--- /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/fields/".
+
+include "attic/rings.ma".
+
+record is_field (R:ring) (inv:∀x:R.x ≠ 0 → R) : Prop
+≝
+ { (* multiplicative abelian properties *)
+ mult_comm_: symmetric ? (mult R);
+ (* multiplicative group properties *)
+ inv_inverse_: ∀x.∀p: x ≠ 0. inv x p * x = 1
+ }.
+
+lemma opp_opp: ∀R:ring. ∀x:R. --x=x.
+intros;
+apply (cancellationlaw ? (-x) ? ?);
+rewrite > (opp_inverse R x);
+rewrite > plus_comm;
+rewrite > opp_inverse;
+reflexivity.
+qed.
+
+let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n ≝
+ match n with
+ [ O ⇒ zero
+ | (S m) ⇒ plus one (sum C plus zero one m)
+ ].
+
+record field : Type \def
+ { f_ring:> ring;
+ inv: ∀x:f_ring. x ≠ 0 → f_ring;
+ field_properties: is_field f_ring inv
+ }.
+
+theorem mult_comm: ∀F:field.symmetric ? (mult F).
+ intro;
+ apply (mult_comm_ ? ? (field_properties F)).
+qed.
+
+theorem inv_inverse: ∀F:field.∀x:F.∀p: x ≠ 0. (inv ? x p)*x = 1.
+ intro;
+ apply (inv_inverse_ ? ? (field_properties F)).
+qed.
+
+(*CSC: qua funzionava anche mettendo ? al posto della prima F*)
+definition sum_field ≝
+ λF:field. sum F (plus F) 0 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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/integration_algebras/".
+
+include "attic/vector_spaces.ma".
+include "lattice.ma".
+
+(**************** Riesz Spaces ********************)
+
+record pre_riesz_space (K:ordered_field_ch0) : Type \def
+ { rs_vector_space:> vector_space K;
+ rs_lattice_: lattice;
+ rs_ordered_abelian_group_: ordered_abelian_group;
+ rs_with1:
+ og_abelian_group rs_ordered_abelian_group_ = vs_abelian_group ? rs_vector_space;
+ rs_with2:
+ og_ordered_set rs_ordered_abelian_group_ = ordered_set_of_lattice rs_lattice_
+ }.
+
+lemma rs_lattice: ∀K.pre_riesz_space K → lattice.
+ intros (K V);
+ cut (os_carrier (rs_lattice_ ? V) = V);
+ [ apply mk_lattice;
+ [ apply (carrier V)
+ | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut);
+ apply l_join
+ | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut);
+ apply l_meet
+ | apply
+ (eq_rect' ? ?
+ (λa:Type.λH:os_carrier (rs_lattice_ ? V)=a.
+ is_lattice a
+ (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
+ (l_join (rs_lattice_ K V)) a H)
+ (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
+ (l_meet (rs_lattice_ K V)) a H))
+ ? ? Hcut);
+ simplify;
+ apply l_lattice_properties
+ ]
+ | transitivity (os_carrier (rs_ordered_abelian_group_ ? V));
+ [ apply (eq_f ? ? os_carrier);
+ symmetry;
+ apply rs_with2
+ | apply (eq_f ? ? carrier);
+ apply rs_with1
+ ]
+ ].
+qed.
+
+coercion cic:/matita/integration_algebras/rs_lattice.con.
+
+lemma rs_ordered_abelian_group: ∀K.pre_riesz_space K → ordered_abelian_group.
+ intros (K V);
+ apply mk_ordered_abelian_group;
+ [ apply mk_pre_ordered_abelian_group;
+ [ apply (vs_abelian_group ? (rs_vector_space ? V))
+ | apply (ordered_set_of_lattice (rs_lattice ? V))
+ | reflexivity
+ ]
+ | simplify;
+ generalize in match
+ (og_ordered_abelian_group_properties (rs_ordered_abelian_group_ ? V));
+ intro P;
+ unfold in P;
+ elim daemon(*
+ apply
+ (eq_rect ? ?
+ (λO:ordered_set.
+ ∀f,g,h.
+ os_le O f g →
+ os_le O
+ (plus (abelian_group_OF_pre_riesz_space K V) f h)
+ (plus (abelian_group_OF_pre_riesz_space K V) g h))
+ ? ? (rs_with2 ? V));
+ apply
+ (eq_rect ? ?
+ (λG:abelian_group.
+ ∀f,g,h.
+ os_le (ordered_set_OF_pre_riesz_space K V) f g →
+ os_le (ordered_set_OF_pre_riesz_space K V)
+ (plus (abelian_group_OF_pre_riesz_space K V) f h)
+ (plus (abelian_group_OF_pre_riesz_space K V) g h))
+ ? ? (rs_with1 ? V));
+ simplify;
+ apply og_ordered_abelian_group_properties*)
+ ]
+qed.
+
+coercion cic:/matita/integration_algebras/rs_ordered_abelian_group.con.
+
+record is_riesz_space (K:ordered_field_ch0) (V:pre_riesz_space K) : Prop ≝
+ { rs_compat_le_times: ∀a:K.∀f:V. 0≤a → 0≤f → 0≤a*f
+ }.
+
+record riesz_space (K:ordered_field_ch0) : Type \def
+ { rs_pre_riesz_space:> pre_riesz_space K;
+ rs_riesz_space_properties: is_riesz_space ? rs_pre_riesz_space
+ }.
+
+record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
+ { positive: ∀u:V. 0≤u → 0≤T u;
+ linear1: ∀u,v:V. T (u+v) = T u + T v;
+ linear2: ∀u:V.∀k:K. T (k*u) = k*(T u)
+ }.
+
+record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝
+ { soc_incr:
+ ∀a:nat→V.∀l:V.is_increasing ? a → is_sup V a l →
+ is_increasing K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
+ }.
+
+definition absolute_value ≝ λK.λS:riesz_space K.λf:S.f ∨ -f.
+
+(**************** Normed Riesz spaces ****************************)
+
+definition is_riesz_norm ≝
+ λR:real.λV:riesz_space R.λnorm:norm R V.
+ ∀f,g:V. absolute_value ? V f ≤ absolute_value ? V g →
+ n_function R V norm f ≤ n_function R V norm g.
+
+record riesz_norm (R:real) (V:riesz_space R) : Type ≝
+ { rn_norm:> norm R V;
+ rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
+ }.
+
+(*CSC: non fa la chiusura delle coercion verso funclass *)
+definition rn_function ≝
+ λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
+ n_function R V (rn_norm ? ? norm).
+
+coercion cic:/matita/integration_algebras/rn_function.con 1.
+
+(************************** L-SPACES *************************************)
+(*
+record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
+ { ls_banach: is_complete ? V (induced_distance ? ? norm);
+ ls_linear: ∀f,g:V. le ? V 0 f → le ? V 0 g → norm (f+g) = norm f + norm g
+ }.
+*)
+(******************** ARCHIMEDEAN RIESZ SPACES ***************************)
+
+record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
+\def
+ { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
+ absolute_value ? S a ≤
+ (inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u →
+ a = 0
+ }.
+
+record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
+ { ars_riesz_space:> riesz_space K;
+ ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
+ }.
+
+definition is_weak_unit ≝
+(* This definition is by Spitters. He cites Fremlin 353P, but:
+ 1. that theorem holds only in f-algebras (as in Spitters, but we are
+ defining it on Riesz spaces)
+ 2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value?
+ λR:real.λV:archimedean_riesz_space R.λunit: V.
+ ∀x:V. meet x unit = 0 → u = 0.
+ 3. Fremlin proves u > 0 implies x /\ u > 0 > 0 for Archimedean spaces
+ only. We pick this definition for now.
+*) λR:real.λV:archimedean_riesz_space R.λe:V.
+ ∀v:V. 0<v → 0 < (v ∧ e).
+
+(* Here we are avoiding a construction (the quotient space to define
+ f=g iff I(|f-g|)=0 *)
+record integration_riesz_space (R:real) : Type \def
+ { irs_archimedean_riesz_space:> archimedean_riesz_space R;
+ irs_unit: irs_archimedean_riesz_space;
+ irs_weak_unit: is_weak_unit ? ? irs_unit;
+ integral: irs_archimedean_riesz_space → R;
+ irs_positive_linear: is_positive_linear ? ? integral;
+ irs_limit1:
+ ∀f:irs_archimedean_riesz_space.
+ tends_to ?
+ (λn.integral (f ∧ ((sum_field R n)*irs_unit)))
+ (integral f);
+ irs_limit2:
+ ∀f:irs_archimedean_riesz_space.
+ tends_to ?
+ (λn.
+ integral (f ∧
+ ((inv ? (sum_field R (S n))
+ (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
+ ) * irs_unit))) 0;
+ irs_quotient_space1:
+ ∀f,g:irs_archimedean_riesz_space.
+ integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
+ }.
+
+definition induced_norm_fun ≝
+ λR:real.λV:integration_riesz_space R.λf:V.
+ integral ? V (absolute_value ? ? f).
+
+lemma induced_norm_is_norm:
+ ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V).
+ elim daemon.(*
+ intros;
+ apply mk_is_norm;
+ [ apply mk_is_semi_norm;
+ [ unfold induced_norm_fun;
+ intros;
+ apply positive;
+ [ apply (irs_positive_linear ? V)
+ | (* difficile *)
+ elim daemon
+ ]
+ | intros;
+ unfold induced_norm_fun;
+ (* facile *)
+ elim daemon
+ | intros;
+ unfold induced_norm_fun;
+ (* difficile *)
+ elim daemon
+ ]
+ | intros;
+ unfold induced_norm_fun in H;
+ apply irs_quotient_space1;
+ unfold minus;
+ rewrite < plus_comm;
+ rewrite < eq_zero_opp_zero;
+ rewrite > zero_neutral;
+ assumption
+ ].*)
+qed.
+
+definition induced_norm ≝
+ λR:real.λV:integration_riesz_space R.
+ mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
+
+lemma is_riesz_norm_induced_norm:
+ ∀R:real.∀V:integration_riesz_space R.
+ is_riesz_norm ? ? (induced_norm ? V).
+ intros;
+ unfold is_riesz_norm;
+ intros;
+ unfold induced_norm;
+ simplify;
+ unfold induced_norm_fun;
+ (* difficile *)
+ elim daemon.
+qed.
+
+definition induced_riesz_norm ≝
+ λR:real.λV:integration_riesz_space R.
+ mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V).
+
+definition distance_induced_by_integral ≝
+ λR:real.λV:integration_riesz_space R.
+ induced_distance ? ? (induced_norm R V).
+
+definition is_complete_integration_riesz_space ≝
+ λR:real.λV:integration_riesz_space R.
+ is_complete ? ? (distance_induced_by_integral ? V).
+
+record complete_integration_riesz_space (R:real) : Type ≝
+ { cirz_integration_riesz_space:> integration_riesz_space R;
+ cirz_complete_integration_riesz_space_property:
+ is_complete_integration_riesz_space ? cirz_integration_riesz_space
+ }.
+
+(* now we prove that any complete integration riesz space is an L-space *)
+
+(*theorem is_l_space_l_space_induced_by_integral:
+ ∀R:real.∀V:complete_integration_riesz_space R.
+ is_l_space ? ? (induced_riesz_norm ? V).
+ intros;
+ constructor 1;
+ [ apply cirz_complete_integration_riesz_space_property
+ | intros;
+ unfold induced_riesz_norm;
+ simplify;
+ unfold induced_norm;
+ simplify;
+ unfold induced_norm_fun;
+ (* difficile *)
+ elim daemon
+ ].
+qed.*)
+
+(**************************** f-ALGEBRAS ********************************)
+
+record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
+≝
+ { (* ring properties *)
+ a_ring: is_ring V mult one;
+ (* algebra properties *)
+ a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
+ a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
+ }.
+
+record algebra (K: field) : Type \def
+ { a_vector_space:> vector_space K;
+ a_one: a_vector_space;
+ a_mult: a_vector_space → a_vector_space → a_vector_space;
+ a_algebra_properties: is_algebra ? ? a_mult a_one
+ }.
+
+interpretation "Algebra product" 'times a b =
+ (cic:/matita/integration_algebras/a_mult.con _ a b).
+
+definition ring_of_algebra ≝
+ λK.λA:algebra K.
