record excedence : Type ≝ {
exc_carr:> Type;
- exc_relation: exc_carr → exc_carr → Type; (* Big bug: era in Prop!!! *)
+ exc_relation: exc_carr → exc_carr → Type;
exc_coreflexive: coreflexive ? exc_relation;
exc_cotransitive: cotransitive ? exc_relation
}.
cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)
qed.
-(* CSC: lo avevi gia' dimostrato; ho messo apply! *)
-theorem le_le_to_eq: ∀E:excedence.∀x,y:E. x ≤ y → y ≤ x → x ≈ y.
-apply le_antisymmetric;
-qed.
-
-(* CSC: perche' quel casino: bastava intros; assumption! *)
lemma unfold_apart: ∀E:excedence. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
intros; assumption;
qed.
lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x.
intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
apply ap_symmetric; assumption;
-qed.
\ No newline at end of file
+qed.
+
+lemma exc_rewl: ∀A:excedence.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z.
+intros (A x z y Exy Ayz); elim (exc_cotransitive ???x Ayz); [2:assumption]
+cases Exy; right; assumption;
+qed.
+
+lemma exc_rewr: ∀A:excedence.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x.
+intros (A x z y Exy Azy); elim (exc_cotransitive ???x Azy); [assumption]
+elim (Exy); left; assumption;
+qed.
*)
record ogroup : Type ≝ {
og_carr:> pre_ogroup;
- fle_plusr: ∀f,g,h:og_carr. f≤g → f+h≤g+h
+ exc_canc_plusr: ∀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));
+apply (exc_rewl ??? (x + (z + -z)) (plus_assoc ????));
+apply (exc_rewl ??? (x + (-z + z)) (plus_comm ??z));
+apply (exc_rewl ??? (x+0) (opp_inverse ??));
+apply (exc_rewl ??? (0+x) (plus_comm ???));
+apply (exc_rewl ??? x (zero_neutral ??));
+apply (exc_rewr ??? (y + (z + -z)) (plus_assoc ????));
+apply (exc_rewr ??? (y + (-z + z)) (plus_comm ??z));
+apply (exc_rewr ??? (y+0) (opp_inverse ??));
+apply (exc_rewr ??? (0+y) (plus_comm ???));
+apply (exc_rewr ??? 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_rewr ??? (y+(-z+z)) (opp_inverse ??));
apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
-apply (fle_plusr ??? (-z) L);
+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.
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_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);
--- /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/valued_lattice/".
+
+include "ordered_groups.ma".
+
+record vlattice (R : ogroup) : Type ≝ {
+ vl_carr:> Type;
+ value: vl_carr → R;
+ join: vl_carr → vl_carr → vl_carr;
+ meet: vl_carr → vl_carr → vl_carr;
+ meet_refl: ∀x. value (meet x x) ≈ value x;
+ join_refl: ∀x. value (join x x) ≈ value x;
+ meet_comm: ∀x,y. value (meet x y) ≈ value (meet y x);
+ join_comm: ∀x,y. value (join x y) ≈ value (join y x);
+ join_assoc: ∀x,y,z. value (join x (join y z)) ≈ value (join (join x y) z);
+ meet_assoc: ∀x,y,z. value (meet x (meet y z)) ≈ value (meet (meet x y) z);
+ meet_wins1: ∀x,y. value (join x (meet x y)) ≈ value x;
+ meet_wins2: ∀x,y. value (meet x (join x y)) ≈ value x;
+ meet_join_plus: ∀x,y. value (join x y) + value (meet x y) ≈ value x + value y;
+ join_meet_le: ∀x,y,z. value (join x (meet y z)) ≤ value (join x y);
+ meet_join_le: ∀x,y,z. value (meet x (join y z)) ≤ value (meet x y)
+}.
+
+interpretation "valued lattice meet" 'and a b =
+ (cic:/matita/valued_lattice/meet.con _ _ a b).
+
+interpretation "valued lattice join" 'or a b =
+ (cic:/matita/valued_lattice/join.con _ _ a b).
+
+notation < "\nbsp \mu a" non associative with precedence 80 for @{ 'value2 $a}.
+interpretation "lattice value" 'value2 a = (cic:/matita/valued_lattice/value.con _ _ a).
+
+notation "\mu" non associative with precedence 80 for @{ 'value }.
+interpretation "lattice value" 'value = (cic:/matita/valued_lattice/value.con _ _).
+
+axiom foo: ∀R.∀L:vlattice R.∀x,y,z:L.
+ μ(x ∧ (y ∨ z)) ≈ (μ x) + (μ y) + μ z + -μ (y ∧ z) + -μ (z ∨ (x ∨ y)).
+
+lemma meet_join_le1: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∧ z) ≤ μ (x ∧ (y ∨ z)).
+intros (R L x y z);
+apply (le_rewr ??? (μ x + μ y + μ z + -μ (y ∧ z) + -μ(z ∨ (x ∨ y))) (foo ?????));
+apply (le_rewr ??? (μ x + μ y + μ z + -μ (y ∧ z) + -μ((z ∨ x) ∨ y)));
+ [ apply feq_plusl; apply eq_opp_sym; apply join_assoc;]
+lapply (meet_join_le ?? z x y);
+cut (- μ (z ∨ x ∨ y) ≈ - μ (z ∨ x) - μ y + μ (y ∧ (z ∨ x)));
+ [2:
+
+
+lemma join_meet_le1: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∨ (y ∧ z)) ≤ μ (x ∨ z).
+(* hint per duplicati? *)
+intros (R L x y z);
+apply (le_rewr ??? (0 + μ (x ∨ z)) (zero_neutral ??));
+apply (le_rewr ??? (μ (x ∨ z) + 0) (plus_comm ???));
+apply (le_rewr ??? (μ (x ∨ z) + (-μ(y ∨ z) + μ(y ∨ z))) (opp_inverse ? ?));
+
+
+
projects / helm.git / commitdiff