Nuprl Lemma : order_split
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Order(T;x,y.R[x;y])
⇒ (∀x,y:T. Dec(x = y ∈ T))
⇒ (∀a,b:T. (R[a;b] ⇐⇒ strict_part(x,y.R[x;y];a;b) ∨ (a = b ∈ T))))
Proof
Definitions occuring in Statement :
order: Order(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,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
strict_part: strict_part(x,y.R[x; y];a;b),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
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]),
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B,
not: ¬A,
false: False,
guard: {T}
Lemmas referenced :
or_wf,
subtype_rel_self,
not_wf,
equal_wf,
all_wf,
decidable_wf,
order_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
applyEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
productEquality,
hypothesis,
instantiate,
universeEquality,
lambdaEquality,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
dependent_functionElimination,
unionElimination,
inrFormation,
inlFormation,
independent_functionElimination,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Order(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}a,b:T. (R[a;b] \mLeftarrow{}{}\mRightarrow{} strict\_part(x,y.R[x;y];a;b) \mvee{} (a = b))))
Date html generated:
2019_06_20-PM-00_29_53
Last ObjectModification:
2018_09_26-PM-00_04_55
Theory : rel_1
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