "and" = "constructive and".
alias symbol "exists" = "constructive exists (Type)".
-theorem carabinieri: (* non trova la coercion .... *)
- ∀R.∀ml:mlattice R.∀an,bn,xn:sequence (pordered_set_of_excedence ml).
+theorem carabinieri:
+ ∀R.∀ml:mlattice R.∀an,bn,xn:sequence ml.
(∀n. (an n ≤ xn n) ∧ (xn n ≤ bn n)) →
∀x:ml. tends0 ? (d2s ? ml an x) → tends0 ? (d2s ? ml bn x) →
tends0 ? (d2s ? ml xn x).
intros (R ml an bn xn H x Ha Hb); unfold tends0 in Ha Hb ⊢ %. unfold d2s in Ha Hb ⊢ %.
intros (e He);
-elim (Ha ? He) (n1 H1); clear Ha; elim (Hb e He) (n2 H2); clear Hb;
+alias num (instance 0) = "natural number".
+elim (Ha (e/2) (divide_preserves_lt ??? He)) (n1 H1); clear Ha;
+elim (Hb (e/3) (divide_preserves_lt ??? He)) (n2 H2); clear Hb;
constructor 1; [apply (n1 + n2);] intros (n3 Hn3);
lapply (ltwr ??? Hn3) as Hn1n3; lapply (ltwl ??? Hn3) as Hn2n3;
elim (H1 ? Hn1n3) (H3 H4); elim (H2 ? Hn2n3) (H5 H6); clear Hn1n3 Hn2n3;
right; apply exc_opp_x_zero_to_exc_zero_x;
apply (Ex≪ ? (eq_opp_opp_x_x ??)); assumption;]
clear Dxm Dym H7 H8 Hn3 H5 H3 n1 n2;
+apply (le_lt_transitive ???? ? (core1 ?? He));
+apply (le_transitive ???? Hcut);
+apply (le_transitive ?? (e/3+ δ(an n3) x+ δ(an n3) x)); [
+ repeat apply fle_plusr; cases H6; assumption;]
+apply (le_transitive ?? (e/3+ e/2 + δ(an n3) x)); [
+ apply fle_plusr; apply fle_plusl; cases H4; assumption;]
+apply (le_transitive ?? (e/3+ e/2 + e/2)); [
+ apply fle_plusl; cases H4; assumption;]
+apply le_reflexive;
+qed.
(* 3.17 conclusione: δ x y ≈ 0 *)
set "baseuri" "cic:/matita/metric_space/".
-include "ordered_group.ma".
+include "ordered_divisible_group.ma".
-record metric_space (R : ogroup) : Type ≝ {
+record metric_space (R : todgroup) : Type ≝ {
ms_carr :> Type;
metric: ms_carr → ms_carr → R;
mpositive: ∀a,b:ms_carr. 0 ≤ metric a b;
definition apart_metric:=
λR.λms:metric_space R.λa,b:ms.0 < δ a b.
-lemma apart_of_metric_space: ∀R:ogroup.metric_space R → apartness.
+lemma apart_of_metric_space: ∀R:todgroup.metric_space R → apartness.
intros (R ms); apply (mk_apartness ? (apart_metric ? ms)); unfold apart_metric; unfold;
[1: intros 2 (x H); cases H (H1 H2);
lapply (ap_rewr ???? (eq_sym ??? (mreflexive ??x)) H2);
--- /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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/ordered_divisible_group/".
+
+include "nat/orders.ma".
+include "nat/times.ma".
+include "ordered_group.ma".
+include "divisible_group.ma".
+
+record todgroup : Type ≝ {
+ todg_order:> togroup;
+ todg_division_: dgroup;
+ todg_with_: dg_carr todg_division_ = og_abelian_group todg_order
+}.
+
+lemma todg_division: todgroup → dgroup.
+intro G; apply (mk_dgroup G); unfold abelian_group_OF_todgroup;
+cases (todg_with_ G); exact (dg_prop (todg_division_ G));
+qed.
+
+coercion cic:/matita/ordered_divisible_group/todg_division.con.
+
+lemma pow_lt: ∀G:todgroup.∀x:G.∀n.0 < x → 0 < x + pow ? x n.