+ mk_ring A (a_mult ? A) (a_one ? A)
+ (a_ring ? ? ? ? (a_algebra_properties ? A)).
+
+coercion cic:/matita/integration_algebras/ring_of_algebra.con.
+
+record pre_f_algebra (K:ordered_field_ch0) : Type ≝
+ { fa_archimedean_riesz_space:> archimedean_riesz_space K;
+ fa_algebra_: algebra K;
+ fa_with: a_vector_space ? fa_algebra_ = rs_vector_space ? fa_archimedean_riesz_space
+ }.
+
+lemma fa_algebra: ∀K:ordered_field_ch0.pre_f_algebra K → algebra K.
+ intros (K A);
+ apply mk_algebra;
+ [ apply (rs_vector_space ? A)
+ | elim daemon
+ | elim daemon
+ | elim daemon
+ ]
+ qed.
+
+coercion cic:/matita/integration_algebras/fa_algebra.con.
+
+record is_f_algebra (K) (A:pre_f_algebra K) : Prop ≝
+{ compat_mult_le: ∀f,g:A.0 ≤ f → 0 ≤ g → 0 ≤ f*g;
+ compat_mult_meet:
+ ∀f,g,h:A.(f ∧ g) = 0 → ((h*f) ∧ g) = 0
+}.
+
+record f_algebra (K:ordered_field_ch0) : Type ≝
+{ fa_pre_f_algebra:> pre_f_algebra K;
+ fa_f_algebra_properties: is_f_algebra ? fa_pre_f_algebra
+}.
+
+(* to be proved; see footnote 2 in the paper by Spitters *)
+axiom symmetric_a_mult:
+ ∀K.∀A:f_algebra K. symmetric ? (a_mult ? A).
+
+record integration_f_algebra (R:real) : Type \def
+ { ifa_integration_riesz_space:> integration_riesz_space R;
+ ifa_f_algebra_: f_algebra R;
+ ifa_with:
+ fa_archimedean_riesz_space ? ifa_f_algebra_ =
+ irs_archimedean_riesz_space ? ifa_integration_riesz_space
+ }.
+
+axiom ifa_f_algebra: ∀R:real.integration_f_algebra R → f_algebra R.
+
+coercion cic:/matita/integration_algebras/ifa_f_algebra.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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/ordered_fields_ch0/".
+
+include "attic/fields.ma".
+include "ordered_group.ma".
+
+(*CSC: non capisco questi alias! Una volta non servivano*)
+alias id "plus" = "cic:/matita/groups/plus.con".
+alias symbol "plus" = "Abelian group plus".
+
+record pre_ordered_field_ch0: Type ≝
+ { of_field:> field;
+ of_ordered_abelian_group_: ordered_abelian_group;
+ of_cotransitively_ordered_set_: cotransitively_ordered_set;
+ of_with1_:
+ cos_ordered_set of_cotransitively_ordered_set_ =
+ og_ordered_set of_ordered_abelian_group_;
+ of_with2:
+ og_abelian_group of_ordered_abelian_group_ = r_abelian_group of_field
+ }.
+
+lemma of_ordered_abelian_group: pre_ordered_field_ch0 → ordered_abelian_group.
+ intro F;
+ apply mk_ordered_abelian_group;
+ [ apply mk_pre_ordered_abelian_group;
+ [ apply (r_abelian_group F)
+ | apply (og_ordered_set (of_ordered_abelian_group_ F))
+ | apply (eq_f ? ? carrier);
+ apply (of_with2 F)
+ ]
+ |
+ apply
+ (eq_rect' ? ?
+ (λG:abelian_group.λH:og_abelian_group (of_ordered_abelian_group_ F)=G.
+ is_ordered_abelian_group
+ (mk_pre_ordered_abelian_group G (of_ordered_abelian_group_ F)
+ (eq_f abelian_group Type carrier (of_ordered_abelian_group_ F) G
+ H)))
+ ? ? (of_with2 F));
+ simplify;
+ apply (og_ordered_abelian_group_properties (of_ordered_abelian_group_ F))
+ ]
+qed.
+
+coercion cic:/matita/ordered_fields_ch0/of_ordered_abelian_group.con.
+
+(*CSC: I am not able to prove this since unfold is undone by coercion composition*)
+axiom of_with1:
+ ∀G:pre_ordered_field_ch0.
+ cos_ordered_set (of_cotransitively_ordered_set_ G) =
+ og_ordered_set (of_ordered_abelian_group G).
+
+lemma of_cotransitively_ordered_set : pre_ordered_field_ch0 → cotransitively_ordered_set.
+ intro F;
+ apply mk_cotransitively_ordered_set;
+ [ apply (og_ordered_set F)
+ | apply
+ (eq_rect ? ? (λa:ordered_set.cotransitive (os_carrier a) (os_le a))
+ ? ? (of_with1 F));
+ apply cos_cotransitive
+ ]
+qed.
+
+coercion cic:/matita/ordered_fields_ch0/of_cotransitively_ordered_set.con.
+
+record is_ordered_field_ch0 (F:pre_ordered_field_ch0) : Type \def
+ { of_mult_compat: ∀a,b:F. 0≤a → 0≤b → 0≤a*b;
+ of_weak_tricotomy : ∀a,b:F. a≠b → a≤b ∨ b≤a;
+ (* 0 characteristics *)
+ of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
+ }.
+
+record ordered_field_ch0 : Type \def
+ { of_pre_ordered_field_ch0:> pre_ordered_field_ch0;
+ of_ordered_field_properties:> is_ordered_field_ch0 of_pre_ordered_field_ch0
+ }.
+
+(*
+lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
+ intros;
+
+lemma not_eq_x_zero_to_lt_zero_mult_x_x:
+ ∀F:ordered_field_ch0.∀x:F. x ≠ 0 → 0 < x * x.
+ intros;
+ elim (of_weak_tricotomy ? ? ? ? ? ? ? ? F ? ? H);
+ [ generalize in match (le_x_zero_to_le_zero_opp_x F ? H1); intro;
+ generalize in match (of_mult_compat ? ? ? ? ? ? ? ? F ? ? H2 H2); intro;
+*)
+
+axiom lt_zero_to_lt_inv_zero:
+ ∀F:ordered_field_ch0.∀x:F.∀p:x≠0. lt F 0 x → lt F 0 (inv ? x p).
+
+alias symbol "lt" = "natural 'less than'".
+
+(* The ordering is not necessary. *)
+axiom not_eq_sum_field_zero: ∀F:ordered_field_ch0.∀n. O<n → sum_field F n ≠ 0.
+axiom le_zero_sum_field: ∀F:ordered_field_ch0.∀n. O<n → lt F 0 (sum_field F n).
+
+axiom lt_zero_to_le_inv_zero:
+ ∀F:ordered_field_ch0.∀n:nat.∀p:sum_field F n ≠ 0. 0 ≤ inv ? (sum_field ? n) p.
+
+definition tends_to : ∀F:ordered_field_ch0.∀f:nat→F.∀l:F.Prop.
+ apply
+ (λF:ordered_field_ch0.λf:nat → F.λl:F.
+ ∀n:nat.∃m:nat.∀j:nat.m ≤ j →
+ l - (inv F (sum_field ? (S n)) ?) ≤ f j ∧
+ f j ≤ l + (inv F (sum_field ? (S n)) ?));
+ apply not_eq_sum_field_zero;
+ unfold;
+ autobatch.
+qed.
+
+(*
+definition is_cauchy_seq ≝
+ λF:ordered_field_ch0.λf:nat→F.
+ ∀eps:F. 0 < eps →
+ ∃n:nat.∀M. n ≤ M →
+ -eps ≤ f M - f n ∧ f M - f n ≤ eps.
+*)
+
+
+
+definition is_cauchy_seq : ∀F:ordered_field_ch0.∀f:nat→F.Prop.
+ apply
+ (λF:ordered_field_ch0.λf:nat→F.
+ ∀m:nat.
+ ∃n:nat.∀N.n ≤ N →
+ -(inv ? (sum_field F (S m)) ?) ≤ f N - f n ∧
+ f N - f n ≤ inv ? (sum_field F (S m)) ?);
+ apply not_eq_sum_field_zero;
+ unfold;
+ autobatch.
+qed.
+
+definition is_complete ≝
+ λF:ordered_field_ch0.
+ ∀f:nat→F. is_cauchy_seq ? f →
+ ex F (λl:F. tends_to ? f l).
--- /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/reals/".
+
+include "attic/ordered_fields_ch0.ma".
+
+record is_real (F:ordered_field_ch0) : Type
+≝
+ { r_archimedean: ∀x:F. ∃n:nat. x ≤ (sum_field ? n);
+ r_complete: is_complete F
+ }.
+
+record real: Type \def
+ { r_ordered_field_ch0:> ordered_field_ch0;
+ r_real_properties: is_real r_ordered_field_ch0
+ }.
+
+definition lim: ∀R:real.∀f:nat→R.is_cauchy_seq ? f → R.
+ intros;
+ elim (r_complete ? (r_real_properties R) ? H);
+ exact a.
+qed.
+
+definition max_seq: ∀R:real.∀x,y:R. nat → R.
+ intros (R x y);
+ elim (cos_cotransitive R 0 (inv ? (sum_field ? (S n)) ?) (x-y));
+ [ apply x
+ | apply not_eq_sum_field_zero ;
+ unfold;
+ autobatch
+ | apply y
+ | apply lt_zero_to_le_inv_zero
+ ].
+qed.
+
+axiom daemon: False.
+
+theorem cauchy_max_seq: ∀R:real.∀x,y:R. is_cauchy_seq ? (max_seq ? x y).
+elim daemon.
+(*
+ intros;
+ unfold;
+ intros;
+ exists; [ exact m | ]; (* apply (ex_intro ? ? m); *)
+ intros;
+ unfold max_seq;
+ elim (of_cotransitive R 0
+(inv R (sum_field R (S N))
+ (not_eq_sum_field_zero R (S N) (le_S_S O N (le_O_n N)))) (x-y)
+(lt_zero_to_le_inv_zero R (S N)
+ (not_eq_sum_field_zero R (S N) (le_S_S O N (le_O_n N)))));
+ [ simplify;
+ elim (of_cotransitive R 0
+(inv R (1+sum R (plus R) 0 1 m)
+ (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))) (x-y)
+(lt_zero_to_le_inv_zero R (S m)
+ (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))));
+ [ simplify;
+ rewrite > (plus_comm ? x (-x));
+ rewrite > opp_inverse;
+ split;
+ [ apply (le_zero_x_to_le_opp_x_zero R ?);
+ apply lt_zero_to_le_inv_zero
+ | apply lt_zero_to_le_inv_zero
+ ]
+ | simplify;
+ split;
+ [ apply (or_transitive ? ? R ? 0);
+ [ apply (le_zero_x_to_le_opp_x_zero R ?)
+ | assumption
+ ]
+ | assumption
+ ]
+ ]
+ | simplify;
+ elim (of_cotransitive R 0
+(inv R (1+sum R (plus R) 0 1 m)
+ (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))) (x-y)
+(lt_zero_to_le_inv_zero R (S m)
+ (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))));
+ [ simplify;
+ split;
+ [ elim daemon
+ | generalize in match (le_zero_x_to_le_opp_x_zero R ? t1);
+ intro;
+ unfold minus in H1;
+ rewrite > eq_opp_plus_plus_opp_opp in H1;
+ rewrite > eq_opp_opp_x_x in H1;
+ rewrite > plus_comm in H1;
+ apply (or_transitive ? ? R ? 0);
+ [ assumption
+ | apply lt_zero_to_le_inv_zero
+ ]
+ ]
+ | simplify;
+ rewrite > (plus_comm ? y (-y));
+ rewrite > opp_inverse;
+ split;
+ [ elim daemon
+ | apply lt_zero_to_le_inv_zero
+ ]
+ ]
+ ].
+ elim daemon.*)
+qed.
+
+definition max: ∀R:real.R → R → R.
+ intros (R x y);
+ apply (lim R (max_seq R x y));
+ apply cauchy_max_seq.
+qed.
+
+definition abs \def λR:real.λx:R. max R x (-x).
+
+lemma comparison:
+ ∀R:real.∀f,g:nat→R. is_cauchy_seq ? f → is_cauchy_seq ? g →
+ (∀n:nat.f n ≤ g n) → lim ? f ? ≤ lim ? g ?.
