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This theory file is about properties of loops (the algebraic structures introduced in IsarMathLib in the Loop_ZF theory) with an additional order relation that is in a way compatible with the loop's binary operation. The oldest reference I have found on the subject is \cite{Zelinski1948}.
Definition and notation
An ordered loop \((G,A)\) is a loop with a partial order relation r that is "translation invariant" with respect to the loop operation \(A\).
A triple \((G,A,r)\) is an ordered loop if \((G,A)\) is a loop and \(r\) is a relation on \(G\) (i.e. a subset of \(G\times G\) with is a partial order and for all elements \(x,y,z \in G\) the condition \(\langle x,y\rangle \in r\) is equivalent to both \(\langle A\langle x,z\rangle, A\langle x,z\rangle\rangle \in r\) and \(\langle A\langle z,x\rangle, A\langle z,x\rangle\rangle \in r\). This looks a bit awkward in the basic set theory notation, but using the additive notation for the group operation and \(x\leq y\) to instead of \(\langle x,y \rangle \in r\) this just means that \(x\leq y\) if and only if \(x+z\leq y+z\) and \(x\leq y\) if and only if \(z+x\leq z+y\).
definition
\( \text{IsAnOrdLoop}(L,A,r) \equiv \) \( \text{IsAloop}(L,A) \wedge r\subseteq L\times L \wedge \text{IsPartOrder}(L,r) \wedge (\forall x\in L.\ \forall y\in L.\ \forall z\in L.\ \) \( ((\langle x,y\rangle \in r \longleftrightarrow \langle A\langle x,z\rangle ,A\langle y,z\rangle \rangle \in r) \wedge (\langle x,y\rangle \in r \longleftrightarrow \langle A\langle z,x\rangle ,A\langle z,y\rangle \rangle \in r ))) \)
We define the set of nonnegative elements in the obvious way as \(L^+ =\{x\in L: 0 \leq x\}\).
definition
\( \text{Nonnegative}(L,A,r) \equiv \{x\in L.\ \langle \text{ TheNeutralElement}(L,A),x\rangle \in r\} \)
The \( \text{PositiveSet}(L,A,r) \) is a set similar to \( \text{Nonnegative}(L,A,r) \), but without the neutral element.
definition
\( \text{PositiveSet}(L,A,r) \equiv \) \( \{x\in L.\ \langle \text{ TheNeutralElement}(L,A),x\rangle \in r \wedge \text{ TheNeutralElement}(L,A)\neq x\} \)
We will use the additive notation for ordered loops.
locale loop1
assumes ordLoopAssum: \( \text{IsAnOrdLoop}(L,A,r) \)
defines \( 0 \equiv \text{ TheNeutralElement}(L,A) \)
defines \( x + y \equiv A\langle x,y\rangle \)
defines \( x \leq y \equiv \langle x,y\rangle \in r \)
defines \( x \lt y \equiv x\leq y \wedge x\neq y \)
defines \( L^+ \equiv \text{Nonnegative}(L,A,r) \)
defines \( L_+ \equiv \text{PositiveSet}(L,A,r) \)
defines \( - x + y \equiv \text{ LeftDiv}(L,A)\langle x,y\rangle \)
defines \( x - y \equiv \text{ RightDiv}(L,A)\langle y,x\rangle \)
Theorems proven in the loop0 locale are valid in the loop1 locale
sublocale loop1 < loop0
using ordLoopAssum, loop_loop0_valid unfolding IsAnOrdLoop_def
In this context \(x \leq y\) implies that both \(x\) and \(y\) belong to \(L\).
lemma (in loop1) lsq_members:
assumes \( x\leq y \)
shows \( x\in L \) and \( y\in L \) using ordLoopAssum, assms, IsAnOrdLoop_defIn this context \(x < y\) implies that both \(x\) and \(y\) belong to \(L\).
lemma (in loop1) less_members:
assumes \( x \lt y \)
shows \( x\in L \) and \( y\in L \) using ordLoopAssum, assms, IsAnOrdLoop_defIn an ordered loop the order is translation invariant.
