Nuprl Lemma : Rice-theorem-for-Type_1
∀F:Type ⟶ 𝔹
((∀X,Y:Type. (X ~ Y ⇒ F X = F Y)) ⇒ (∀X,Y:Type. (F X = F Y ∨ (∃p:ℕ∞ ⟶ 𝔹. ((∀n:ℕ. (¬↑(p n∞))) ∧ (↑(p ∞)))))))
Proof
Definitions occuring in Statement :
nat-inf-infinity: ∞,
nat2inf: n∞,
nat-inf: ℕ∞,
equipollent: A ~ B,
nat: ℕ,
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
guard: {T},
exists: ∃x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
cand: A c∧ B,
not: ¬A,
false: False,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
squash: ↓T,
true: True,
compose: f o g,
assert: ↑b,
sq_type: SQType(T)
Lemmas referenced :
decidable__equal_bool,
exists_wf,
nat-inf_wf,
bool_wf,
all_wf,
nat_wf,
not_wf,
assert_wf,
nat2inf_wf,
nat-inf-infinity_wf,
equal_wf,
equipollent_wf,
nat-inf-attach,
bnot_wf,
assert_of_bnot,
assert_elim,
bfalse_wf,
and_wf,
btrue_neq_bfalse,
assert_functionality_wrt_uiff,
squash_wf,
true_wf,
compose_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
applyEquality,
functionExtensionality,
hypothesisEquality,
universeEquality,
cumulativity,
hypothesis,
unionElimination,
inlFormation,
isectElimination,
functionEquality,
sqequalRule,
lambdaEquality,
productEquality,
inrFormation,
instantiate,
productElimination,
independent_functionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
voidElimination,
independent_pairFormation,
independent_isectElimination,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
impliesFunctionality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
allFunctionality,
promote_hyp,
because_Cache
Latex:
\mforall{}F:Type {}\mrightarrow{} \mBbbB{}
((\mforall{}X,Y:Type. (X \msim{} Y {}\mRightarrow{} F X = F Y))
{}\mRightarrow{} (\mforall{}X,Y:Type. (F X = F Y \mvee{} (\mexists{}p:\mBbbN{}\minfty{} {}\mrightarrow{} \mBbbB{}. ((\mforall{}n:\mBbbN{}. (\mneg{}\muparrow{}(p n\minfty{}))) \mwedge{} (\muparrow{}(p \minfty{})))))))
Date html generated:
2017_10_01-AM-08_29_37
Last ObjectModification:
2017_07_26-PM-04_24_06
Theory : basic
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