+intros (G x n H); elim n; [
+ simplify; apply (lt_rewr ???? (plus_comm ???));
+ apply (lt_rewr ???x (zero_neutral ??)); assumption]
+simplify; apply (lt_transitive ?? (x+(x)\sup(n1))); [assumption]
+apply flt_plusl; apply (lt_rewr ???? (plus_comm ???));
+apply (lt_rewl ??? (0 + (x \sup n1)) (eq_sym ??? (zero_neutral ??)));
+apply (lt_rewl ???? (plus_comm ???));
+apply flt_plusl; assumption;
+qed.
+
+lemma pow_ge: ∀G:todgroup.∀x:G.∀n.0 ≤ x → 0 ≤ pow ? x n.
+intros (G x n); elim n; simplify; [apply le_reflexive]
+apply (le_transitive ???? H1);
+apply (le_rewl ??? (0+(x\sup n1)) (zero_neutral ??));
+apply fle_plusr; assumption;
+qed.
+
+lemma ge_pow: ∀G:todgroup.∀x:G.∀n.0 < pow ? x n → 0 < x.
+intros 3; elim n; [
+ simplify in l; cases (lt_coreflexive ?? l);]
+simplify in l;
+cut (0+0<x+(x)\sup(n1));[2:
+ apply (lt_rewl ??? 0 (zero_neutral ??)); assumption].
+cases (pippo4 ????? Hcut); [assumption]
+apply f; assumption;
+qed.
+
+lemma divide_preserves_lt: ∀G:todgroup.∀e:G.∀n.0<e → 0 < e/n.
+intros; elim n; [apply (lt_rewr ???? (div1 ??));assumption]
+unfold divide; elim (dg_prop G e (S n1)); simplify; simplify in f;
+apply (ge_pow ?? (S (S n1))); apply (lt_rewr ???? f); assumption;
+qed.
+
+alias num (instance 0) = "natural number".
+axiom core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/ordered_gorup/".
+set "baseuri" "cic:/matita/ordered_group/".
-include "ordered_set.ma".
include "group.ma".
-record pre_ogroup : Type ≝ {
+record pogroup_ : Type ≝ {
og_abelian_group_: abelian_group;
- og_tordered_set:> tordered_set;
- og_with: carr og_abelian_group_ = og_tordered_set
+ og_excedence:> excedence;
+ og_with: carr og_abelian_group_ = apart_of_excedence og_excedence
}.
-lemma og_abelian_group: pre_ogroup → abelian_group.
+lemma og_abelian_group: pogroup_ → abelian_group.
intro G; apply (mk_abelian_group G); [1,2,3: rewrite < (og_with G)]
[apply (plus (og_abelian_group_ G));|apply zero;|apply opp]
-unfold apartness_OF_pre_ogroup; cases (og_with G); simplify;
+unfold apartness_OF_pogroup_; cases (og_with G); simplify;
[apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext]
qed.
-coercion cic:/matita/ordered_gorup/og_abelian_group.con.
+coercion cic:/matita/ordered_group/og_abelian_group.con.
-record ogroup : Type ≝ {
- og_carr:> pre_ogroup;
- exc_canc_plusr: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
+record pogroup : Type ≝ {
+ og_carr:> pogroup_;
+ canc_plusr_exc: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
}.
lemma fexc_plusr:
- ∀G:ogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
-intros 5 (G x y z L); apply (exc_canc_plusr ??? (-z));
+ ∀G:pogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
+intros 5 (G x y z L); apply (canc_plusr_exc ??? (-z));
apply (Ex≪ (x + (z + -z)) (plus_assoc ????));
apply (Ex≪ (x + (-z + z)) (plus_comm ??z));
apply (Ex≪ (x+0) (opp_inverse ??));
apply (Ex≫ y (zero_neutral ??) L);
qed.
-coercion cic:/matita/ordered_gorup/fexc_plusr.con nocomposites.
+coercion cic:/matita/ordered_group/fexc_plusr.con nocomposites.