lemma (in loop1) ord_trans_inv:
assumes \( x\leq y \), \( z\in L \)
shows \( x + z \leq y + z \) and \( z + x \leq z + y \)proof from ordLoopAssum, assms have \( (\langle x,y\rangle \in r \longleftrightarrow \langle A\langle x,z\rangle ,A\langle y,z\rangle \rangle \in r) \wedge (\langle x,y\rangle \in r \longleftrightarrow \langle A\langle z,x\rangle ,A\langle z,y\rangle \rangle \in r ) \) using lsq_members unfolding IsAnOrdLoop_def
with assms(1) show \( x + z \leq y + z \) and \( z + x \leq z + y \)
qed
In an ordered loop the strict order is translation invariant.
lemma (in loop1) strict_ord_trans_inv:
assumes \( x \lt y \), \( z\in L \)
shows \( x + z \lt y + z \) and \( z + x \lt z + y \)proof from assms have \( x + z \leq y + z \) and \( z + x \leq z + y \) using ord_trans_inv
moreover
qed
have \( x + z \neq y + z \) and \( z + x \neq z + y \)proof
ultimately show \( x + z \lt y + z \) and \( z + x \lt z + y \)
{
}
}
assume \( x + z = y + z \)
with assms have \( x=y \) using less_members, qg_cancel_right
with assms(1) have \( False \)
thus \( x + z \neq y + z \)
{
assume \( z + x = z + y \)
with assms have \( x=y \) using less_members, qg_cancel_left
with assms(1) have \( False \)
thus \( z + x \neq z + y \)
qed
We can cancel an element from both sides of an inequality on the right side.
lemma (in loop1) ineq_cancel_right:
assumes \( x\in L \), \( y\in L \), \( z\in L \) and \( x + z \leq y + z \)
shows \( x\leq y \)proof from ordLoopAssum, assms(1,2,3) have \( \langle x,y\rangle \in r \longleftrightarrow \langle A\langle x,z\rangle ,A\langle y,z\rangle \rangle \in r \) unfolding IsAnOrdLoop_def
with assms(4) show \( x\leq y \)
qed
We can cancel an element from both sides of a strict inequality on the right side.
lemma (in loop1) strict_ineq_cancel_right:
assumes \( x\in L \), \( y\in L \), \( z\in L \) and \( x + z \lt y + z \)
shows \( x \lt y \) using assms, ineq_cancel_rightWe can cancel an element from both sides of an inequality on the left side.
lemma (in loop1) ineq_cancel_left:
assumes \( x\in L \), \( y\in L \), \( z\in L \) and \( z + x \leq z + y \)
shows \( x\leq y \)proof from ordLoopAssum, assms(1,2,3) have \( \langle x,y\rangle \in r \longleftrightarrow \langle A\langle z,x\rangle ,A\langle z,y\rangle \rangle \in r \) unfolding IsAnOrdLoop_def
with assms(4) show \( x\leq y \)
qed
We can cancel an element from both sides of a strict inequality on the left side.
lemma (in loop1) strict_ineq_cancel_left:
assumes \( x\in L \), \( y\in L \), \( z\in L \) and \( z + x \lt z + y \)
shows \( x \lt y \) using assms, ineq_cancel_leftThe definition of the nonnegative set in the notation used in the loop1 locale:
lemma (in loop1) nonneg_definition:
shows \( x \in L^+ \longleftrightarrow 0 \leq x \) using ordLoopAssum, IsAnOrdLoop_def, Nonnegative_defThe nonnegative set is contained in the loop.
lemma (in loop1) nonneg_subset:
shows \( L^+ \subseteq L \) using Nonnegative_defThe positive set is contained in the loop.
lemma (in loop1) positive_subset:
shows \( L_+ \subseteq L \) using PositiveSet_defThe definition of the positive set in the notation used in the loop1 locale:
lemma (in loop1) posset_definition:
shows \( x \in L_+ \longleftrightarrow ( 0 \leq x \wedge x\neq 0 ) \) using ordLoopAssum, IsAnOrdLoop_def, PositiveSet_defAnother form of the definition of the positive set in the notation used in the loop1 locale:
lemma (in loop1) posset_definition1:
shows \( x \in L_+ \longleftrightarrow 0 \lt x \) using ordLoopAssum, IsAnOrdLoop_def, PositiveSet_defThe order in an ordered loop is antisymmeric.