+ [ assumption
+ | assumption
+ | intros;
+ elim daemon
+ ].
+qed.
+
+definition to_zero ≝
+ λR:real.λn.
+ -(inv R (sum_field R (S n))
+ (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))).
+
+axiom is_cauchy_seq_to_zero: ∀R:real. is_cauchy_seq ? (to_zero R).
+
+lemma technical1: ∀R:real.lim R (to_zero R) (is_cauchy_seq_to_zero R) = 0.
+ intros;
+ unfold lim;
+ elim daemon.
+qed.
+
+lemma abs_x_ge_O: ∀R:real.∀x:R. 0 ≤ abs ? x.
+ intros;
+ unfold abs;
+ unfold max;
+ rewrite < technical1;
+ apply comparison;
+ intros;
+ unfold to_zero;
+ unfold max_seq;
+ elim
+ (cos_cotransitive R 0
+(inv R (sum_field R (S n))
+ (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))) (x--x)
+(lt_zero_to_le_inv_zero R (S n)
+ (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))));
+ [ simplify;
+ (* facile *)
+ elim daemon
+ | simplify;
+ (* facile *)
+ elim daemon
+ ].
+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/rings/".
+
+include "group.ma".
+
+record is_ring (G:abelian_group) (mult:G→G→G) (one:G) : Prop
+≝
+ { (* multiplicative monoid properties *)
+ mult_assoc_: associative ? mult;
+ one_neutral_left_: left_neutral ? mult one;
+ one_neutral_right_: right_neutral ? mult one;
+ (* ring properties *)
+ mult_plus_distr_left_: distributive_left ? mult (plus G);
+ mult_plus_distr_right_: distributive_right ? mult (plus G);
+ not_eq_zero_one_: (0 ≠ one)
+ }.
+
+record ring : Type \def
+ { r_abelian_group:> abelian_group;
+ mult: r_abelian_group → r_abelian_group → r_abelian_group;
+ one: r_abelian_group;
+ r_ring_properties: is_ring r_abelian_group mult one
+ }.
+
+theorem mult_assoc: ∀R:ring.associative ? (mult R).
+ intros;
+ apply (mult_assoc_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R).
+ intros;
+ apply (one_neutral_left_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem one_neutral_right: ∀R:ring.right_neutral ? (mult R) (one R).
+ intros;
+ apply (one_neutral_right_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem mult_plus_distr_left: ∀R:ring.distributive_left ? (mult R) (plus R).
+ intros;
+ apply (mult_plus_distr_left_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem mult_plus_distr_right: ∀R:ring.distributive_right ? (mult R) (plus R).
+ intros;
+ apply (mult_plus_distr_right_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem not_eq_zero_one: ∀R:ring.0 ≠ one R.
+ intros;
+ apply (not_eq_zero_one_ ? ? ? (r_ring_properties R)).
+qed.
+
+interpretation "Ring mult" 'times a b =
+ (cic:/matita/rings/mult.con _ a b).
+
+notation "1" with precedence 89
+for @{ 'one }.
+
+interpretation "Ring one" 'one =
+ (cic:/matita/rings/one.con _).
+
+lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0.
+ intros;
+ generalize in match (zero_neutral R 0); intro;
+ generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H;
+ rewrite > mult_plus_distr_right in H1;
+ generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
+ rewrite < plus_assoc in H;
+ rewrite > opp_inverse in H;
+ rewrite > zero_neutral in H;
+ assumption.
+qed.
+
+lemma eq_mult_x_zero_zero: ∀R:ring.∀x:R.x*0=0.
+intros;
+generalize in match (zero_neutral R 0);
+intro;
+generalize in match (eq_f ? ? (λy.x*y) ? ? H); intro; clear H;
+(*CSC: qua funzionava prima della patch all'unificazione!*)
+rewrite > (mult_plus_distr_left R) in H1;
+generalize in match (eq_f ? ? (λy. (-(x*0)) +y) ? ? H1);intro;
+clear H1;
+rewrite < plus_assoc in H;
+rewrite > opp_inverse in H;
+rewrite > zero_neutral in H;
+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/vector_spaces/".
+
+include "attic/reals.ma".
+
+record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
+≝
+ { vs_nilpotent: ∀v. emult 0 v = 0;
+ vs_neutral: ∀v. emult 1 v = v;
+ vs_distributive: ∀a,b,v. emult (a + b) v = (emult a v) + (emult b v);
+ vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v)
+ }.
+
+record vector_space (K:field): Type \def
+{ vs_abelian_group :> abelian_group;
+ emult: K → vs_abelian_group → vs_abelian_group;
+ vs_vector_space_properties :> is_vector_space ? vs_abelian_group emult
+}.
+
+interpretation "Vector space external product" 'times a b =
+ (cic:/matita/vector_spaces/emult.con _ _ a b).
+
+record is_semi_norm (R:real) (V: vector_space R) (semi_norm:V→R) : Prop \def
+ { sn_positive: ∀x:V. zero R ≤ semi_norm x;
+ sn_omogeneous: ∀a:R.∀x:V. semi_norm (a*x) = (abs ? a) * semi_norm x;
+ sn_triangle_inequality: ∀x,y:V. semi_norm (x + y) ≤ semi_norm x + semi_norm y
+ }.
+
+theorem eq_semi_norm_zero_zero:
+ ∀R:real.∀V:vector_space R.∀semi_norm:V→R.
+ is_semi_norm ? ? semi_norm →
+ semi_norm 0 = 0.
+ intros;
+ (* facile *)
+ elim daemon.
+qed.
+
+record is_norm (R:real) (V:vector_space R) (norm:V → R) : Prop ≝
+ { n_semi_norm:> is_semi_norm ? ? norm;
+ n_properness: ∀x:V. norm x = 0 → x = 0
+ }.
+
+record norm (R:real) (V:vector_space R) : Type ≝
+ { n_function:1> V→R;
+ n_norm_properties: is_norm ? ? n_function
+ }.
+
+record is_semi_distance (R:real) (C:Type) (semi_d: C→C→R) : Prop ≝
+ { sd_positive: ∀x,y:C. zero R ≤ semi_d x y;
+ sd_properness: ∀x:C. semi_d x x = 0;
+ sd_triangle_inequality: ∀x,y,z:C. semi_d x z ≤ semi_d x y + semi_d z y
+ }.
+
+record is_distance (R:real) (C:Type) (d:C→C→R) : Prop ≝
+ { d_semi_distance:> is_semi_distance ? ? d;
+ d_properness: ∀x,y:C. d x y = 0 → x=y
+ }.
+
+record distance (R:real) (V:vector_space R) : Type ≝
+ { d_function:2> V→V→R;
+ d_distance_properties: is_distance ? ? d_function
+ }.
+
+definition induced_distance_fun ≝
+ λR:real.λV:vector_space R.λnorm:norm ? V.
+ λf,g:V.norm (f - g).
+
+theorem induced_distance_is_distance:
+ ∀R:real.∀V:vector_space R.∀norm:norm ? V.
+ is_distance ? ? (induced_distance_fun ? ? norm).
+elim daemon.(*
+ intros;
+ apply mk_is_distance;
+ [ apply mk_is_semi_distance;
+ [ unfold induced_distance_fun;
+ intros;
+ apply sn_positive;
+ apply n_semi_norm;
+ apply (n_norm_properties ? ? norm)
+ | unfold induced_distance_fun;
+ intros;
+ unfold minus;
+ rewrite < plus_comm;
+ rewrite > opp_inverse;
+ apply eq_semi_norm_zero_zero;
+ apply n_semi_norm;
+ apply (n_norm_properties ? ? norm)
+ | unfold induced_distance_fun;
+ intros;
+ (* ??? *)
+ elim daemon
+ ]
+ | unfold induced_distance_fun;
+ intros;
+ generalize in match (n_properness ? ? norm ? ? H);
+ [ intro;
+ (* facile *)
+ elim daemon
+ | apply (n_norm_properties ? ? norm)
+ ]
+ ].*)
+qed.
+
+definition induced_distance ≝
+ λR:real.λV:vector_space R.λnorm:norm ? V.
+ mk_distance ? ? (induced_distance_fun ? ? norm)
+ (induced_distance_is_distance ? ? norm).
+
+definition tends_to :
+ ∀R:real.∀V:vector_space R.∀d:distance ? V.∀f:nat→V.∀l:V.Prop.
+apply
+ (λR:real.λV:vector_space R.λd:distance ? V.λf:nat→V.λl:V.
+ ∀n:nat.∃m:nat.∀j:nat. m ≤ j →
+ d (f j) l ≤ inv R (sum_field ? (S n)) ?);
+ apply not_eq_sum_field_zero;
+ unfold;
+ autobatch.
+qed.
+
+definition is_cauchy_seq : ∀R:real.\forall V:vector_space R.
+\forall d:distance ? V.∀f:nat→V.Prop.
+ apply
+ (λR:real.λV: vector_space R. \lambda d:distance ? V.
+ \lambda f:nat→V.
+ ∀m:nat.
+ ∃n:nat.∀N. n ≤ N →
+ -(inv R (sum_field ? (S m)) ?) ≤ d (f N) (f n) ∧
+ d (f N) (f n)≤ inv R (sum_field R (S m)) ?);
+ apply not_eq_sum_field_zero;
+ unfold;
+ autobatch.
+qed.
+
+definition is_complete ≝
+ λR:real.λV:vector_space R.
+ λd:distance ? V.
+ ∀f:nat→V. is_cauchy_seq ? ? d f→
+ ex V (λl:V. tends_to ? ? d f l).
+++ /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/fields/".
-
-include "rings.ma".
-
-record is_field (R:ring) (inv:∀x:R.x ≠ 0 → R) : Prop
-≝
- { (* multiplicative abelian properties *)
- mult_comm_: symmetric ? (mult R);
- (* multiplicative group properties *)
- inv_inverse_: ∀x.∀p: x ≠ 0. inv x p * x = 1
- }.
-
-lemma opp_opp: ∀R:ring. ∀x:R. --x=x.
-intros;
-apply (cancellationlaw ? (-x) ? ?);
-rewrite > (opp_inverse R x);
-rewrite > plus_comm;
-rewrite > opp_inverse;
-reflexivity.
-qed.
-
-let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n ≝
- match n with
- [ O ⇒ zero
- | (S m) ⇒ plus one (sum C plus zero one m)
- ].
-
-record field : Type \def
- { f_ring:> ring;
- inv: ∀x:f_ring. x ≠ 0 → f_ring;
- field_properties: is_field f_ring inv
- }.
-
-theorem mult_comm: ∀F:field.symmetric ? (mult F).
- intro;
- apply (mult_comm_ ? ? (field_properties F)).
-qed.
-
-theorem inv_inverse: ∀F:field.∀x:F.∀p: x ≠ 0. (inv ? x p)*x = 1.
- intro;
- apply (inv_inverse_ ? ? (field_properties F)).
-qed.
-
-(*CSC: qua funzionava anche mettendo ? al posto della prima F*)
-definition sum_field ≝
- λF:field. sum F (plus F) 0 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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/group/".
+
+include "excedence.ma".
+
+definition left_neutral ≝ λC:apartness.λop.λe:C. ∀x:C. op e x ≈ x.
+definition right_neutral ≝ λC:apartness.λop. λe:C. ∀x:C. op x e ≈ x.
+definition left_inverse ≝ λC:apartness.λop.λe:C.λinv:C→C. ∀x:C. op (inv x) x ≈ e.
+definition right_inverse ≝ λC:apartness.λop.λe:C.λ inv: C→ C. ∀x:C. op x (inv x) ≈ e.
+definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
+(* ALLOW DEFINITION WITH SOME METAS *)
+
+definition distributive_left ≝
+ λA:apartness.λf:A→A→A.λg:A→A→A.
+ ∀x,y,z. f x (g y z) ≈ g (f x y) (f x z).
+
+definition distributive_right ≝
+ λA:apartness.λf:A→A→A.λg:A→A→A.
+ ∀x,y,z. f (g x y) z ≈ g (f x z) (f y z).
+
+record abelian_group : Type ≝
+ { carr:> apartness;
+ plus: carr → carr → carr;
+ zero: carr;
+ opp: carr → carr;
+ plus_assoc_: associative ? plus (eq carr);
+ plus_comm_: commutative ? plus (eq carr);
+ zero_neutral_: left_neutral ? plus zero;
+ opp_inverse_: left_inverse ? plus zero opp;
+ plus_strong_ext: ∀z.strong_ext ? (plus z)
+}.
