Nuprl Lemma : connex_iff_trichot
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀a,b:T. Dec(R[a;b]))
⇒ (Connex(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 :
connex: Connex(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],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
connex: Connex(T;x,y.R[x; y]),
symmetrize: Symmetrize(x,y.R[x; y];a;b),
strict_part: strict_part(x,y.R[x; y];a;b),
guard: {T},
or: P ∨ Q,
false: False,
not: ¬A,
decidable: Dec(P)
Lemmas referenced :
decidable_wf,
not_wf,
subtype_rel_self,
or_wf,
all_wf
Rules used in proof :
Error :inhabitedIsType,
Error :universeIsType,
Error :functionIsType,
because_Cache,
universeEquality,
instantiate,
productEquality,
hypothesis,
applyEquality,
lambdaEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
hypothesisEquality,
independent_pairFormation,
lambdaFormation,
Error :isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
unionElimination,
dependent_functionElimination,
voidElimination,
independent_functionElimination,
functionExtensionality,
inlFormation,
inrFormation,
productElimination,
Error :inlFormation_alt,
Error :inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}a,b:T. Dec(R[a;b]))
{}\mRightarrow{} (Connex(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_23
Last ObjectModification:
2018_10_04-PM-04_36_32
Theory : rel_1
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