lemma (in loop1) loop_ord_antisym:
assumes \( x\leq y \) and \( y\leq x \)
shows \( x=y \)proof from ordLoopAssum, assms have \( \text{antisym}(r) \), \( \langle x,y\rangle \in r \), \( \langle y,x\rangle \in r \) unfolding IsAnOrdLoop_def, IsPartOrder_def
then show \( x=y \) by (rule Fol1_L4 )
qed
The loop order is transitive.
lemma (in loop1) loop_ord_trans:
assumes \( x\leq y \) and \( y\leq z \)
shows \( x\leq z \)proof from ordLoopAssum, assms have \( \text{trans}(r) \) and \( \langle x,y\rangle \in r \wedge \langle y,z\rangle \in r \) unfolding IsAnOrdLoop_def, IsPartOrder_def
then have \( \langle x,z\rangle \in r \) by (rule Fol1_L3 )
thus \( thesis \)
qed
The loop order is reflexive.
lemma (in loop1) loop_ord_refl:
assumes \( x\in L \)
shows \( x\leq x \) using assms, ordLoopAssum unfolding IsAnOrdLoop_def, IsPartOrder_def, refl_defA form of mixed transitivity for the strict order:
lemma (in loop1) loop_strict_ord_trans:
assumes \( x\leq y \) and \( y \lt z \)
shows \( x \lt z \)proof from assms have \( x\leq y \) and \( y\leq z \)
then have \( x\leq z \) by (rule loop_ord_trans )
with assms show \( x \lt z \) using loop_ord_antisym
qed
Another form of mixed transitivity for the strict order:
lemma (in loop1) loop_strict_ord_trans1:
assumes \( x \lt y \) and \( y\leq z \)
shows \( x \lt z \)proof from assms have \( x\leq y \) and \( y\leq z \)
then have \( x\leq z \) by (rule loop_ord_trans )
with assms show \( x \lt z \) using loop_ord_antisym
qed
Yet another form of mixed transitivity for the strict order:
lemma (in loop1) loop_strict_ord_trans2:
assumes \( x \lt y \) and \( y \lt z \)
shows \( x \lt z \)proof from assms have \( x\leq y \) and \( y\leq z \)
then have \( x\leq z \) by (rule loop_ord_trans )
with assms show \( x \lt z \) using loop_ord_antisym
qed
We can move an element to the other side of an inequality. Well, not exactly, but our notation creates an illusion to that effect.
lemma (in loop1) lsq_other_side:
assumes \( x\leq y \)
shows \( 0 \leq - x + y \), \( ( - x + y) \in L^+ \), \( 0 \leq y - x \), \( (y - x) \in L^+ \)proof from assms have \( x\in L \), \( y\in L \), \( 0 \in L \), \( ( - x + y) \in L \), \( (y - x) \in L \) using lsq_members, neut_props_loop(1), lrdiv_props(2,5)
then have \( x = x + 0 \) and \( y = x + ( - x + y) \) using neut_props_loop(2), lrdiv_props(6)
with assms have \( x + 0 \leq x + ( - x + y) \)
with \( x\in L \), \( 0 \in L \), \( ( - x + y) \in L \) show \( 0 \leq - x + y \) using ineq_cancel_left
then show \( ( - x + y) \in L^+ \) using nonneg_definition
from \( x\in L \), \( y\in L \) have \( x = 0 + x \) and \( y = (y - x) + x \) using neut_props_loop(2), lrdiv_props(3)
with assms have \( 0 + x \leq (y - x) + x \)
with \( x\in L \), \( 0 \in L \), \( (y - x) \in L \) show \( 0 \leq y - x \) using ineq_cancel_right
then show \( (y - x) \in L^+ \) using nonneg_definition
qed
We can move an element to the other side of a strict inequality.