-lemma exc_canc_plusl: ∀G:ogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
-intros 5 (G x y z L); apply (exc_canc_plusr ??? z);
+lemma canc_plusl_exc: ∀G:pogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
+intros 5 (G x y z L); apply (canc_plusr_exc ??? z);
apply (exc_rewl ??? (z+x) (plus_comm ???));
apply (exc_rewr ??? (z+y) (plus_comm ???) L);
qed.
lemma fexc_plusl:
- ∀G:ogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
-intros 5 (G x y z L); apply (exc_canc_plusl ??? (-z));
+ ∀G:pogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
+intros 5 (G x y z L); apply (canc_plusl_exc ??? (-z));
apply (exc_rewl ???? (plus_assoc ??z x));
apply (exc_rewr ???? (plus_assoc ??z y));
apply (exc_rewl ??? (0+x) (opp_inverse ??));
apply (exc_rewr ???? (zero_neutral ??) L);
qed.
-coercion cic:/matita/ordered_gorup/fexc_plusl.con nocomposites.
+coercion cic:/matita/ordered_group/fexc_plusl.con nocomposites.
lemma plus_cancr_le:
- ∀G:ogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
+ ∀G:pogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
intros 5 (G x y z L);
apply (le_rewl ??? (0+x) (zero_neutral ??));
apply (le_rewl ??? (x+0) (plus_comm ???));
apply (le_rewr ??? (y+(-z+z)) (opp_inverse ??));
apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
-intro H; apply L; clear L; apply (exc_canc_plusr ??? (-z) H);
+intro H; apply L; clear L; apply (canc_plusr_exc ??? (-z) H);
qed.
-lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
+lemma fle_plusl: ∀G:pogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
intros (G f g h);
apply (plus_cancr_le ??? (-h));
apply (le_rewl ??? (f+h+ -h) (plus_comm ? f h));
apply (le_rewr ??? (g) (zero_neutral ??) H);
qed.
-lemma fle_plusr: ∀G:ogroup. ∀f,g,h:G. f≤g → f+h≤g+h.
+lemma fle_plusr: ∀G:pogroup. ∀f,g,h:G. f≤g → f+h≤g+h.
intros (G f g h H); apply (le_rewl ???? (plus_comm ???));
apply (le_rewr ???? (plus_comm ???)); apply fle_plusl; assumption;
qed.
lemma plus_cancl_le:
- ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
+ ∀G:pogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
intros 5 (G x y z L);
apply (le_rewl ??? (0+x) (zero_neutral ??));
apply (le_rewl ??? ((-z+z)+x) (opp_inverse ??));
qed.
lemma plus_cancl_lt:
- ∀G:ogroup.∀x,y,z:G.z+x < z+y → x < y.
+ ∀G:pogroup.∀x,y,z:G.z+x < z+y → x < y.
intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancl_le; assumption]
apply (plus_cancl_ap ???? LE);
qed.
lemma plus_cancr_lt:
- ∀G:ogroup.∀x,y,z:G.x+z < y+z → x < y.
+ ∀G:pogroup.∀x,y,z:G.x+z < y+z → x < y.
intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancr_le; assumption]
apply (plus_cancr_ap ???? LE);
qed.
lemma exc_opp_x_zero_to_exc_zero_x:
- ∀G:ogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
-intros (G x H); apply (exc_canc_plusr ??? (-x));
+ ∀G:pogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
+intros (G x H); apply (canc_plusr_exc ??? (-x));
apply (exc_rewr ???? (plus_comm ???));
apply (exc_rewr ???? (opp_inverse ??));
apply (exc_rewl ???? (zero_neutral ??) H);
qed.
lemma le_zero_x_to_le_opp_x_zero:
- ∀G:ogroup.∀x:G.0 ≤ x → -x ≤ 0.
+ ∀G:pogroup.∀x:G.0 ≤ x → -x ≤ 0.
intros (G x Px); apply (plus_cancr_le ??? x);
apply (le_rewl ??? 0 (opp_inverse ??));
apply (le_rewr ??? x (zero_neutral ??) Px);
qed.
lemma lt_zero_x_to_lt_opp_x_zero:
- ∀G:ogroup.∀x:G.0 < x → -x < 0.
