Nuprl Lemma : linorder_lt_neg
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀x,y:T. Dec(R[x;y])) ⇒ Linorder(T;x,y.R[x;y]) ⇒ (∀a,b:T. (¬strict_part(x,y.R[x;y];a;b) ⇐⇒ R[b;a])))
Proof
Definitions occuring in Statement :
linorder: Linorder(T;x,y.R[x; y]),
strict_part: strict_part(x,y.R[x; y];a;b),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
strict_part: strict_part(x,y.R[x; y];a;b),
decidable: Dec(P),
or: P ∨ Q,
linorder: Linorder(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
trans: Trans(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
not_wf,
strict_part_wf,
linorder_wf,
all_wf,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
voidElimination,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
universeEquality,
productElimination,
dependent_functionElimination,
unionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. Dec(R[x;y]))
{}\mRightarrow{} Linorder(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a,b:T. (\mneg{}strict\_part(x,y.R[x;y];a;b) \mLeftarrow{}{}\mRightarrow{} R[b;a])))
Date html generated:
2019_06_20-PM-00_30_02
Last ObjectModification:
2018_09_26-PM-00_04_59
Theory : rel_1
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