lemma (in loop1) ls_other_side:
assumes \( x \lt y \)
shows \( 0 \lt - x + y \), \( ( - x + y) \in L_+ \), \( 0 \lt y - x \), \( (y - x) \in L_+ \)proof from assms have \( x\in L \), \( y\in L \), \( 0 \in L \), \( ( - x + y) \in L \), \( (y - x) \in L \) using lsq_members, neut_props_loop(1), lrdiv_props(2,5)
then have \( x = x + 0 \) and \( y = x + ( - x + y) \) using neut_props_loop(2), lrdiv_props(6)
with assms have \( x + 0 \lt x + ( - x + y) \)
with \( x\in L \), \( 0 \in L \), \( ( - x + y) \in L \) show \( 0 \lt - x + y \) using strict_ineq_cancel_left
then show \( ( - x + y) \in L_+ \) using posset_definition1
from \( x\in L \), \( y\in L \) have \( x = 0 + x \) and \( y = (y - x) + x \) using neut_props_loop(2), lrdiv_props(3)
with assms have \( 0 + x \lt (y - x) + x \)
with \( x\in L \), \( 0 \in L \), \( (y - x) \in L \) show \( 0 \lt y - x \) using strict_ineq_cancel_right
then show \( (y - x) \in L_+ \) using posset_definition1
qed
We can add sides of inequalities.
lemma (in loop1) add_ineq:
assumes \( x\leq y \), \( z\leq t \)
shows \( x + z \leq y + t \)proof from assms have \( x + z \leq y + z \) using lsq_members(1), ord_trans_inv(1)
with assms show \( thesis \) using lsq_members(2), ord_trans_inv(2), loop_ord_trans
qed
We can add sides of strict inequalities. The proof uses a lemma that relies on the antisymmetry of the order relation.
lemma (in loop1) add_ineq_strict:
assumes \( x \lt y \), \( z \lt t \)
shows \( x + z \lt y + t \)proof from assms have \( x + z \lt y + z \) using less_members(1), strict_ord_trans_inv(1)
moreover
qed
from assms have \( y + z \lt y + t \) using less_members(2), strict_ord_trans_inv(2)
ultimately show \( thesis \) by (rule loop_strict_ord_trans2 )
We can add sides of inequalities one of which is strict.
lemma (in loop1) add_ineq_strict1:
assumes \( x\leq y \), \( z \lt t \)
shows \( x + z \lt y + t \) and \( z + x \lt t + y \)proof from assms have \( x + z \leq y + z \) using less_members(1), ord_trans_inv(1)
with assms show \( x + z \lt y + t \) using lsq_members(2), strict_ord_trans_inv(2), loop_strict_ord_trans
from assms have \( z + x \lt t + x \) using lsq_members(1), strict_ord_trans_inv(1)
with assms show \( z + x \lt t + y \) using less_members(2), ord_trans_inv(2), loop_strict_ord_trans1
qed
Subtracting a positive element decreases the value.