+
+notation "0" with precedence 89 for @{ 'zero }.
+
+interpretation "Abelian group zero" 'zero =
+ (cic:/matita/group/zero.con _).
+
+interpretation "Abelian group plus" 'plus a b =
+ (cic:/matita/group/plus.con _ a b).
+
+interpretation "Abelian group opp" 'uminus a =
+ (cic:/matita/group/opp.con _ a).
+
+definition minus ≝
+ λG:abelian_group.λa,b:G. a + -b.
+
+interpretation "Abelian group minus" 'minus a b =
+ (cic:/matita/group/minus.con _ a b).
+
+lemma plus_assoc: ∀G:abelian_group.∀x,y,z:G.x+(y+z)≈x+y+z ≝ plus_assoc_.
+lemma plus_comm: ∀G:abelian_group.∀x,y:G.x+y≈y+x ≝ plus_comm_.
+lemma zero_neutral: ∀G:abelian_group.∀x:G.0+x≈x ≝ zero_neutral_.
+lemma opp_inverse: ∀G:abelian_group.∀x:G.-x+x≈0 ≝ opp_inverse_.
+
+definition ext ≝ λA:apartness.λop:A→A. ∀x,y. x ≈ y → op x ≈ op y.
+
+lemma strong_ext_to_ext: ∀A:apartness.∀op:A→A. strong_ext ? op → ext ? op.
+intros 6 (A op SEop x y Exy); intro Axy; apply Exy; apply SEop; assumption;
+qed.
+
+lemma feq_plusl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x+y ≈ x+z.
+intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_ext ? x));
+assumption;
+qed.
+
+coercion cic:/matita/group/feq_plusl.con nocomposites.
+
+lemma plus_strong_extr: ∀G:abelian_group.∀z:G.strong_ext ? (λx.x + z).
+intros 5 (G z x y A); simplify in A;
+lapply (plus_comm ? z x) as E1; lapply (plus_comm ? z y) as E2;
+lapply (ap_rewl ???? E1 A) as A1; lapply (ap_rewr ???? E2 A1) as A2;
+apply (plus_strong_ext ???? A2);
+qed.
+
+lemma feq_plusr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → y+x ≈ z+x.
+intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_extr ? x));
+assumption;
+qed.
+
+coercion cic:/matita/group/feq_plusr.con nocomposites.
+
+(* generation of coercions to make *_rew[lr] easier *)
+lemma feq_plusr_sym_: ∀G:abelian_group.∀x,y,z:G.z ≈ y → y+x ≈ z+x.
+compose feq_plusr with eq_sym (H); apply H; assumption;
+qed.
+coercion cic:/matita/group/feq_plusr_sym_.con nocomposites.
+lemma feq_plusl_sym_: ∀G:abelian_group.∀x,y,z:G.z ≈ y → x+y ≈ x+z.
+compose feq_plusl with eq_sym (H); apply H; assumption;
+qed.
+coercion cic:/matita/group/feq_plusl_sym_.con nocomposites.
+
+lemma fap_plusl: ∀G:abelian_group.∀x,y,z:G. y # z → x+y # x+z.
+intros (G x y z Ayz); apply (plus_strong_ext ? (-x));
+apply (ap_rewl ??? ((-x + x) + y));
+[1: apply plus_assoc;
+|2: apply (ap_rewr ??? ((-x +x) +z));
+ [1: apply plus_assoc;
+ |2: apply (ap_rewl ??? (0 + y));
+ [1: apply (feq_plusr ???? (opp_inverse ??));
+ |2: apply (ap_rewl ???? (zero_neutral ? y));
+ apply (ap_rewr ??? (0 + z) (opp_inverse ??));
+ apply (ap_rewr ???? (zero_neutral ??)); assumption;]]]
+qed.
+
+lemma fap_plusr: ∀G:abelian_group.∀x,y,z:G. y # z → y+x # z+x.
+intros (G x y z Ayz); apply (plus_strong_extr ? (-x));
+apply (ap_rewl ??? (y + (x + -x)));
+[1: apply (eq_sym ??? (plus_assoc ????));
+|2: apply (ap_rewr ??? (z + (x + -x)));
+ [1: apply (eq_sym ??? (plus_assoc ????));
+ |2: apply (ap_rewl ??? (y + (-x+x)) (plus_comm ? x (-x)));
+ apply (ap_rewl ??? (y + 0) (opp_inverse ??));
+ apply (ap_rewl ??? (0 + y) (plus_comm ???));
+ apply (ap_rewl ??? y (zero_neutral ??));
+ apply (ap_rewr ??? (z + (-x+x)) (plus_comm ? x (-x)));
+ apply (ap_rewr ??? (z + 0) (opp_inverse ??));
+ apply (ap_rewr ??? (0 + z) (plus_comm ???));
+ apply (ap_rewr ??? z (zero_neutral ??));
+ assumption]]
+qed.
+
+lemma plus_cancl: ∀G:abelian_group.∀y,z,x:G. x+y ≈ x+z → y ≈ z.
+intros 6 (G y z x E Ayz); apply E; apply fap_plusl; assumption;
+qed.
+
+lemma plus_cancr: ∀G:abelian_group.∀y,z,x:G. y+x ≈ z+x → y ≈ z.
+intros 6 (G y z x E Ayz); apply E; apply fap_plusr; assumption;
+qed.
+
+theorem eq_opp_plus_plus_opp_opp:
+ ∀G:abelian_group.∀x,y:G. -(x+y) ≈ -x + -y.
+intros (G x y); apply (plus_cancr ??? (x+y));
+apply (eq_trans ?? 0 ? (opp_inverse ??));
+apply (eq_trans ?? (-x + -y + x + y)); [2: apply (eq_sym ??? (plus_assoc ????))]
+apply (eq_trans ?? (-y + -x + x + y)); [2: repeat apply feq_plusr; apply plus_comm]
+apply (eq_trans ?? (-y + (-x + x) + y)); [2: apply feq_plusr; apply plus_assoc;]
+apply (eq_trans ?? (-y + 0 + y));
+ [2: apply feq_plusr; apply feq_plusl; apply eq_sym; apply opp_inverse]
+apply (eq_trans ?? (-y + y));
+ [2: apply feq_plusr; apply eq_sym;
+ apply (eq_trans ?? (0+-y)); [apply plus_comm|apply zero_neutral]]
+apply eq_sym; apply opp_inverse.
+qed.
+
+theorem eq_opp_opp_x_x: ∀G:abelian_group.∀x:G.--x ≈ x.
+intros (G x); apply (plus_cancl ??? (-x));
+apply (eq_trans ?? (--x + -x)); [apply plus_comm]
+apply (eq_trans ?? 0); [apply opp_inverse]
+apply eq_sym; apply opp_inverse;
+qed.
+
+theorem eq_zero_opp_zero: ∀G:abelian_group.0 ≈ -0. [assumption]
+intro G; apply (plus_cancr ??? 0);
+apply (eq_trans ?? 0); [apply zero_neutral;]
+apply eq_sym; apply opp_inverse;
+qed.
+
+lemma feq_oppr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x ≈ -y → x ≈ -z.
+intros (G x y z H1 H2); apply (plus_cancr ??? z);
+apply (eq_trans ?? 0 ?? (opp_inverse ?z));
+apply (eq_trans ?? (-y + z) ? H2);
+apply (eq_trans ?? (-y + y) ? H1);
+apply (eq_trans ?? 0 ? (opp_inverse ??));
+apply eq_reflexive;
+qed.
+
+lemma feq_oppl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → -y ≈ x → -z ≈ x.
+intros (G x y z H1 H2); apply eq_sym; apply (feq_oppr ??y);
+[2:apply eq_sym] assumption;
+qed.
+
+lemma feq_opp: ∀G:abelian_group.∀x,y:G. x ≈ y → -x ≈ -y.
+intros (G x y H); apply (feq_oppl ??y ? H); apply eq_reflexive;
+qed.
+
+coercion cic:/matita/group/feq_opp.con nocomposites.
+
+lemma eq_opp_sym: ∀G:abelian_group.∀x,y:G. y ≈ x → -x ≈ -y.
+compose feq_opp with eq_sym (H); apply H; assumption;
+qed.
+
+coercion cic:/matita/group/eq_opp_sym.con nocomposites.
+
+lemma eq_opp_plusr: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(x + z) ≈ -(y + z).
+compose feq_plusr with feq_opp(H); apply H; assumption;
+qed.
+
+coercion cic:/matita/group/eq_opp_plusr.con nocomposites.
+
+lemma eq_opp_plusl: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(z + x) ≈ -(z + y).
+compose feq_plusl with feq_opp(H); apply H; assumption;
+qed.
+
+coercion cic:/matita/group/eq_opp_plusl.con nocomposites.
+
+lemma plus_cancr_ap: ∀G:abelian_group.∀x,y,z:G. x+z # y+z → x # y.
+intros (G x y z H); lapply (fap_plusr ? (-z) ?? H) as H1; clear H;
+lapply (ap_rewl ? (x + (z + -z)) ?? (plus_assoc ? x z (-z)) H1) as H2; clear H1;
+lapply (ap_rewl ? (x + (-z + z)) ?? (plus_comm ?z (-z)) H2) as H1; clear H2;
+lapply (ap_rewl ? (x + 0) ?? (opp_inverse ?z) H1) as H2; clear H1;
+lapply (ap_rewl ? (0+x) ?? (plus_comm ?x 0) H2) as H1; clear H2;
+lapply (ap_rewl ? x ?? (zero_neutral ?x) H1) as H2; clear H1;
+lapply (ap_rewr ? (y + (z + -z)) ?? (plus_assoc ? y z (-z)) H2) as H3;
+lapply (ap_rewr ? (y + (-z + z)) ?? (plus_comm ?z (-z)) H3) as H4;
+lapply (ap_rewr ? (y + 0) ?? (opp_inverse ?z) H4) as H5;
+lapply (ap_rewr ? (0+y) ?? (plus_comm ?y 0) H5) as H6;
+lapply (ap_rewr ? y ?? (zero_neutral ?y) H6);
+assumption;
+qed.
+
+lemma pluc_cancl_ap: ∀G:abelian_group.∀x,y,z:G. z+x # z+y → x # y.
+intros (G x y z H); apply (plus_cancr_ap ??? z);
+apply (ap_rewl ???? (plus_comm ???));
+apply (ap_rewr ???? (plus_comm ???));
+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/groups/".
-
-include "excedence.ma".
-
-definition left_neutral ≝ λC:apartness.λop.λe:C. ∀x:C. op e x ≈ x.
-definition right_neutral ≝ λC:apartness.λop. λe:C. ∀x:C. op x e ≈ x.
-definition left_inverse ≝ λC:apartness.λop.λe:C.λinv:C→C. ∀x:C. op (inv x) x ≈ e.
-definition right_inverse ≝ λC:apartness.λop.λe:C.λ inv: C→ C. ∀x:C. op x (inv x) ≈ e.
-definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
-(* ALLOW DEFINITION WITH SOME METAS *)
-
-definition distributive_left ≝
- λA:apartness.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f x (g y z) ≈ g (f x y) (f x z).
-
-definition distributive_right ≝
- λA:apartness.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f (g x y) z ≈ g (f x z) (f y z).
-
-record abelian_group : Type ≝
- { carr:> apartness;
- plus: carr → carr → carr;
- zero: carr;
- opp: carr → carr;
- plus_assoc_: associative ? plus (eq carr);
- plus_comm_: commutative ? plus (eq carr);
- zero_neutral_: left_neutral ? plus zero;
- opp_inverse_: left_inverse ? plus zero opp;
- plus_strong_ext: ∀z.strong_ext ? (plus z)
-}.
-
-notation "0" with precedence 89 for @{ 'zero }.
-
-interpretation "Abelian group zero" 'zero =
- (cic:/matita/groups/zero.con _).
-
-interpretation "Abelian group plus" 'plus a b =
- (cic:/matita/groups/plus.con _ a b).
-
-interpretation "Abelian group opp" 'uminus a =
- (cic:/matita/groups/opp.con _ a).
-
-definition minus ≝
- λG:abelian_group.λa,b:G. a + -b.
