Nuprl Lemma : ulinorder_functionality_wrt_iff
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ].
((∀[x,y:T]. uiff(R[x;y];R'[x;y])) ⇒ (UniformLinorder(T;x,y.R[x;y]) ⇐⇒ UniformLinorder(T;x,y.R'[x;y])))
Proof
Definitions occuring in Statement :
ulinorder: UniformLinorder(T;x,y.R[x; y]),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
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]),
iff: P ⇐⇒ Q,
and: P ∧ Q,
uorder: UniformOrder(T;x,y.R[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
member: t ∈ T,
so_apply: x[s1;s2],
utrans: UniformlyTrans(T;x,y.E[x; y]),
uimplies: b supposing a,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
all: ∀x:A. B[x],
or: P ∨ Q,
prop: ℙ,
guard: {T},
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
uorder_wf,
connex_wf,
uall_wf,
isect_wf,
subtype_rel_self,
uiff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
promote_hyp,
cut,
hypothesis,
isectElimination,
hypothesisEquality,
independent_functionElimination,
independent_isectElimination,
applyEquality,
because_Cache,
sqequalRule,
introduction,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
inlFormation,
inrFormation,
productEquality,
extract_by_obid,
lambdaEquality,
instantiate,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}[x,y:T]. uiff(R[x;y];R'[x;y]))
{}\mRightarrow{} (UniformLinorder(T;x,y.R[x;y]) \mLeftarrow{}{}\mRightarrow{} UniformLinorder(T;x,y.R'[x;y])))
Date html generated:
2019_06_20-PM-00_29_38
Last ObjectModification:
2018_08_25-AM-08_23_09
Theory : rel_1
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