lemma (in loop1) subtract_pos:
assumes \( x\in L \), \( 0 \lt y \)
shows \( x - y \lt x \) and \( ( - y + x) \lt x \)proof from assms(2) have \( y\in L \) using less_members(2)
from assms(1) have \( x\leq x \) using ordLoopAssum unfolding IsAnOrdLoop_def, IsPartOrder_def, refl_def
with assms(2) have \( x + 0 \lt x + y \) using add_ineq_strict1(1)
with assms, \( y\in L \) show \( x - y \lt x \) using neut_props_loop(2), lrdiv_props(3), lrdiv_props(2), strict_ineq_cancel_right
from assms(2), \( x\leq x \) have \( 0 + x \lt y + x \) using add_ineq_strict1(2)
with assms, \( y\in L \) show \( ( - y + x) \lt x \) using neut_props_loop(2), lrdiv_props(6), lrdiv_props(5), strict_ineq_cancel_left
qed
end
lemma loop_loop0_valid:
assumes \( \text{IsAloop}(G,A) \)
shows \( loop0(G,A) \)Definition of IsAnOrdLoop:
\( \text{IsAnOrdLoop}(L,A,r) \equiv \)
\( \text{IsAloop}(L,A) \wedge r\subseteq L\times L \wedge \text{IsPartOrder}(L,r) \wedge (\forall x\in L.\ \forall y\in L.\ \forall z\in L.\ \)
\( ((\langle x,y\rangle \in r \longleftrightarrow \langle A\langle x,z\rangle ,A\langle y,z\rangle \rangle \in r) \wedge (\langle x,y\rangle \in r \longleftrightarrow \langle A\langle z,x\rangle ,A\langle z,y\rangle \rangle \in r ))) \)
lemma (in loop1) lsq_members:
assumes \( x\leq y \)
shows \( x\in L \) and \( y\in L \) lemma (in loop1) ord_trans_inv:
assumes \( x\leq y \), \( z\in L \)
shows \( x + z \leq y + z \) and \( z + x \leq z + y \) lemma (in loop1) less_members:
assumes \( x \lt y \)
shows \( x\in L \) and \( y\in L \) lemma (in quasigroup0) qg_cancel_right:
assumes \( x\in G \), \( y\in G \), \( z\in G \) and \( y\cdot x = z\cdot x \)
shows \( y=z \) lemma (in quasigroup0) qg_cancel_left:
assumes \( x\in G \), \( y\in G \), \( z\in G \) and \( x\cdot y = x\cdot z \)
shows \( y=z \) lemma (in group3) ineq_cancel_right:
assumes \( a\in G \), \( b\in G \), \( c\in G \) and \( a\cdot b \leq c\cdot b \)
shows \( a\leq c \) lemma (in loop1) ineq_cancel_left:
assumes \( x\in L \), \( y\in L \), \( z\in L \) and \( z + x \leq z + y \)
shows \( x\leq y \)Definition of Nonnegative:
\( \text{Nonnegative}(L,A,r) \equiv \{x\in L.\ \langle \text{ TheNeutralElement}(L,A),x\rangle \in r\} \)
Definition of PositiveSet:
\( \text{PositiveSet}(L,A,r) \equiv \)
\( \{x\in L.\ \langle \text{ TheNeutralElement}(L,A),x\rangle \in r \wedge \text{ TheNeutralElement}(L,A)\neq x\} \)
Definition of IsPartOrder:
\( \text{IsPartOrder}(X,r) \equiv \text{refl}(X,r) \wedge \text{antisym}(r) \wedge \text{trans}(r) \)
lemma Fol1_L4:
assumes \( \text{antisym}(r) \) and \( \langle a,b\rangle \in r \), \( \langle b,a\rangle \in r \)
shows \( a=b \) lemma Fol1_L3:
assumes \( \text{trans}(r) \) and \( \langle a,b\rangle \in r \wedge \langle b,c\rangle \in r \)
shows \( \langle a,c\rangle \in r \) lemma (in loop1) loop_ord_trans:
assumes \( x\leq y \) and \( y\leq z \)
shows \( x\leq z \) lemma (in loop1) loop_ord_antisym:
assumes \( x\leq y \) and \( y\leq x \)
shows \( x=y \) lemma (in loop0) neut_props_loop: shows \( 1 \in G \) and \( \forall x\in G.\ 1 \cdot x =x \wedge x\cdot 1 = x \)
lemma (in quasigroup0) lrdiv_props:
assumes \( x\in G \), \( y\in G \)