-
-interpretation "Abelian group minus" 'minus a b =
- (cic:/matita/groups/minus.con _ a b).
-
-lemma plus_assoc: ∀G:abelian_group.∀x,y,z:G.x+(y+z)≈x+y+z ≝ plus_assoc_.
-lemma plus_comm: ∀G:abelian_group.∀x,y:G.x+y≈y+x ≝ plus_comm_.
-lemma zero_neutral: ∀G:abelian_group.∀x:G.0+x≈x ≝ zero_neutral_.
-lemma opp_inverse: ∀G:abelian_group.∀x:G.-x+x≈0 ≝ opp_inverse_.
-
-definition ext ≝ λA:apartness.λop:A→A. ∀x,y. x ≈ y → op x ≈ op y.
-
-lemma strong_ext_to_ext: ∀A:apartness.∀op:A→A. strong_ext ? op → ext ? op.
-intros 6 (A op SEop x y Exy); intro Axy; apply Exy; apply SEop; assumption;
-qed.
-
-lemma feq_plusl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x+y ≈ x+z.
-intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_ext ? x));
-assumption;
-qed.
-
-coercion cic:/matita/groups/feq_plusl.con nocomposites.
-
-lemma plus_strong_extr: ∀G:abelian_group.∀z:G.strong_ext ? (λx.x + z).
-intros 5 (G z x y A); simplify in A;
-lapply (plus_comm ? z x) as E1; lapply (plus_comm ? z y) as E2;
-lapply (ap_rewl ???? E1 A) as A1; lapply (ap_rewr ???? E2 A1) as A2;
-apply (plus_strong_ext ???? A2);
-qed.
-
-lemma feq_plusr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → y+x ≈ z+x.
-intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_extr ? x));
-assumption;
-qed.
-
-coercion cic:/matita/groups/feq_plusr.con nocomposites.
-
-(* generation of coercions to make *_rew[lr] easier *)
-lemma feq_plusr_sym_: ∀G:abelian_group.∀x,y,z:G.z ≈ y → y+x ≈ z+x.
-compose feq_plusr with eq_sym (H); apply H; assumption;
-qed.
-coercion cic:/matita/groups/feq_plusr_sym_.con nocomposites.
-lemma feq_plusl_sym_: ∀G:abelian_group.∀x,y,z:G.z ≈ y → x+y ≈ x+z.
-compose feq_plusl with eq_sym (H); apply H; assumption;
-qed.
-coercion cic:/matita/groups/feq_plusl_sym_.con nocomposites.
-
-lemma fap_plusl: ∀G:abelian_group.∀x,y,z:G. y # z → x+y # x+z.
-intros (G x y z Ayz); apply (plus_strong_ext ? (-x));
-apply (ap_rewl ??? ((-x + x) + y));
-[1: apply plus_assoc;
-|2: apply (ap_rewr ??? ((-x +x) +z));
- [1: apply plus_assoc;
- |2: apply (ap_rewl ??? (0 + y));
- [1: apply (feq_plusr ???? (opp_inverse ??));
- |2: apply (ap_rewl ???? (zero_neutral ? y));
- apply (ap_rewr ??? (0 + z) (opp_inverse ??));
- apply (ap_rewr ???? (zero_neutral ??)); assumption;]]]
-qed.
-
-lemma fap_plusr: ∀G:abelian_group.∀x,y,z:G. y # z → y+x # z+x.
-intros (G x y z Ayz); apply (plus_strong_extr ? (-x));
-apply (ap_rewl ??? (y + (x + -x)));
-[1: apply (eq_sym ??? (plus_assoc ????));
-|2: apply (ap_rewr ??? (z + (x + -x)));
- [1: apply (eq_sym ??? (plus_assoc ????));
- |2: apply (ap_rewl ??? (y + (-x+x)) (plus_comm ? x (-x)));
- apply (ap_rewl ??? (y + 0) (opp_inverse ??));
- apply (ap_rewl ??? (0 + y) (plus_comm ???));
- apply (ap_rewl ??? y (zero_neutral ??));
- apply (ap_rewr ??? (z + (-x+x)) (plus_comm ? x (-x)));
- apply (ap_rewr ??? (z + 0) (opp_inverse ??));
- apply (ap_rewr ??? (0 + z) (plus_comm ???));
- apply (ap_rewr ??? z (zero_neutral ??));
- assumption]]
-qed.
-
-lemma plus_cancl: ∀G:abelian_group.∀y,z,x:G. x+y ≈ x+z → y ≈ z.
-intros 6 (G y z x E Ayz); apply E; apply fap_plusl; assumption;
-qed.
-
-lemma plus_cancr: ∀G:abelian_group.∀y,z,x:G. y+x ≈ z+x → y ≈ z.
-intros 6 (G y z x E Ayz); apply E; apply fap_plusr; assumption;
-qed.
-
-theorem eq_opp_plus_plus_opp_opp:
- ∀G:abelian_group.∀x,y:G. -(x+y) ≈ -x + -y.
-intros (G x y); apply (plus_cancr ??? (x+y));
-apply (eq_trans ?? 0 ? (opp_inverse ??));
-apply (eq_trans ?? (-x + -y + x + y)); [2: apply (eq_sym ??? (plus_assoc ????))]
-apply (eq_trans ?? (-y + -x + x + y)); [2: repeat apply feq_plusr; apply plus_comm]
-apply (eq_trans ?? (-y + (-x + x) + y)); [2: apply feq_plusr; apply plus_assoc;]
-apply (eq_trans ?? (-y + 0 + y));
- [2: apply feq_plusr; apply feq_plusl; apply eq_sym; apply opp_inverse]
-apply (eq_trans ?? (-y + y));
- [2: apply feq_plusr; apply eq_sym;
- apply (eq_trans ?? (0+-y)); [apply plus_comm|apply zero_neutral]]
-apply eq_sym; apply opp_inverse.
-qed.
-
-theorem eq_opp_opp_x_x: ∀G:abelian_group.∀x:G.--x ≈ x.
-intros (G x); apply (plus_cancl ??? (-x));
-apply (eq_trans ?? (--x + -x)); [apply plus_comm]
-apply (eq_trans ?? 0); [apply opp_inverse]
-apply eq_sym; apply opp_inverse;
-qed.
-
-theorem eq_zero_opp_zero: ∀G:abelian_group.0 ≈ -0. [assumption]
-intro G; apply (plus_cancr ??? 0);
-apply (eq_trans ?? 0); [apply zero_neutral;]
-apply eq_sym; apply opp_inverse;
-qed.
-
-lemma feq_oppr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x ≈ -y → x ≈ -z.
-intros (G x y z H1 H2); apply (plus_cancr ??? z);
-apply (eq_trans ?? 0 ?? (opp_inverse ?z));
-apply (eq_trans ?? (-y + z) ? H2);
-apply (eq_trans ?? (-y + y) ? H1);
-apply (eq_trans ?? 0 ? (opp_inverse ??));
-apply eq_reflexive;
-qed.
-
-lemma feq_oppl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → -y ≈ x → -z ≈ x.
-intros (G x y z H1 H2); apply eq_sym; apply (feq_oppr ??y);
-[2:apply eq_sym] assumption;
-qed.
-
-lemma feq_opp: ∀G:abelian_group.∀x,y:G. x ≈ y → -x ≈ -y.
-intros (G x y H); apply (feq_oppl ??y ? H); apply eq_reflexive;
-qed.
-
-coercion cic:/matita/groups/feq_opp.con nocomposites.
-
-lemma eq_opp_sym: ∀G:abelian_group.∀x,y:G. y ≈ x → -x ≈ -y.
-compose feq_opp with eq_sym (H); apply H; assumption;
-qed.
-
-coercion cic:/matita/groups/eq_opp_sym.con nocomposites.
-
-lemma eq_opp_plusr: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(x + z) ≈ -(y + z).
-compose feq_plusr with feq_opp(H); apply H; assumption;
-qed.
-
-coercion cic:/matita/groups/eq_opp_plusr.con nocomposites.
-
-lemma eq_opp_plusl: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(z + x) ≈ -(z + y).
-compose feq_plusl with feq_opp(H); apply H; assumption;
-qed.
-
-coercion cic:/matita/groups/eq_opp_plusl.con nocomposites.
-
-lemma plus_cancr_ap: ∀G:abelian_group.∀x,y,z:G. x+z # y+z → x # y.
-intros (G x y z H); lapply (fap_plusr ? (-z) ?? H) as H1; clear H;
-lapply (ap_rewl ? (x + (z + -z)) ?? (plus_assoc ? x z (-z)) H1) as H2; clear H1;
-lapply (ap_rewl ? (x + (-z + z)) ?? (plus_comm ?z (-z)) H2) as H1; clear H2;
-lapply (ap_rewl ? (x + 0) ?? (opp_inverse ?z) H1) as H2; clear H1;
-lapply (ap_rewl ? (0+x) ?? (plus_comm ?x 0) H2) as H1; clear H2;
-lapply (ap_rewl ? x ?? (zero_neutral ?x) H1) as H2; clear H1;
-lapply (ap_rewr ? (y + (z + -z)) ?? (plus_assoc ? y z (-z)) H2) as H3;
-lapply (ap_rewr ? (y + (-z + z)) ?? (plus_comm ?z (-z)) H3) as H4;
-lapply (ap_rewr ? (y + 0) ?? (opp_inverse ?z) H4) as H5;
-lapply (ap_rewr ? (0+y) ?? (plus_comm ?y 0) H5) as H6;
-lapply (ap_rewr ? y ?? (zero_neutral ?y) H6);
-assumption;
-qed.
-
-lemma pluc_cancl_ap: ∀G:abelian_group.∀x,y,z:G. z+x # z+y → x # y.
-intros (G x y z H); apply (plus_cancr_ap ??? z);
-apply (ap_rewl ???? (plus_comm ???));
-apply (ap_rewr ???? (plus_comm ???));
-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/integration_algebras/".
-
-include "vector_spaces.ma".
-include "lattice.ma".
-
-(**************** Riesz Spaces ********************)
-
-record pre_riesz_space (K:ordered_field_ch0) : Type \def
- { rs_vector_space:> vector_space K;
- rs_lattice_: lattice;
- rs_ordered_abelian_group_: ordered_abelian_group;
- rs_with1:
- og_abelian_group rs_ordered_abelian_group_ = vs_abelian_group ? rs_vector_space;
- rs_with2:
- og_ordered_set rs_ordered_abelian_group_ = ordered_set_of_lattice rs_lattice_
- }.
-
-lemma rs_lattice: ∀K.pre_riesz_space K → lattice.
- intros (K V);
- cut (os_carrier (rs_lattice_ ? V) = V);
- [ apply mk_lattice;
- [ apply (carrier V)
- | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut);
- apply l_join
- | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut);
- apply l_meet
- | apply
- (eq_rect' ? ?
- (λa:Type.λH:os_carrier (rs_lattice_ ? V)=a.
- is_lattice a
- (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
- (l_join (rs_lattice_ K V)) a H)
- (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C)
- (l_meet (rs_lattice_ K V)) a H))
- ? ? Hcut);
- simplify;
- apply l_lattice_properties
- ]
- | transitivity (os_carrier (rs_ordered_abelian_group_ ? V));
- [ apply (eq_f ? ? os_carrier);
- symmetry;
- apply rs_with2
- | apply (eq_f ? ? carrier);
- apply rs_with1
- ]
- ].
-qed.
-
-coercion cic:/matita/integration_algebras/rs_lattice.con.
-
-lemma rs_ordered_abelian_group: ∀K.pre_riesz_space K → ordered_abelian_group.
- intros (K V);
- apply mk_ordered_abelian_group;
- [ apply mk_pre_ordered_abelian_group;
- [ apply (vs_abelian_group ? (rs_vector_space ? V))
- | apply (ordered_set_of_lattice (rs_lattice ? V))
- | reflexivity
- ]
- | simplify;
- generalize in match
- (og_ordered_abelian_group_properties (rs_ordered_abelian_group_ ? V));
- intro P;
- unfold in P;
- elim daemon(*
- apply
- (eq_rect ? ?
- (λO:ordered_set.
- ∀f,g,h.
- os_le O f g →
- os_le O
- (plus (abelian_group_OF_pre_riesz_space K V) f h)
- (plus (abelian_group_OF_pre_riesz_space K V) g h))
- ? ? (rs_with2 ? V));
- apply
- (eq_rect ? ?