+ ∀G:pogroup.∀x:G.0 < x → -x < 0.
intros (G x Px); apply (plus_cancr_lt ??? x);
apply (lt_rewl ??? 0 (opp_inverse ??));
apply (lt_rewr ??? x (zero_neutral ??) Px);
qed.
lemma exc_zero_opp_x_to_exc_x_zero:
- ∀G:ogroup.∀x:G. 0 ≰ -x → x ≰ 0.
-intros (G x H); apply (exc_canc_plusl ??? (-x));
+ ∀G:pogroup.∀x:G. 0 ≰ -x → x ≰ 0.
+intros (G x H); apply (canc_plusl_exc ??? (-x));
apply (exc_rewr ???? (plus_comm ???));
apply (exc_rewl ???? (opp_inverse ??));
apply (exc_rewr ???? (zero_neutral ??) H);
qed.
lemma le_x_zero_to_le_zero_opp_x:
- ∀G:ogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
+ ∀G:pogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
intros (G x Lx0); apply (plus_cancr_le ??? x);
apply (le_rewr ??? 0 (opp_inverse ??));
apply (le_rewl ??? x (zero_neutral ??));
qed.
lemma lt_x_zero_to_lt_zero_opp_x:
- ∀G:ogroup.∀x:G. x < 0 → 0 < -x.
+ ∀G:pogroup.∀x:G. x < 0 → 0 < -x.
intros (G x Lx0); apply (plus_cancr_lt ??? x);
apply (lt_rewr ??? 0 (opp_inverse ??));
apply (lt_rewl ??? x (zero_neutral ??));
lemma lt0plus_orlt:
- ∀G:ogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y.
+ ∀G:pogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y.
intros (G x y LEx LEy LT); cases LT (H1 H2); cases (ap_cotransitive ??? y H2);
[right; split; assumption|left;split;[assumption]]
apply (plus_cancr_ap ??? y); apply (ap_rewl ???? (zero_neutral ??));
qed.
lemma le0plus_le:
- ∀G:ogroup.∀a,b,c:G. 0 ≤ b → a + b ≤ c → a ≤ c.
+ ∀G:pogroup.∀a,b,c:G. 0 ≤ b → a + b ≤ c → a ≤ c.
intros (G a b c L H); apply (le_transitive ????? H);
apply (plus_cancl_le ??? (-a));
apply (le_rewl ??? 0 (opp_inverse ??));
qed.
lemma le_le0plus:
- ∀G:ogroup.∀a,b:G. 0 ≤ a → 0 ≤ b → 0 ≤ a + b.
+ ∀G:pogroup.∀a,b:G. 0 ≤ a → 0 ≤ b → 0 ≤ a + b.
intros (G a b L1 L2); apply (le_transitive ???? L1);
apply (plus_cancl_le ??? (-a));
apply (le_rewl ??? 0 (opp_inverse ??));
assumption;
qed.
-
-
\ No newline at end of file
+lemma flt_plusl:
+ ∀G:pogroup.∀x,y,z:G.x < y → z + x < z + y.
+intros (G x y z H); cases H; split; [apply fle_plusl; assumption]
+apply fap_plusl; assumption;
+qed.
+
+lemma flt_plusr:
+ ∀G:pogroup.∀x,y,z:G.x < y → x + z < y + z.
+intros (G x y z H); cases H; split; [apply fle_plusr; assumption]
+apply fap_plusr; assumption;
+qed.
+
+
+lemma ltxy_ltyyxx: ∀G:pogroup.∀x,y:G. y < x → y+y < x+x.
+intros; apply (lt_transitive ?? (y+x));[2:
+ apply (lt_rewl ???? (plus_comm ???));
+ apply (lt_rewr ???? (plus_comm ???));]
+apply flt_plusl;assumption;
+qed.
+
+lemma lew_opp : ∀O:pogroup.∀a,b,c:O.0 ≤ b → a ≤ c → a + -b ≤ c.
+intros (O a b c L0 L);
+apply (le_transitive ????? L);
+apply (plus_cancl_le ??? (-a));
+apply (le_rewr ??? 0 (opp_inverse ??));
+apply (le_rewl ??? (-a+a+-b) (plus_assoc ????));
+apply (le_rewl ??? (0+-b) (opp_inverse ??));
+apply (le_rewl ??? (-b) (zero_neutral ?(-b)));
+apply le_zero_x_to_le_opp_x_zero;
+assumption;
+qed.
