Nuprl Lemma : ulinorder_le_neg
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (UniformLinorder(T;x,y.R[x;y]) ⇒ (∀[a,b:T]. uiff(¬R[a;b];strict_part(x,y.R[x;y];b;a))))
Proof
Definitions occuring in Statement :
ulinorder: UniformLinorder(T;x,y.R[x; y]),
strict_part: strict_part(x,y.R[x; y];a;b),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
false: False,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
strict_part: strict_part(x,y.R[x; y];a;b),
urefl: UniformlyRefl(T;x,y.E[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
uorder: UniformOrder(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
ulinorder: UniformLinorder(T;x,y.R[x; y]),
or: P ∨ Q,
all: ∀x:A. B[x]
Lemmas referenced :
not_wf,
strict_part_wf,
ulinorder_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
applyEquality,
functionExtensionality,
cumulativity,
hypothesis,
universeEquality,
rename,
extract_by_obid,
isectElimination,
independent_functionElimination,
because_Cache,
functionEquality,
productElimination,
independent_isectElimination,
unionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(UniformLinorder(T;x,y.R[x;y]) {}\mRightarrow{} (\mforall{}[a,b:T]. uiff(\mneg{}R[a;b];strict\_part(x,y.R[x;y];b;a))))
Date html generated:
2016_10_21-AM-09_43_05
Last ObjectModification:
2016_08_01-PM-09_48_48
Theory : rel_1
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