- (λG:abelian_group.
- ∀f,g,h.
- os_le (ordered_set_OF_pre_riesz_space K V) f g →
- os_le (ordered_set_OF_pre_riesz_space K V)
- (plus (abelian_group_OF_pre_riesz_space K V) f h)
- (plus (abelian_group_OF_pre_riesz_space K V) g h))
- ? ? (rs_with1 ? V));
- simplify;
- apply og_ordered_abelian_group_properties*)
- ]
-qed.
-
-coercion cic:/matita/integration_algebras/rs_ordered_abelian_group.con.
-
-record is_riesz_space (K:ordered_field_ch0) (V:pre_riesz_space K) : Prop ≝
- { rs_compat_le_times: ∀a:K.∀f:V. 0≤a → 0≤f → 0≤a*f
- }.
-
-record riesz_space (K:ordered_field_ch0) : Type \def
- { rs_pre_riesz_space:> pre_riesz_space K;
- rs_riesz_space_properties: is_riesz_space ? rs_pre_riesz_space
- }.
-
-record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝
- { positive: ∀u:V. 0≤u → 0≤T u;
- linear1: ∀u,v:V. T (u+v) = T u + T v;
- linear2: ∀u:V.∀k:K. T (k*u) = k*(T u)
- }.
-
-record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝
- { soc_incr:
- ∀a:nat→V.∀l:V.is_increasing ? a → is_sup V a l →
- is_increasing K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l)
- }.
-
-definition absolute_value ≝ λK.λS:riesz_space K.λf:S.f ∨ -f.
-
-(**************** Normed Riesz spaces ****************************)
-
-definition is_riesz_norm ≝
- λR:real.λV:riesz_space R.λnorm:norm R V.
- ∀f,g:V. absolute_value ? V f ≤ absolute_value ? V g →
- n_function R V norm f ≤ n_function R V norm g.
-
-record riesz_norm (R:real) (V:riesz_space R) : Type ≝
- { rn_norm:> norm R V;
- rn_riesz_norm_property: is_riesz_norm ? ? rn_norm
- }.
-
-(*CSC: non fa la chiusura delle coercion verso funclass *)
-definition rn_function ≝
- λR:real.λV:riesz_space R.λnorm:riesz_norm ? V.
- n_function R V (rn_norm ? ? norm).
-
-coercion cic:/matita/integration_algebras/rn_function.con 1.
-
-(************************** L-SPACES *************************************)
-(*
-record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝
- { ls_banach: is_complete ? V (induced_distance ? ? norm);
- ls_linear: ∀f,g:V. le ? V 0 f → le ? V 0 g → norm (f+g) = norm f + norm g
- }.
-*)
-(******************** ARCHIMEDEAN RIESZ SPACES ***************************)
-
-record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop
-\def
- { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O.
- absolute_value ? S a ≤
- (inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u →
- a = 0
- }.
-
-record archimedean_riesz_space (K:ordered_field_ch0) : Type \def
- { ars_riesz_space:> riesz_space K;
- ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space
- }.
-
-definition is_weak_unit ≝
-(* This definition is by Spitters. He cites Fremlin 353P, but:
- 1. that theorem holds only in f-algebras (as in Spitters, but we are
- defining it on Riesz spaces)
- 2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value?
- λR:real.λV:archimedean_riesz_space R.λunit: V.
- ∀x:V. meet x unit = 0 → u = 0.
- 3. Fremlin proves u > 0 implies x /\ u > 0 > 0 for Archimedean spaces
- only. We pick this definition for now.
-*) λR:real.λV:archimedean_riesz_space R.λe:V.
- ∀v:V. 0<v → 0 < (v ∧ e).
-
-(* Here we are avoiding a construction (the quotient space to define
- f=g iff I(|f-g|)=0 *)
-record integration_riesz_space (R:real) : Type \def
- { irs_archimedean_riesz_space:> archimedean_riesz_space R;
- irs_unit: irs_archimedean_riesz_space;
- irs_weak_unit: is_weak_unit ? ? irs_unit;
- integral: irs_archimedean_riesz_space → R;
- irs_positive_linear: is_positive_linear ? ? integral;
- irs_limit1:
- ∀f:irs_archimedean_riesz_space.
- tends_to ?
- (λn.integral (f ∧ ((sum_field R n)*irs_unit)))
- (integral f);
- irs_limit2:
- ∀f:irs_archimedean_riesz_space.
- tends_to ?
- (λn.
- integral (f ∧
- ((inv ? (sum_field R (S n))
- (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))
- ) * irs_unit))) 0;
- irs_quotient_space1:
- ∀f,g:irs_archimedean_riesz_space.
- integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g
- }.
-
-definition induced_norm_fun ≝
- λR:real.λV:integration_riesz_space R.λf:V.
- integral ? V (absolute_value ? ? f).
-
-lemma induced_norm_is_norm:
- ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V).
- elim daemon.(*
- intros;
- apply mk_is_norm;
- [ apply mk_is_semi_norm;
- [ unfold induced_norm_fun;
- intros;
- apply positive;
- [ apply (irs_positive_linear ? V)
- | (* difficile *)
- elim daemon
- ]
- | intros;
- unfold induced_norm_fun;
- (* facile *)
- elim daemon
- | intros;
- unfold induced_norm_fun;
- (* difficile *)
- elim daemon
- ]
- | intros;
- unfold induced_norm_fun in H;
- apply irs_quotient_space1;
- unfold minus;
- rewrite < plus_comm;
- rewrite < eq_zero_opp_zero;
- rewrite > zero_neutral;
- assumption
- ].*)
-qed.
-
-definition induced_norm ≝
- λR:real.λV:integration_riesz_space R.
- mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V).
-
-lemma is_riesz_norm_induced_norm:
- ∀R:real.∀V:integration_riesz_space R.
- is_riesz_norm ? ? (induced_norm ? V).
- intros;
- unfold is_riesz_norm;
- intros;
- unfold induced_norm;
- simplify;
- unfold induced_norm_fun;
- (* difficile *)
- elim daemon.
-qed.
-
-definition induced_riesz_norm ≝
- λR:real.λV:integration_riesz_space R.
- mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V).
-
-definition distance_induced_by_integral ≝
- λR:real.λV:integration_riesz_space R.
- induced_distance ? ? (induced_norm R V).
-
-definition is_complete_integration_riesz_space ≝
- λR:real.λV:integration_riesz_space R.
- is_complete ? ? (distance_induced_by_integral ? V).
-
-record complete_integration_riesz_space (R:real) : Type ≝
- { cirz_integration_riesz_space:> integration_riesz_space R;
- cirz_complete_integration_riesz_space_property:
- is_complete_integration_riesz_space ? cirz_integration_riesz_space
- }.
-
-(* now we prove that any complete integration riesz space is an L-space *)
-
-(*theorem is_l_space_l_space_induced_by_integral:
- ∀R:real.∀V:complete_integration_riesz_space R.
- is_l_space ? ? (induced_riesz_norm ? V).
- intros;
- constructor 1;
- [ apply cirz_complete_integration_riesz_space_property
- | intros;
- unfold induced_riesz_norm;
- simplify;
- unfold induced_norm;
- simplify;
- unfold induced_norm_fun;
- (* difficile *)
- elim daemon
- ].
-qed.*)
-
-(**************************** f-ALGEBRAS ********************************)
-
-record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop
-≝
- { (* ring properties *)
- a_ring: is_ring V mult one;
- (* algebra properties *)
- a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g;
- a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g)
- }.
-
-record algebra (K: field) : Type \def
- { a_vector_space:> vector_space K;
- a_one: a_vector_space;
- a_mult: a_vector_space → a_vector_space → a_vector_space;
- a_algebra_properties: is_algebra ? ? a_mult a_one
- }.
-
-interpretation "Algebra product" 'times a b =
- (cic:/matita/integration_algebras/a_mult.con _ a b).
-
-definition ring_of_algebra ≝
- λK.λA:algebra K.
- mk_ring A (a_mult ? A) (a_one ? A)
- (a_ring ? ? ? ? (a_algebra_properties ? A)).
-
-coercion cic:/matita/integration_algebras/ring_of_algebra.con.
-
-record pre_f_algebra (K:ordered_field_ch0) : Type ≝
- { fa_archimedean_riesz_space:> archimedean_riesz_space K;
- fa_algebra_: algebra K;
- fa_with: a_vector_space ? fa_algebra_ = rs_vector_space ? fa_archimedean_riesz_space
- }.
-
-lemma fa_algebra: ∀K:ordered_field_ch0.pre_f_algebra K → algebra K.
- intros (K A);
- apply mk_algebra;
- [ apply (rs_vector_space ? A)
- | elim daemon
- | elim daemon
- | elim daemon
- ]
- qed.
-
-coercion cic:/matita/integration_algebras/fa_algebra.con.
-
-record is_f_algebra (K) (A:pre_f_algebra K) : Prop ≝
-{ compat_mult_le: ∀f,g:A.0 ≤ f → 0 ≤ g → 0 ≤ f*g;
- compat_mult_meet:
- ∀f,g,h:A.(f ∧ g) = 0 → ((h*f) ∧ g) = 0
-}.
-
-record f_algebra (K:ordered_field_ch0) : Type ≝
-{ fa_pre_f_algebra:> pre_f_algebra K;
- fa_f_algebra_properties: is_f_algebra ? fa_pre_f_algebra
-}.
-
-(* to be proved; see footnote 2 in the paper by Spitters *)
-axiom symmetric_a_mult:
- ∀K.∀A:f_algebra K. symmetric ? (a_mult ? A).
-
-record integration_f_algebra (R:real) : Type \def
- { ifa_integration_riesz_space:> integration_riesz_space R;
- ifa_f_algebra_: f_algebra R;
- ifa_with:
- fa_archimedean_riesz_space ? ifa_f_algebra_ =
- irs_archimedean_riesz_space ? ifa_integration_riesz_space
- }.
-
-axiom ifa_f_algebra: ∀R:real.integration_f_algebra R → f_algebra R.
-
-coercion cic:/matita/integration_algebras/ifa_f_algebra.con.
set "baseuri" "cic:/matita/metric_space/".
-include "ordered_groups.ma".
+include "ordered_group.ma".
record metric_space (R : ogroup) : Type ≝ {
ms_carr :> Type;
+++ /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_fields_ch0/".
-
-include "fields.ma".
-include "ordered_groups.ma".
-
-(*CSC: non capisco questi alias! Una volta non servivano*)
-alias id "plus" = "cic:/matita/groups/plus.con".
-alias symbol "plus" = "Abelian group plus".
-
-record pre_ordered_field_ch0: Type ≝
- { of_field:> field;
- of_ordered_abelian_group_: ordered_abelian_group;
- of_cotransitively_ordered_set_: cotransitively_ordered_set;
- of_with1_:
- cos_ordered_set of_cotransitively_ordered_set_ =
- og_ordered_set of_ordered_abelian_group_;
- of_with2:
- og_abelian_group of_ordered_abelian_group_ = r_abelian_group of_field
- }.
-
-lemma of_ordered_abelian_group: pre_ordered_field_ch0 → ordered_abelian_group.
- intro F;
- apply mk_ordered_abelian_group;
- [ apply mk_pre_ordered_abelian_group;
- [ apply (r_abelian_group F)
- | apply (og_ordered_set (of_ordered_abelian_group_ F))
- | apply (eq_f ? ? carrier);
- apply (of_with2 F)
- ]
- |
- apply
- (eq_rect' ? ?
- (λG:abelian_group.λH:og_abelian_group (of_ordered_abelian_group_ F)=G.
- is_ordered_abelian_group
- (mk_pre_ordered_abelian_group G (of_ordered_abelian_group_ F)
- (eq_f abelian_group Type carrier (of_ordered_abelian_group_ F) G
- H)))
- ? ? (of_with2 F));
- simplify;
- apply (og_ordered_abelian_group_properties (of_ordered_abelian_group_ F))
- ]
-qed.
-
-coercion cic:/matita/ordered_fields_ch0/of_ordered_abelian_group.con.
-
-(*CSC: I am not able to prove this since unfold is undone by coercion composition*)
-axiom of_with1:
- ∀G:pre_ordered_field_ch0.
- cos_ordered_set (of_cotransitively_ordered_set_ G) =
- og_ordered_set (of_ordered_abelian_group G).
