Nuprl Lemma : no-halt-decider
¬(∃h:partial(ℤ) ⟶ 𝔹. (h 0 = tt ∧ h ⊥ = ff))
Proof
Definitions occuring in Statement :
partial: partial(T),
bottom: ⊥,
bfalse: ff,
btrue: tt,
bool: 𝔹,
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
sq_type: SQType(T),
guard: {T},
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
bottom_diverge,
base-member-partial,
int-value-type,
has-value_wf_base,
not-is-exception-bottom,
fixpoint-induction-bottom,
partial_wf,
int-mono,
ifthenelse_wf,
exists_wf,
bool_wf,
equal-wf-T-base,
inclusion-partial,
subtype_base_sq,
bool_subtype_base,
btrue_neq_bfalse,
equal_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
baseClosed,
isect_memberFormation,
independent_functionElimination,
voidElimination,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
productElimination,
because_Cache,
lambdaEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
natural_numberEquality,
functionEquality,
productEquality,
unionElimination,
equalityElimination,
addLevel,
instantiate,
cumulativity,
dependent_functionElimination,
levelHypothesis
Latex:
\mneg{}(\mexists{}h:partial(\mBbbZ{}) {}\mrightarrow{} \mBbbB{}. (h 0 = tt \mwedge{} h \mbot{} = ff))
Date html generated:
2017_04_14-AM-07_40_55
Last ObjectModification:
2017_02_27-PM-03_12_40
Theory : partial_1
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