Nuprl Lemma : fixedpoint-property-iff
∀X:Type. ∀d:metric(X). (mcompact(X;d) ⇒ (FP(X) ⇐⇒ ∀f:FUN(X ⟶ X). (¬(∀x:X. f x # x))))
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
fixedpoint-property: FP(X),
mfun: FUN(X ⟶ Y),
msep: x # y,
metric: metric(X),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
apply: f a,
universe: Type
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
prop: ℙ,
mfun: FUN(X ⟶ Y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
false: False,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
fixedpoint-property: FP(X),
msep: x # y,
pi1: fst(t),
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
mcompact: mcompact(X;d),
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
rgt: x > y,
rge: x ≥ y,
sq_exists: ∃x:A [B[x]],
rless: x < y,
le: A ≤ B,
true: True,
squash: ↓T,
rneq: x ≠ y,
is-mfun: f:FUN(X;Y),
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
stable: Stable{P}
Latex:
\mforall{}X:Type. \mforall{}d:metric(X). (mcompact(X;d) {}\mRightarrow{} (FP(X) \mLeftarrow{}{}\mRightarrow{} \mforall{}f:FUN(X {}\mrightarrow{} X). (\mneg{}(\mforall{}x:X. f x \# x))))
Date html generated:
2020_05_20-PM-00_02_12
Last ObjectModification:
2020_03_19-PM-05_47_58
Theory : reals
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