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new theories in ringoid and ordered_ringoid

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michael.gruninger
michael.gruninger committed Jan 29, 2015
1 parent a6985b9 commit f67e2acc4b682640b66fe8c54440bd6b4ebf7b49
@@ -0,0 +1,30 @@

(cl-text http://colore.oor.net/ordered_ringoid/discrete_ordered_ring.clif

(cl-imports http://colore.oor.net/ringoids/ring.clif)

(cl-imports http://colore.oor.net/orderings/linear_ordering.clif)

(cl-imports http://colore.oor.net/orderings/definitions/ordering_defs.clif)

(forall (x y z)
(if (lt x y)
(lt (sum x z) (sum y z))))

(forall (x y z)
(if (and (lt zero x)
(lt x y))
(lt (prod x z) (prod y z))))

(forall (x y)
(if (lt x y)
(exists (z)
(= y (sum x z)))))

(lt zero one)

(forall (x)
(if (lt zero x)
(leq one x)))

)
@@ -0,0 +1,30 @@

(cl-text http://colore.oor.net/ordered_ringoid/discrete_ordered_semiring.clif

(cl-imports http://colore.oor.net/ringoids/semiring.clif)

(cl-imports http://colore.oor.net/orderings/linear_ordering.clif)

(cl-imports http://colore.oor.net/orderings/definitions/ordering_defs.clif)

(forall (x y z)
(if (lt x y)
(lt (sum x z) (sum y z))))

(forall (x y z)
(if (and (lt zero x)
(lt x y))
(lt (prod x z) (prod y z))))

(forall (x y)
(if (lt x y)
(exists (z)
(= y (sum x z)))))

(lt zero one)

(forall (x)
(if (lt zero x)
(leq one x)))

)
@@ -1,7 +1,7 @@

(cl-text http://colore.oor.net/ordered_ringoid/euclidean_field
(cl-text http://colore.oor.net/ordered_ringoid/euclidean_field.clif

(cl-imports http://colore.oor.net/ordered_ringoid/ordered_field)
(cl-imports http://colore.oor.net/ordered_ringoid/ordered_field.clif)

(forall (x)
(if (leq zero x)
@@ -1,9 +1,9 @@

(cl-text http://colore.oor.net/ordered_ringoid/ordered_field
(cl-text http://colore.oor.net/ordered_ringoid/ordered_field.clif

(cl-imports http://colore.oor.net/ringoids/field)
(cl-imports http://colore.oor.net/ringoids/field.clif)

(cl-imports http://colore.oor.net/orderings/linear_ordering)
(cl-imports http://colore.oor.net/orderings/linear_ordering.clif)

(forall (x y z)
(if (leq x y)
@@ -0,0 +1,9 @@

(cl-text http://colore.oor.net/ringoids/commutative_semiring.clif

(cl-imports http://colore.oor.net/ringoids/semiring.clif)

(forall (x y)
(= (prod x y) (prod y x)))

)
@@ -0,0 +1,9 @@

(cl-text http://colore.oor.net/ringoids/idempotent_semiring.clif

(cl-imports http://colore.oor.net/ringoids/semiring.clif)

(forall (x)
(= x (sum x x)))

)
@@ -0,0 +1,31 @@

(cl-text http://colore.oor.net/ringoids/semiring.clif

(forall (x y z)
(= (sum (sum x y) z) (sum x (sum y z))))

(forall (x)
(and (= x (sum zero x))
(= x (sum x zero))))

(forall (x y)
(= (sum x y) (sum y x)))

(forall (x y z)
(= (prod (prod x y) z) (prod x (prod y z))))

(forall (x)
(and (= x (prod one x))
(= x (prod x one))))

(forall (x)
(and (= zero (prod zero x))
(= zero (prod x zero))))

(forall (x y z)
(= (prod x (sum y z)) (sum (prod x y) (prod x z))))

(forall (x y z)
(= (prod (sum x y) z) (sum (prod x z) (prod y z))))

)

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