+
+lemma ltw_opp : ∀O:pogroup.∀a,b,c:O.0 < b → a < c → a + -b < c.
+intros (O a b c P L);
+apply (lt_transitive ????? L);
+apply (plus_cancl_lt ??? (-a));
+apply (lt_rewr ??? 0 (opp_inverse ??));
+apply (lt_rewl ??? (-a+a+-b) (plus_assoc ????));
+apply (lt_rewl ??? (0+-b) (opp_inverse ??));
+apply (lt_rewl ??? ? (zero_neutral ??));
+apply lt_zero_x_to_lt_opp_x_zero;
+assumption;
+qed.
+
+record togroup : Type ≝ {
+ tog_carr:> pogroup;
+ tog_total: ∀x,y:tog_carr.x≰y → y < x
+}.
+
+lemma lexxyy_lexy: ∀G:togroup. ∀x,y:G. x+x ≤ y+y → x ≤ y.
+intros (G x y H); intro H1; lapply (tog_total ??? H1) as H2;
+lapply (ltxy_ltyyxx ??? H2) as H3; lapply (lt_to_excede ??? H3) as H4;
+cases (H H4);
+qed.
+
+lemma eqxxyy_eqxy: ∀G:togroup.∀x,y:G. x + x ≈ y + y → x ≈ y.
+intros (G x y H); cases (eq_le_le ??? H); apply le_le_eq;
+apply lexxyy_lexy; assumption;
+qed.
+
+lemma bar: ∀G:abelian_group. ∀x,y:G. 0 # x + y → 0 #x ∨ 0#y.
+intros; cases (ap_cotransitive ??? y a); [right; assumption]
+left; apply (plus_cancr_ap ??? y); apply (ap_rewl ???y (zero_neutral ??));
+assumption;
+qed.
+
+lemma pippo: ∀G:pogroup.∀a,b,c,d:G. a < b → c < d → a+c < b + d.
+intros (G a b c d H1 H2);
+lapply (flt_plusr ??? c H1) as H3;
+apply (lt_transitive ???? H3);
+apply flt_plusl; assumption;
+qed.
+
+lemma pippo2: ∀G:pogroup.∀a,b,c,d:G. a+c ≰ b + d → a ≰ b ∨ c ≰ d.
+intros (G a b c d H1 H2);
+cases (exc_cotransitive ??? (a + d) H1); [
+ right; apply (canc_plusl_exc ??? a); assumption]
+left; apply (canc_plusr_exc ??? d); assumption;
+qed.
+
+lemma pippo3: ∀G:pogroup.∀a,b,c,d:G. a ≤ b → c ≤ d → a+c ≤ b + d.
+intros (G a b c d H1 H2); intro H3; cases (pippo2 ????? H3);
+[apply H1|apply H2] assumption;
+qed.
+
+lemma foo: ∀G:togroup.∀x,y:G. 0 ≤ x + y → x < 0 → 0 ≤ y.
+intros; intro; apply H; lapply (lt_to_excede ??? l);
+lapply (tog_total ??? e);
+lapply (tog_total ??? Hletin);
+lapply (pippo ????? Hletin2 Hletin1);
+apply (exc_rewl ??? (0+0)); [apply eq_sym; apply zero_neutral]
+apply lt_to_excede; assumption;
+qed.
+
+lemma pippo4: ∀G:togroup.∀a,b,c,d:G. a+c < b + d → a < b ∨ c < d.
+intros (G a b c d H1 H2); lapply (lt_to_excede ??? H1);
+cases (pippo2 ????? Hletin); [left|right] apply tog_total; assumption;
+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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/ordered_set/".
-
-include "excedence.ma".
-
-record is_porder_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ {
- por_reflexive: reflexive ? le;
- por_transitive: transitive ? le;
- por_antisimmetric: antisymmetric ? le eq
-}.
-
-record pordered_set: Type ≝ {
- pos_carr:> excedence;
- pos_order_relation_properties:> is_porder_relation ? (le pos_carr) (eq pos_carr)
-}.