-
-lemma of_cotransitively_ordered_set : pre_ordered_field_ch0 → cotransitively_ordered_set.
- intro F;
- apply mk_cotransitively_ordered_set;
- [ apply (og_ordered_set F)
- | apply
- (eq_rect ? ? (λa:ordered_set.cotransitive (os_carrier a) (os_le a))
- ? ? (of_with1 F));
- apply cos_cotransitive
- ]
-qed.
-
-coercion cic:/matita/ordered_fields_ch0/of_cotransitively_ordered_set.con.
-
-record is_ordered_field_ch0 (F:pre_ordered_field_ch0) : Type \def
- { of_mult_compat: ∀a,b:F. 0≤a → 0≤b → 0≤a*b;
- of_weak_tricotomy : ∀a,b:F. a≠b → a≤b ∨ b≤a;
- (* 0 characteristics *)
- of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
- }.
-
-record ordered_field_ch0 : Type \def
- { of_pre_ordered_field_ch0:> pre_ordered_field_ch0;
- of_ordered_field_properties:> is_ordered_field_ch0 of_pre_ordered_field_ch0
- }.
-
-(*
-lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
- intros;
-
-lemma not_eq_x_zero_to_lt_zero_mult_x_x:
- ∀F:ordered_field_ch0.∀x:F. x ≠ 0 → 0 < x * x.
- intros;
- elim (of_weak_tricotomy ? ? ? ? ? ? ? ? F ? ? H);
- [ generalize in match (le_x_zero_to_le_zero_opp_x F ? H1); intro;
- generalize in match (of_mult_compat ? ? ? ? ? ? ? ? F ? ? H2 H2); intro;
-*)
-
-axiom lt_zero_to_lt_inv_zero:
- ∀F:ordered_field_ch0.∀x:F.∀p:x≠0. lt F 0 x → lt F 0 (inv ? x p).
-
-alias symbol "lt" = "natural 'less than'".
-
-(* The ordering is not necessary. *)
-axiom not_eq_sum_field_zero: ∀F:ordered_field_ch0.∀n. O<n → sum_field F n ≠ 0.
-axiom le_zero_sum_field: ∀F:ordered_field_ch0.∀n. O<n → lt F 0 (sum_field F n).
-
-axiom lt_zero_to_le_inv_zero:
- ∀F:ordered_field_ch0.∀n:nat.∀p:sum_field F n ≠ 0. 0 ≤ inv ? (sum_field ? n) p.
-
-definition tends_to : ∀F:ordered_field_ch0.∀f:nat→F.∀l:F.Prop.
- apply
- (λF:ordered_field_ch0.λf:nat → F.λl:F.
- ∀n:nat.∃m:nat.∀j:nat.m ≤ j →
- l - (inv F (sum_field ? (S n)) ?) ≤ f j ∧
- f j ≤ l + (inv F (sum_field ? (S n)) ?));
- apply not_eq_sum_field_zero;
- unfold;
- autobatch.
-qed.
-
-(*
-definition is_cauchy_seq ≝
- λF:ordered_field_ch0.λf:nat→F.
- ∀eps:F. 0 < eps →
- ∃n:nat.∀M. n ≤ M →
- -eps ≤ f M - f n ∧ f M - f n ≤ eps.
-*)
-
-
-
-definition is_cauchy_seq : ∀F:ordered_field_ch0.∀f:nat→F.Prop.
- apply
- (λF:ordered_field_ch0.λf:nat→F.
- ∀m:nat.
- ∃n:nat.∀N.n ≤ N →
- -(inv ? (sum_field F (S m)) ?) ≤ f N - f n ∧
- f N - f n ≤ inv ? (sum_field F (S m)) ?);
- apply not_eq_sum_field_zero;
- unfold;
- autobatch.
-qed.
-
-definition is_complete ≝
- λF:ordered_field_ch0.
- ∀f:nat→F. is_cauchy_seq ? f →
- ex F (λl:F. tends_to ? f l).
--- /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_gorup/".
+
+include "ordered_set.ma".
+include "group.ma".
+
+record pre_ogroup : Type ≝ {
+ og_abelian_group_: abelian_group;
+ og_tordered_set:> tordered_set;
+ og_with: carr og_abelian_group_ = og_tordered_set
+}.
+
+lemma og_abelian_group: pre_ogroup → 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;
+[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.
+
+record ogroup : Type ≝ {
+ og_carr:> pre_ogroup;
+ exc_canc_plusr: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
+}.
+
+notation > "'Ex'≪" non associative with precedence 50 for
+ @{'excedencerewritel}.
+
+interpretation "exc_rewl" 'excedencerewritel =
+ (cic:/matita/excedence/exc_rewl.con _ _ _).
+
+notation > "'Ex'≫" non associative with precedence 50 for
+ @{'excedencerewriter}.
+
+interpretation "exc_rewr" 'excedencerewriter =
+ (cic:/matita/excedence/exc_rewr.con _ _ _).
+
+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));
+apply (Ex≪ (x + (z + -z)) (plus_assoc ????));
+apply (Ex≪ (x + (-z + z)) (plus_comm ??z));
+apply (Ex≪ (x+0) (opp_inverse ??));
+apply (Ex≪ (0+x) (plus_comm ???));
+apply (Ex≪ x (zero_neutral ??));
+apply (Ex≫ (y + (z + -z)) (plus_assoc ????));
+apply (Ex≫ (y + (-z + z)) (plus_comm ??z));
+apply (Ex≫ (y+0) (opp_inverse ??));
+apply (Ex≫ (0+y) (plus_comm ???));
+apply (Ex≫ y (zero_neutral ??) L);
+qed.
+
+coercion cic:/matita/ordered_gorup/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);
+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));
+apply (exc_rewl ???? (plus_assoc ??z x));
+apply (exc_rewr ???? (plus_assoc ??z y));
+apply (exc_rewl ??? (0+x) (opp_inverse ??));
+apply (exc_rewr ??? (0+y) (opp_inverse ??));
+apply (exc_rewl ???? (zero_neutral ??));
+apply (exc_rewr ???? (zero_neutral ??) L);
+qed.
+
+coercion cic:/matita/ordered_gorup/fexc_plusl.con nocomposites.
+
+lemma plus_cancr_le:
+ ∀G:ogroup.∀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_rewl ??? (x+(-z+z)) (opp_inverse ??));
+apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
+apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
+apply (le_rewr ??? (0+y) (zero_neutral ??));
+apply (le_rewr ??? (y+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);
+qed.
+
+lemma fle_plusl: ∀G:ogroup. ∀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_rewl ??? (f+(h+ -h)) (plus_assoc ????));
+apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
+apply (le_rewl ??? (f+0) (opp_inverse ??));
+apply (le_rewl ??? (0+f) (plus_comm ???));
+apply (le_rewl ??? (f) (zero_neutral ??));
+apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
+apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
+apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
+apply (le_rewr ??? (g+0) (opp_inverse ??));
+apply (le_rewr ??? (0+g) (plus_comm ???));
+apply (le_rewr ??? (g) (zero_neutral ??) H);
+qed.
+
+lemma plus_cancl_le:
+ ∀G:ogroup.∀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 ??));
+apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
+apply (le_rewr ??? (0+y) (zero_neutral ??));
+apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
+apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
+apply (fle_plusl ??? (-z) L);
+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));
+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.
+intros (G x Px); apply (plus_cancr_le ??? x);
+apply (le_rewl ??? 0 (opp_inverse ??));
+apply (le_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));
+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.
+intros (G x Lx0); apply (plus_cancr_le ??? x);
+apply (le_rewr ??? 0 (opp_inverse ??));
+apply (le_rewl ??? x (zero_neutral ??));
+assumption;
+qed.
+
+lemma lt0plus_orlt:
+ ∀G:ogroup. ∀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 ??));
+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_groups/".
-
-include "ordered_set.ma".
-include "groups.ma".
-
-record pre_ogroup : Type ≝ {
- og_abelian_group_: abelian_group;
- og_tordered_set:> tordered_set;
- og_with: carr og_abelian_group_ = og_tordered_set
-}.
-
-lemma og_abelian_group: pre_ogroup → 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;
-[apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext]
-qed.
-
-coercion cic:/matita/ordered_groups/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
-}.
-
-notation > "'Ex'≪" non associative with precedence 50 for
- @{'excedencerewritel}.
-
-interpretation "exc_rewl" 'excedencerewritel =
- (cic:/matita/excedence/exc_rewl.con _ _ _).
-
-notation > "'Ex'≫" non associative with precedence 50 for
- @{'excedencerewriter}.
-
-interpretation "exc_rewr" 'excedencerewriter =
- (cic:/matita/excedence/exc_rewr.con _ _ _).
-
-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));
-apply (Ex≪ (x + (z + -z)) (plus_assoc ????));
-apply (Ex≪ (x + (-z + z)) (plus_comm ??z));
-apply (Ex≪ (x+0) (opp_inverse ??));
-apply (Ex≪ (0+x) (plus_comm ???));
-apply (Ex≪ x (zero_neutral ??));
-apply (Ex≫ (y + (z + -z)) (plus_assoc ????));
-apply (Ex≫ (y + (-z + z)) (plus_comm ??z));
-apply (Ex≫ (y+0) (opp_inverse ??));
-apply (Ex≫ (0+y) (plus_comm ???));
-apply (Ex≫ y (zero_neutral ??) L);
-qed.
-
-coercion cic:/matita/ordered_groups/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);
-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));
-apply (exc_rewl ???? (plus_assoc ??z x));
-apply (exc_rewr ???? (plus_assoc ??z y));
-apply (exc_rewl ??? (0+x) (opp_inverse ??));
-apply (exc_rewr ??? (0+y) (opp_inverse ??));
-apply (exc_rewl ???? (zero_neutral ??));
-apply (exc_rewr ???? (zero_neutral ??) L);
-qed.
-
-coercion cic:/matita/ordered_groups/fexc_plusl.con nocomposites.
-
-lemma plus_cancr_le:
- ∀G:ogroup.∀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_rewl ??? (x+(-z+z)) (opp_inverse ??));
-apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
-apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
-apply (le_rewr ??? (0+y) (zero_neutral ??));
-apply (le_rewr ??? (y+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);
-qed.
-
-lemma fle_plusl: ∀G:ogroup. ∀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_rewl ??? (f+(h+ -h)) (plus_assoc ????));
-apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
-apply (le_rewl ??? (f+0) (opp_inverse ??));
-apply (le_rewl ??? (0+f) (plus_comm ???));
-apply (le_rewl ??? (f) (zero_neutral ??));
-apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
-apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
-apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
-apply (le_rewr ??? (g+0) (opp_inverse ??));
-apply (le_rewr ??? (0+g) (plus_comm ???));
-apply (le_rewr ??? (g) (zero_neutral ??) H);
-qed.
-
-lemma plus_cancl_le:
- ∀G:ogroup.∀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 ??));
-apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
-apply (le_rewr ??? (0+y) (zero_neutral ??));
-apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
-apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
-apply (fle_plusl ??? (-z) L);
-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));
-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.
-intros (G x Px); apply (plus_cancr_le ??? x);
-apply (le_rewl ??? 0 (opp_inverse ??));
-apply (le_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));
-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.
-intros (G x Lx0); apply (plus_cancr_le ??? x);
-apply (le_rewr ??? 0 (opp_inverse ??));
-apply (le_rewl ??? x (zero_neutral ??));
-assumption;
-qed.
-
-lemma lt0plus_orlt:
- ∀G:ogroup. ∀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 ??));
-assumption;
-qed.
set "baseuri" "cic:/matita/preweighted_lattice/".
-include "ordered_groups.ma".
+include "ordered_group.ma".
record wlattice (R : ogroup) : Type ≝ {
wl_carr:> Type;
+++ /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/reals/".
-
-include "ordered_fields_ch0.ma".
-
-record is_real (F:ordered_field_ch0) : Type
-≝
- { r_archimedean: ∀x:F. ∃n:nat. x ≤ (sum_field ? n);
- r_complete: is_complete F
- }.
-
-record real: Type \def
- { r_ordered_field_ch0:> ordered_field_ch0;
- r_real_properties: is_real r_ordered_field_ch0
- }.
-
-definition lim: ∀R:real.∀f:nat→R.is_cauchy_seq ? f → R.
