Nuprl Lemma : uconnex_iff_trichot
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀[a,b:T]. Dec(R[a;b]))
⇒ (uconnex(T; x,y.R[x;y])
⇐⇒ {∀[a,b:T]. (strict_part(x,y.R[x;y];a;b) ∨ Symmetrize(x,y.R[x;y];a;b) ∨ strict_part(x,y.R[x;y];b;a))}))
Proof
Definitions occuring in Statement :
uconnex: uconnex(T; x,y.R[x; y]),
strict_part: strict_part(x,y.R[x; y];a;b),
symmetrize: Symmetrize(x,y.R[x; y];a;b),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
strict_part: strict_part(x,y.R[x; y];a;b),
symmetrize: Symmetrize(x,y.R[x; y];a;b),
uconnex: uconnex(T; x,y.R[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
false: False
Lemmas referenced :
uall_wf,
or_wf,
subtype_rel_self,
not_wf,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
independent_pairFormation,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
applyEquality,
hypothesis,
productEquality,
instantiate,
universeEquality,
because_Cache,
Error :functionIsType,
unionElimination,
inrFormation,
inlFormation,
functionExtensionality,
independent_functionElimination,
voidElimination,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}[a,b:T]. Dec(R[a;b]))
{}\mRightarrow{} (uconnex(T; x,y.R[x;y])
\mLeftarrow{}{}\mRightarrow{} \{\mforall{}[a,b:T].
(strict\_part(x,y.R[x;y];a;b)
\mvee{} Symmetrize(x,y.R[x;y];a;b)
\mvee{} strict\_part(x,y.R[x;y];b;a))\}))
Date html generated:
2019_06_20-PM-00_29_25
Last ObjectModification:
2018_09_26-PM-00_01_01
Theory : rel_1
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