Nuprl Lemma : mk-ss_wf
∀[P:Type]. ∀[Sep:{s:P ⟶ P ⟶ ℙ| ∀x:P. (¬(s x x))} ]. ∀[C:∀x,y,z:P. ((Sep x y) ⇒ ((Sep x z) ∨ (Sep y z)))].
(Point=P #=Sep cotrans=C ∈ SeparationSpace)
Proof
Definitions occuring in Statement :
mk-ss: Point=P #=Sep cotrans=C,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
mk-ss: Point=P #=Sep cotrans=C,
separation-space: SeparationSpace,
record+: record+,
record-update: r[x := v],
record: record(x.T[x]),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
record-select: r.x,
top: Top,
eq_atom: x =a y,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
or: P ∨ Q
Lemmas referenced :
eq_atom_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
atom_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
rec_select_update_lemma,
istype-void,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-assert,
istype-atom,
subtype_rel_self,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
dependentIntersection_memberEquality,
because_Cache,
functionExtensionality_alt,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
tokenEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
atomEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype,
sqequalBase,
functionIsType,
functionExtensionality,
axiomEquality,
universeIsType,
setElimination,
rename,
universeEquality,
unionIsType,
isectIsTypeImplies,
setIsType
Latex:
\mforall{}[P:Type]. \mforall{}[Sep:\{s:P {}\mrightarrow{} P {}\mrightarrow{} \mBbbP{}| \mforall{}x:P. (\mneg{}(s x x))\} ]. \mforall{}[C:\mforall{}x,y,z:P.
((Sep x y) {}\mRightarrow{} ((Sep x z) \mvee{} (Sep y z)))].
(Point=P \#=Sep cotrans=C \mmember{} SeparationSpace)
Date html generated:
2019_10_31-AM-07_26_25
Last ObjectModification:
2019_09_19-PM-04_07_38
Theory : constructive!algebra
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