-
-lemma pordered_set_of_excedence: excedence → pordered_set.
-intros (E); apply (mk_pordered_set E); apply (mk_is_porder_relation);
-[apply le_reflexive|apply le_transitive|apply le_antisymmetric]
-qed.
-
-coercion cic:/matita/ordered_set/pordered_set_of_excedence.con.
-
-alias id "transitive" = "cic:/matita/higher_order_defs/relations/transitive.con".
-alias id "cotransitive" = "cic:/matita/higher_order_defs/relations/cotransitive.con".
-alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con".
-
-definition total_order_property : ∀E:excedence. Type ≝
- λE:excedence. ∀a,b:E. a ≰ b → b < a.
-
-record tordered_set : Type ≝ {
- tos_poset:> pordered_set;
- tos_totality: total_order_property tos_poset
-}.
set "baseuri" "cic:/matita/sequence/".
-include "ordered_set.ma".
+include "excedence.ma".
-definition sequence := λO:pordered_set.nat → O.
+definition sequence := λO:excedence.nat → O.
-definition fun_of_sequence: ∀O:pordered_set.sequence O → nat → O.
+definition fun_of_sequence: ∀O:excedence.sequence O → nat → O.
intros; apply s; assumption;
qed.
coercion cic:/matita/sequence/fun_of_sequence.con 1.
definition upper_bound ≝
- λO:pordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+ λO:excedence.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
definition lower_bound ≝
- λO:pordered_set.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+ λO:excedence.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
definition strong_sup ≝
- λO:pordered_set.λs:sequence O.λx.
+ λO:excedence.λs:sequence O.λx.
upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y).
definition strong_inf ≝
- λO:pordered_set.λs:sequence O.λx.
+ λO:excedence.λs:sequence O.λx.
lower_bound ? s x ∧ (∀y:O.y ≰ x → ∃n.y ≰ s n).
definition weak_sup ≝
- λO:pordered_set.λs:sequence O.λx.
+ λO:excedence.λs:sequence O.λx.
upper_bound ? s x ∧ (∀y:O.upper_bound ? s y → x ≤ y).
definition weak_inf ≝
- λO:pordered_set.λs:sequence O.λx.
+ λO:excedence.λs:sequence O.λx.
lower_bound ? s x ∧ (∀y:O.lower_bound ? s y → y ≤ x).
lemma strong_sup_is_weak:
- ∀O:pordered_set.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
+ ∀O:excedence.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x.
intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
qed.
lemma strong_inf_is_weak:
- ∀O:pordered_set.∀s:sequence O.∀x:O.strong_inf ? s x → weak_inf ? s x.
+ ∀O:excedence.∀s:sequence O.∀x:O.strong_inf ? s x → weak_inf ? s x.
intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption]
intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En);
qed.
include "nat/orders.ma".
definition tends0 ≝
- λO:ogroup.λs:sequence O.
+ λO:pogroup.λs:sequence O.
∀e:O.0 < e → ∃N.∀n.N < n → -e < s n ∧ s n < e.
definition increasing ≝
- λO:pordered_set.λa:sequence O.∀n:nat.a n ≤ a (S n).
+ λO:excedence.λa:sequence O.∀n:nat.a n ≤ a (S n).
definition decreasing ≝
- λO:pordered_set.λa:sequence O.∀n:nat.a (S n) ≤ a n.
+ λO:excedence.λa:sequence O.∀n:nat.a (S n) ≤ a n.
(*
-definition is_upper_bound ≝ λO:pordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
-definition is_lower_bound ≝ λO:pordered_set.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
+definition is_upper_bound ≝ λO:excedence.λa:sequence O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:excedence.λa:sequence O.λu:O.∀n:nat.u ≤ a n.
-record is_sup (O:pordered_set) (a:sequence O) (u:O) : Prop ≝
+record is_sup (O:excedence) (a:sequence O) (u:O) : Prop ≝
{ sup_upper_bound: is_upper_bound O a u;
sup_least_upper_bound: ∀v:O. is_upper_bound O a v → u≤v
}.
