Nuprl Lemma : ulinorder_lt_neg
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀x,y:T. Dec(R[x;y])) ⇒ UniformLinorder(T;x,y.R[x;y]) ⇒ (∀[a,b:T]. uiff(¬strict_part(x,y.R[x;y];a;b);R[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),
decidable: Dec(P),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
ulinorder: UniformLinorder(T;x,y.R[x; y]),
and: P ∧ Q,
connex: Connex(T;x,y.R[x; y]),
uorder: UniformOrder(T;x,y.R[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
strict_part: strict_part(x,y.R[x; y];a;b),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
uimplies: b supposing a,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B
Lemmas referenced :
ulinorder_wf,
all_wf,
decidable_wf,
subtype_rel_self,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
Error :functionIsType,
universeEquality,
independent_pairFormation,
dependent_functionElimination,
voidElimination,
productEquality,
instantiate,
rename,
because_Cache,
independent_functionElimination,
independent_isectElimination,
unionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. Dec(R[x;y]))
{}\mRightarrow{} UniformLinorder(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}[a,b:T]. uiff(\mneg{}strict\_part(x,y.R[x;y];a;b);R[b;a])))
Date html generated:
2019_06_20-PM-00_30_05
Last ObjectModification:
2018_09_26-PM-00_06_21
Theory : rel_1
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