- intros;
- elim (r_complete ? (r_real_properties R) ? H);
- exact a.
-qed.
-
-definition max_seq: ∀R:real.∀x,y:R. nat → R.
- intros (R x y);
- elim (cos_cotransitive R 0 (inv ? (sum_field ? (S n)) ?) (x-y));
- [ apply x
- | apply not_eq_sum_field_zero ;
- unfold;
- autobatch
- | apply y
- | apply lt_zero_to_le_inv_zero
- ].
-qed.
-
-axiom daemon: False.
-
-theorem cauchy_max_seq: ∀R:real.∀x,y:R. is_cauchy_seq ? (max_seq ? x y).
-elim daemon.
-(*
- intros;
- unfold;
- intros;
- exists; [ exact m | ]; (* apply (ex_intro ? ? m); *)
- intros;
- unfold max_seq;
- elim (of_cotransitive R 0
-(inv R (sum_field R (S N))
- (not_eq_sum_field_zero R (S N) (le_S_S O N (le_O_n N)))) (x-y)
-(lt_zero_to_le_inv_zero R (S N)
- (not_eq_sum_field_zero R (S N) (le_S_S O N (le_O_n N)))));
- [ simplify;
- elim (of_cotransitive R 0
-(inv R (1+sum R (plus R) 0 1 m)
- (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))) (x-y)
-(lt_zero_to_le_inv_zero R (S m)
- (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))));
- [ simplify;
- rewrite > (plus_comm ? x (-x));
- rewrite > opp_inverse;
- split;
- [ apply (le_zero_x_to_le_opp_x_zero R ?);
- apply lt_zero_to_le_inv_zero
- | apply lt_zero_to_le_inv_zero
- ]
- | simplify;
- split;
- [ apply (or_transitive ? ? R ? 0);
- [ apply (le_zero_x_to_le_opp_x_zero R ?)
- | assumption
- ]
- | assumption
- ]
- ]
- | simplify;
- elim (of_cotransitive R 0
-(inv R (1+sum R (plus R) 0 1 m)
- (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))) (x-y)
-(lt_zero_to_le_inv_zero R (S m)
- (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))));
- [ simplify;
- split;
- [ elim daemon
- | generalize in match (le_zero_x_to_le_opp_x_zero R ? t1);
- intro;
- unfold minus in H1;
- rewrite > eq_opp_plus_plus_opp_opp in H1;
- rewrite > eq_opp_opp_x_x in H1;
- rewrite > plus_comm in H1;
- apply (or_transitive ? ? R ? 0);
- [ assumption
- | apply lt_zero_to_le_inv_zero
- ]
- ]
- | simplify;
- rewrite > (plus_comm ? y (-y));
- rewrite > opp_inverse;
- split;
- [ elim daemon
- | apply lt_zero_to_le_inv_zero
- ]
- ]
- ].
- elim daemon.*)
-qed.
-
-definition max: ∀R:real.R → R → R.
- intros (R x y);
- apply (lim R (max_seq R x y));
- apply cauchy_max_seq.
-qed.
-
-definition abs \def λR:real.λx:R. max R x (-x).
-
-lemma comparison:
- ∀R:real.∀f,g:nat→R. is_cauchy_seq ? f → is_cauchy_seq ? g →
- (∀n:nat.f n ≤ g n) → lim ? f ? ≤ lim ? g ?.
- [ assumption
- | assumption
- | intros;
- elim daemon
- ].
-qed.
-
-definition to_zero ≝
- λR:real.λn.
- -(inv R (sum_field R (S n))
- (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))).
-
-axiom is_cauchy_seq_to_zero: ∀R:real. is_cauchy_seq ? (to_zero R).
-
-lemma technical1: ∀R:real.lim R (to_zero R) (is_cauchy_seq_to_zero R) = 0.
- intros;
- unfold lim;
- elim daemon.
-qed.
-
-lemma abs_x_ge_O: ∀R:real.∀x:R. 0 ≤ abs ? x.
- intros;
- unfold abs;
- unfold max;
- rewrite < technical1;
- apply comparison;
- intros;
- unfold to_zero;
- unfold max_seq;
- elim
- (cos_cotransitive R 0
-(inv R (sum_field R (S n))
- (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))) (x--x)
-(lt_zero_to_le_inv_zero R (S n)
- (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))));
- [ simplify;
- (* facile *)
- elim daemon
- | simplify;
- (* facile *)
- elim daemon
- ].
-qed.
\ No newline at end of file
+++ /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/rings/".
-
-include "groups.ma".
-
-record is_ring (G:abelian_group) (mult:G→G→G) (one:G) : Prop
-≝
- { (* multiplicative monoid properties *)
- mult_assoc_: associative ? mult;
- one_neutral_left_: left_neutral ? mult one;
- one_neutral_right_: right_neutral ? mult one;
- (* ring properties *)
- mult_plus_distr_left_: distributive_left ? mult (plus G);
- mult_plus_distr_right_: distributive_right ? mult (plus G);
- not_eq_zero_one_: (0 ≠ one)
- }.
-
-record ring : Type \def
- { r_abelian_group:> abelian_group;
- mult: r_abelian_group → r_abelian_group → r_abelian_group;
- one: r_abelian_group;
- r_ring_properties: is_ring r_abelian_group mult one
- }.
-
-theorem mult_assoc: ∀R:ring.associative ? (mult R).
- intros;
- apply (mult_assoc_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_left_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem one_neutral_right: ∀R:ring.right_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_right_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem mult_plus_distr_left: ∀R:ring.distributive_left ? (mult R) (plus R).
- intros;
- apply (mult_plus_distr_left_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem mult_plus_distr_right: ∀R:ring.distributive_right ? (mult R) (plus R).
- intros;
- apply (mult_plus_distr_right_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem not_eq_zero_one: ∀R:ring.0 ≠ one R.
- intros;
- apply (not_eq_zero_one_ ? ? ? (r_ring_properties R)).
-qed.
-
-interpretation "Ring mult" 'times a b =
- (cic:/matita/rings/mult.con _ a b).
-
-notation "1" with precedence 89
-for @{ 'one }.
-
-interpretation "Ring one" 'one =
- (cic:/matita/rings/one.con _).
-
-lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0.
- intros;
- generalize in match (zero_neutral R 0); intro;
- generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H;
- rewrite > mult_plus_distr_right in H1;
- generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
- rewrite < plus_assoc in H;
- rewrite > opp_inverse in H;
- rewrite > zero_neutral in H;
- assumption.
-qed.
-
-lemma eq_mult_x_zero_zero: ∀R:ring.∀x:R.x*0=0.
-intros;
-generalize in match (zero_neutral R 0);
-intro;
-generalize in match (eq_f ? ? (λy.x*y) ? ? H); intro; clear H;
-(*CSC: qua funzionava prima della patch all'unificazione!*)
-rewrite > (mult_plus_distr_left R) in H1;
-generalize in match (eq_f ? ? (λy. (-(x*0)) +y) ? ? H1);intro;
-clear H1;
-rewrite < plus_assoc in H;
-rewrite > opp_inverse in H;
-rewrite > zero_neutral in H;
-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/vector_spaces/".
-
-include "reals.ma".
-
-record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
-≝
- { vs_nilpotent: ∀v. emult 0 v = 0;
- vs_neutral: ∀v. emult 1 v = v;
- vs_distributive: ∀a,b,v. emult (a + b) v = (emult a v) + (emult b v);
- vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v)
- }.
-
-record vector_space (K:field): Type \def
-{ vs_abelian_group :> abelian_group;
- emult: K → vs_abelian_group → vs_abelian_group;
- vs_vector_space_properties :> is_vector_space ? vs_abelian_group emult
-}.
-
-interpretation "Vector space external product" 'times a b =
- (cic:/matita/vector_spaces/emult.con _ _ a b).
-
-record is_semi_norm (R:real) (V: vector_space R) (semi_norm:V→R) : Prop \def
- { sn_positive: ∀x:V. zero R ≤ semi_norm x;
- sn_omogeneous: ∀a:R.∀x:V. semi_norm (a*x) = (abs ? a) * semi_norm x;
- sn_triangle_inequality: ∀x,y:V. semi_norm (x + y) ≤ semi_norm x + semi_norm y
- }.
-
-theorem eq_semi_norm_zero_zero:
- ∀R:real.∀V:vector_space R.∀semi_norm:V→R.
- is_semi_norm ? ? semi_norm →
- semi_norm 0 = 0.
- intros;
- (* facile *)
- elim daemon.
-qed.
-
-record is_norm (R:real) (V:vector_space R) (norm:V → R) : Prop ≝
- { n_semi_norm:> is_semi_norm ? ? norm;
- n_properness: ∀x:V. norm x = 0 → x = 0
- }.
-
-record norm (R:real) (V:vector_space R) : Type ≝
- { n_function:1> V→R;
- n_norm_properties: is_norm ? ? n_function
- }.
-
-record is_semi_distance (R:real) (C:Type) (semi_d: C→C→R) : Prop ≝
- { sd_positive: ∀x,y:C. zero R ≤ semi_d x y;
- sd_properness: ∀x:C. semi_d x x = 0;
- sd_triangle_inequality: ∀x,y,z:C. semi_d x z ≤ semi_d x y + semi_d z y
- }.
-
-record is_distance (R:real) (C:Type) (d:C→C→R) : Prop ≝
- { d_semi_distance:> is_semi_distance ? ? d;
- d_properness: ∀x,y:C. d x y = 0 → x=y
- }.
-
-record distance (R:real) (V:vector_space R) : Type ≝
- { d_function:2> V→V→R;
- d_distance_properties: is_distance ? ? d_function
- }.
-
-definition induced_distance_fun ≝
- λR:real.λV:vector_space R.λnorm:norm ? V.
- λf,g:V.norm (f - g).
-
-theorem induced_distance_is_distance:
- ∀R:real.∀V:vector_space R.∀norm:norm ? V.
- is_distance ? ? (induced_distance_fun ? ? norm).
-elim daemon.(*
- intros;
- apply mk_is_distance;
- [ apply mk_is_semi_distance;
- [ unfold induced_distance_fun;
- intros;
- apply sn_positive;
- apply n_semi_norm;
- apply (n_norm_properties ? ? norm)
- | unfold induced_distance_fun;
- intros;
- unfold minus;
- rewrite < plus_comm;
- rewrite > opp_inverse;
- apply eq_semi_norm_zero_zero;
- apply n_semi_norm;
- apply (n_norm_properties ? ? norm)
- | unfold induced_distance_fun;
- intros;
- (* ??? *)
- elim daemon
- ]
- | unfold induced_distance_fun;
- intros;
- generalize in match (n_properness ? ? norm ? ? H);
- [ intro;
- (* facile *)
- elim daemon
- | apply (n_norm_properties ? ? norm)
- ]
- ].*)
-qed.
-
-definition induced_distance ≝
- λR:real.λV:vector_space R.λnorm:norm ? V.
- mk_distance ? ? (induced_distance_fun ? ? norm)
- (induced_distance_is_distance ? ? norm).
-
-definition tends_to :
- ∀R:real.∀V:vector_space R.∀d:distance ? V.∀f:nat→V.∀l:V.Prop.
-apply
- (λR:real.λV:vector_space R.λd:distance ? V.λf:nat→V.λl:V.
- ∀n:nat.∃m:nat.∀j:nat. m ≤ j →
- d (f j) l ≤ inv R (sum_field ? (S n)) ?);
- apply not_eq_sum_field_zero;
- unfold;
- autobatch.
-qed.
-
-definition is_cauchy_seq : ∀R:real.\forall V:vector_space R.
-\forall d:distance ? V.∀f:nat→V.Prop.
- apply
- (λR:real.λV: vector_space R. \lambda d:distance ? V.
- \lambda f:nat→V.
- ∀m:nat.
- ∃n:nat.∀N. n ≤ N →
- -(inv R (sum_field ? (S m)) ?) ≤ d (f N) (f n) ∧
- d (f N) (f n)≤ inv R (sum_field R (S m)) ?);
- apply not_eq_sum_field_zero;
- unfold;
- autobatch.
-qed.
-
-definition is_complete ≝
- λR:real.λV:vector_space R.
- λd:distance ? V.
- ∀f:nat→V. is_cauchy_seq ? ? d f→
- ex V (λl:V. tends_to ? ? d f l).
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