shows \( \exists !z.\ z\in G \wedge z\cdot x = y \), \( y/ x \in G \), \( (y/ x)\cdot x = y \) and \( \exists !z.\ z\in G \wedge x\cdot z = y \), \( x\backslash y \in G \), \( x\cdot (x\backslash y) = y \) lemma (in loop0) neut_props_loop: shows \( 1 \in G \) and \( \forall x\in G.\ 1 \cdot x =x \wedge x\cdot 1 = x \)
lemma (in quasigroup0) lrdiv_props:
assumes \( x\in G \), \( y\in G \)
shows \( \exists !z.\ z\in G \wedge z\cdot x = y \), \( y/ x \in G \), \( (y/ x)\cdot x = y \) and \( \exists !z.\ z\in G \wedge x\cdot z = y \), \( x\backslash y \in G \), \( x\cdot (x\backslash y) = y \) lemma (in loop1) nonneg_definition: shows \( x \in L^+ \longleftrightarrow 0 \leq x \)
lemma (in quasigroup0) lrdiv_props:
assumes \( x\in G \), \( y\in G \)
shows \( \exists !z.\ z\in G \wedge z\cdot x = y \), \( y/ x \in G \), \( (y/ x)\cdot x = y \) and \( \exists !z.\ z\in G \wedge x\cdot z = y \), \( x\backslash y \in G \), \( x\cdot (x\backslash y) = y \) lemma (in loop1) strict_ineq_cancel_left:
assumes \( x\in L \), \( y\in L \), \( z\in L \) and \( z + x \lt z + y \)
shows \( x \lt y \) lemma (in loop1) posset_definition1: shows \( x \in L_+ \longleftrightarrow 0 \lt x \)
lemma (in loop1) strict_ineq_cancel_right:
assumes \( x\in L \), \( y\in L \), \( z\in L \) and \( x + z \lt y + z \)
shows \( x \lt y \) lemma (in loop1) lsq_members:
assumes \( x\leq y \)
shows \( x\in L \) and \( y\in L \) lemma (in loop1) ord_trans_inv:
assumes \( x\leq y \), \( z\in L \)
shows \( x + z \leq y + z \) and \( z + x \leq z + y \) lemma (in loop1) lsq_members:
assumes \( x\leq y \)
shows \( x\in L \) and \( y\in L \) lemma (in loop1) ord_trans_inv:
assumes \( x\leq y \), \( z\in L \)
shows \( x + z \leq y + z \) and \( z + x \leq z + y \) lemma (in loop1) less_members:
assumes \( x \lt y \)
shows \( x\in L \) and \( y\in L \) lemma (in loop1) strict_ord_trans_inv:
assumes \( x \lt y \), \( z\in L \)
shows \( x + z \lt y + z \) and \( z + x \lt z + y \) lemma (in loop1) less_members:
assumes \( x \lt y \)
shows \( x\in L \) and \( y\in L \) lemma (in loop1) strict_ord_trans_inv:
assumes \( x \lt y \), \( z\in L \)
shows \( x + z \lt y + z \) and \( z + x \lt z + y \) lemma (in loop1) loop_strict_ord_trans2:
assumes \( x \lt y \) and \( y \lt z \)
shows \( x \lt z \) lemma (in loop1) loop_strict_ord_trans:
assumes \( x\leq y \) and \( y \lt z \)
shows \( x \lt z \) lemma (in loop1) loop_strict_ord_trans1:
assumes \( x \lt y \) and \( y\leq z \)
shows \( x \lt z \) lemma (in loop1) add_ineq_strict1:
assumes \( x\leq y \), \( z \lt t \)
shows \( x + z \lt y + t \) and \( z + x \lt t + y \) lemma (in quasigroup0) lrdiv_props:
assumes \( x\in G \), \( y\in G \)
shows \( \exists !z.\ z\in G \wedge z\cdot x = y \), \( y/ x \in G \), \( (y/ x)\cdot x = y \) and \( \exists !z.\ z\in G \wedge x\cdot z = y \), \( x\backslash y \in G \), \( x\cdot (x\backslash y) = y \) lemma (in loop1) add_ineq_strict1:
assumes \( x\leq y \), \( z \lt t \)
shows \( x + z \lt y + t \) and \( z + x \lt t + y \) lemma (in quasigroup0) lrdiv_props:
assumes \( x\in G \), \( y\in G \)
shows \( \exists !z.\ z\in G \wedge z\cdot x = y \), \( y/ x \in G \), \( (y/ x)\cdot x = y \) and \( \exists !z.\ z\in G \wedge x\cdot z = y \), \( x\backslash y \in G \), \( x\cdot (x\backslash y) = y \)
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