-record is_inf (O:pordered_set) (a:sequence O) (u:O) : Prop ≝
+record is_inf (O:excedence) (a:sequence O) (u:O) : Prop ≝
{ inf_lower_bound: is_lower_bound O a u;
inf_greatest_lower_bound: ∀v:O. is_lower_bound O a v → v≤u
}.
-record is_bounded_below (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded_below (O:excedence) (a:sequence O) : Type ≝
{ ib_lower_bound: O;
ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound
}.
-record is_bounded_above (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded_above (O:excedence) (a:sequence O) : Type ≝
{ ib_upper_bound: O;
ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound
}.
-record is_bounded (O:pordered_set) (a:sequence O) : Type ≝
+record is_bounded (O:excedence) (a:sequence O) : Type ≝
{ ib_bounded_below:> is_bounded_below ? a;
ib_bounded_above:> is_bounded_above ? a
}.
-record bounded_below_sequence (O:pordered_set) : Type ≝
+record bounded_below_sequence (O:excedence) : Type ≝
{ bbs_seq:> sequence O;
bbs_is_bounded_below:> is_bounded_below ? bbs_seq
}.
-record bounded_above_sequence (O:pordered_set) : Type ≝
+record bounded_above_sequence (O:excedence) : Type ≝
{ bas_seq:> sequence O;
bas_is_bounded_above:> is_bounded_above ? bas_seq
}.
-record bounded_sequence (O:pordered_set) : Type ≝
+record bounded_sequence (O:excedence) : Type ≝
{ bs_seq:> sequence O;
bs_is_bounded_below: is_bounded_below ? bs_seq;
bs_is_bounded_above: is_bounded_above ? bs_seq
}.
definition bounded_below_sequence_of_bounded_sequence ≝
- λO:pordered_set.λb:bounded_sequence O.
+ λO:excedence.λb:bounded_sequence O.
mk_bounded_below_sequence ? b (bs_is_bounded_below ? b).
coercion cic:/matita/sequence/bounded_below_sequence_of_bounded_sequence.con.
definition bounded_above_sequence_of_bounded_sequence ≝
- λO:pordered_set.λb:bounded_sequence O.
+ λO:excedence.λb:bounded_sequence O.
mk_bounded_above_sequence ? b (bs_is_bounded_above ? b).
coercion cic:/matita/sequence/bounded_above_sequence_of_bounded_sequence.con.
definition lower_bound ≝
- λO:pordered_set.λb:bounded_below_sequence O.
+ λO:excedence.λb:bounded_below_sequence O.
ib_lower_bound ? b (bbs_is_bounded_below ? b).
lemma lower_bound_is_lower_bound:
- ∀O:pordered_set.∀b:bounded_below_sequence O.
+ ∀O:excedence.∀b:bounded_below_sequence O.
is_lower_bound ? b (lower_bound ? b).
intros; unfold lower_bound; apply ib_lower_bound_is_lower_bound.
qed.
definition upper_bound ≝
- λO:pordered_set.λb:bounded_above_sequence O.
+ λO:excedence.λb:bounded_above_sequence O.
ib_upper_bound ? b (bas_is_bounded_above ? b).
lemma upper_bound_is_upper_bound:
- ∀O:pordered_set.∀b:bounded_above_sequence O.
+ ∀O:excedence.∀b:bounded_above_sequence O.
is_upper_bound ? b (upper_bound ? b).
intros; unfold upper_bound; apply ib_upper_bound_is_upper_bound.
qed.
|2: intros (x y z); apply Or_symmetric; apply cTf; assumption;]
qed.
-definition reverse_pordered_set: pordered_set → pordered_set.
-intros (p); apply (mk_pordered_set (reverse_excedence p));
+definition reverse_excedence: excedence → excedence.
