Nuprl Lemma : mk-ss_wf
∀[P:Type]. ∀[Sep:{s:P ⟶ P ⟶ ℙ| ∀x:P. (¬(s x x))} ]. ∀[Sym:∀x,y:P. ((Sep x y) ⇒ (Sep y x))]. ∀[C:∀x,y,z:P.
((Sep x y)
⇒ ((Sep x z)
∨ (Sep y z)))].
(Point=P
#=Sep
symm=Sym
cotrans=C ∈ SeparationSpace)
Proof
Definitions occuring in Statement :
mk-ss: mk-ss,
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
prop: ℙ,
not: ¬A,
iff: P ⇐⇒ Q,
bfalse: ff,
eq_atom: x =a y,
top: Top,
record-select: r.x,
guard: {T},
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
record: record(x.T[x]),
record-update: r[x := v],
record+: record+,
separation-space: SeparationSpace,
mk-ss: mk-ss,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
set_wf,
or_wf,
all_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
not_wf,
bnot_wf,
iff_transitivity,
rec_select_update_lemma,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
atom_subtype_base,
assert_wf,
bool_wf,
equal-wf-base,
uiff_transitivity,
eq_atom_wf
Rules used in proof :
universeEquality,
rename,
setElimination,
functionEquality,
lambdaEquality,
axiomEquality,
equalityEquality,
impliesFunctionality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
cumulativity,
instantiate,
independent_isectElimination,
productElimination,
independent_functionElimination,
atomEquality,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
equalityElimination,
unionElimination,
lambdaFormation,
hypothesis,
tokenEquality,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
functionExtensionality,
because_Cache,
dependentIntersection_memberEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[P:Type]. \mforall{}[Sep:\{s:P {}\mrightarrow{} P {}\mrightarrow{} \mBbbP{}| \mforall{}x:P. (\mneg{}(s x x))\} ]. \mforall{}[Sym:\mforall{}x,y:P. ((Sep x y) {}\mRightarrow{} (Sep y x))].
\mforall{}[C:\mforall{}x,y,z:P. ((Sep x y) {}\mRightarrow{} ((Sep x z) \mvee{} (Sep y z)))].
(Point=P
\#=Sep
symm=Sym
cotrans=C \mmember{} SeparationSpace)
Date html generated:
2016_11_08-AM-09_10_55
Last ObjectModification:
2016_11_02-AM-10_51_54
Theory : inner!product!spaces
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