Nuprl Lemma : real-continuity-ext
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ. real-cont(f;a;b) supposing real-fun(f;a;b) supposing a ≤ b
Proof
Definitions occuring in Statement :
real-cont: real-cont(f;a;b),
real-fun: real-fun(f;a;b),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x]
Definitions unfolded in proof :
false: False,
squash: ↓T,
or: P ∨ Q,
guard: {T},
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
strict4: strict4(F),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
uimplies: b supposing a,
so_apply: x[s],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
stable__from_decidable,
sq_stable__from_stable,
iff_preserves_decidability,
any: any x,
sq_stable_from_decidable,
squash_elim,
decidable_functionality,
decidable__less_than',
decidable__and,
decidable__squash,
rleq-iff-not-rless,
rleq_functionality,
sq_stable__rleq,
cantor-to-interval-onto-common,
cantor-to-int-uniform-continuity,
decidable__lt,
decidable-cantor-to-int-ext,
real-continuity,
real-cont-br: real-cont-br(a; b; f; k; N),
real-cont-ps: real-cont-ps(k;a;b;f;x;N),
let: let,
bfalse: ff,
it: ⋅,
btrue: tt,
spreadn: spread3,
absval: |i|,
subtract: n - m,
ifthenelse: if b then t else f fi ,
bottom: ⊥,
member: t ∈ T
Lemmas referenced :
exception-not-value,
is-exception_wf,
base_wf,
has-value_wf_base,
int-value-type,
value-type-has-value,
lifting-strict-less,
strict4-spread,
lifting-strict-decide,
strict4-decide,
lifting-strict-callbyvalue,
real-continuity,
stable__from_decidable,
sq_stable__from_stable,
iff_preserves_decidability,
sq_stable_from_decidable,
squash_elim,
decidable_functionality,
decidable__less_than',
decidable__and,
decidable__squash,
rleq-iff-not-rless,
rleq_functionality,
sq_stable__rleq,
cantor-to-interval-onto-common,
cantor-to-int-uniform-continuity,
decidable__lt,
decidable-cantor-to-int-ext
Rules used in proof :
independent_functionElimination,
sqleReflexivity,
inlFormation,
exceptionSqequal,
imageElimination,
imageMemberEquality,
intEquality,
inrFormation,
lessExceptionCases,
because_Cache,
productElimination,
hypothesisEquality,
closedConclusion,
baseApply,
callbyvalueLess,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
equalitySymmetry,
equalityTransitivity,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. real-cont(f;a;b) supposing real-fun(f;a;b) supposing a \mleq{} b
Date html generated:
2018_05_22-PM-02_11_11
Last ObjectModification:
2018_05_21-AM-00_26_41
Theory : reals
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