+intros (p); apply (mk_excedence (reverse_excedence p));
generalize in match (reverse_excedence p); intros (E);
apply mk_is_porder_relation;
[apply le_reflexive|apply le_transitive|apply le_antisymmetric]
qed.
lemma is_lower_bound_reverse_is_upper_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
- is_lower_bound O a l → is_upper_bound (reverse_pordered_set O) a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+ is_lower_bound O a l → is_upper_bound (reverse_excedence O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence;
unfold reverse_excedence; simplify; fold unfold le (le ? l (a n)); apply H;
qed.
lemma is_upper_bound_reverse_is_lower_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
- is_upper_bound O a l → is_lower_bound (reverse_pordered_set O) a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+ is_upper_bound O a l → is_lower_bound (reverse_excedence O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence;
unfold reverse_excedence; simplify; fold unfold le (le ? (a n) l); apply H;
qed.
lemma reverse_is_lower_bound_is_upper_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
- is_lower_bound (reverse_pordered_set O) a l → is_upper_bound O a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+ is_lower_bound (reverse_excedence O) a l → is_upper_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence in H;
unfold reverse_excedence in H; simplify in H; apply H;
qed.
lemma reverse_is_upper_bound_is_lower_bound:
- ∀O:pordered_set.∀a:sequence O.∀l:O.
- is_upper_bound (reverse_pordered_set O) a l → is_lower_bound O a l.
-intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
+ ∀O:excedence.∀a:sequence O.∀l:O.
+ is_upper_bound (reverse_excedence O) a l → is_lower_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_excedence in H;
unfold reverse_excedence in H; simplify in H; apply H;
qed.
lemma is_inf_to_reverse_is_sup:
- ∀O:pordered_set.∀a:bounded_below_sequence O.∀l:O.
- is_inf O a l → is_sup (reverse_pordered_set O) a l.
-intros (O a l H); apply (mk_is_sup (reverse_pordered_set O));
+ ∀O:excedence.∀a:bounded_below_sequence O.∀l:O.
+ is_inf O a l → is_sup (reverse_excedence O) a l.
+intros (O a l H); apply (mk_is_sup (reverse_excedence O));
[1: apply is_lower_bound_reverse_is_upper_bound; apply inf_lower_bound; assumption
-|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify;
+|2: unfold reverse_excedence; simplify; unfold reverse_excedence; simplify;
intros (m H1); apply (inf_greatest_lower_bound ? ? ? H); apply H1;]
qed.
lemma is_sup_to_reverse_is_inf:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
- is_sup O a l → is_inf (reverse_pordered_set O) a l.
-intros (O a l H); apply (mk_is_inf (reverse_pordered_set O));
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+ is_sup O a l → is_inf (reverse_excedence O) a l.
+intros (O a l H); apply (mk_is_inf (reverse_excedence O));
[1: apply is_upper_bound_reverse_is_lower_bound; apply sup_upper_bound; assumption
-|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify;
+|2: unfold reverse_excedence; simplify; unfold reverse_excedence; simplify;
intros (m H1); apply (sup_least_upper_bound ? ? ? H); apply H1;]
qed.
lemma reverse_is_sup_to_is_inf:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
- is_sup (reverse_pordered_set O) a l → is_inf O a l.
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+ is_sup (reverse_excedence O) a l → is_inf O a l.
intros (O a l H); apply mk_is_inf;
[1: apply reverse_is_upper_bound_is_lower_bound;
- apply (sup_upper_bound (reverse_pordered_set O)); assumption
-|2: intros (v H1); apply (sup_least_upper_bound (reverse_pordered_set O) a l H v);
+ apply (sup_upper_bound (reverse_excedence O)); assumption
+|2: intros (v H1); apply (sup_least_upper_bound (reverse_excedence O) a l H v);
apply is_lower_bound_reverse_is_upper_bound; assumption;]
qed.
lemma reverse_is_inf_to_is_sup:
- ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
- is_inf (reverse_pordered_set O) a l → is_sup O a l.
+ ∀O:excedence.∀a:bounded_above_sequence O.∀l:O.
+ is_inf (reverse_excedence O) a l → is_sup O a l.
intros (O a l H); apply mk_is_sup;
[1: apply reverse_is_lower_bound_is_upper_bound;
- apply (inf_lower_bound (reverse_pordered_set O)); assumption
-|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_pordered_set O) a l H v);
+ apply (inf_lower_bound (reverse_excedence O)); assumption
+|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_excedence O) a l H v);
apply is_upper_bound_reverse_is_lower_bound; assumption;]
qed.
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