Nuprl Lemma : Escardo-Xu
¬(∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((∀i:ℕk. ((g i) = 0 ∈ ℕ)) ⇒ ((F (λi.0)) = (F g) ∈ ℕ)))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
lelt: i ≤ j < k,
int_seg: {i..j-},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
squash: ↓T,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
uimplies: b supposing a,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
pi1: fst(t),
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A
Lemmas referenced :
decidable__le,
int_formula_prop_le_lemma,
intformle_wf,
ifthenelse_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
bool_cases,
not_wf,
bnot_wf,
assert_wf,
int_seg_subtype_nat,
decidable__lt,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformand_wf,
int_seg_properties,
iff_weakening_equal,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
nat_properties,
true_wf,
squash_wf,
int_subtype_base,
set_subtype_base,
less_than_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
le_wf,
false_wf,
equal_wf,
equal-wf-T-base,
int_seg_wf,
exists_wf,
nat_wf,
all_wf
Rules used in proof :
impliesFunctionality,
int_eqEquality,
baseClosed,
imageMemberEquality,
voidEquality,
isect_memberEquality,
approximateComputation,
applyLambdaEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
imageElimination,
intEquality,
voidElimination,
cumulativity,
instantiate,
independent_isectElimination,
equalityElimination,
unionElimination,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
productElimination,
promote_hyp,
independent_pairFormation,
dependent_set_memberEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
rename,
setElimination,
natural_numberEquality,
because_Cache,
lambdaEquality,
sqequalRule,
hypothesis,
functionEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mneg{}(\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mexists{}k:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mforall{}i:\mBbbN{}k. ((g i) = 0)) {}\mRightarrow{} ((F (\mlambda{}i.0)) = (F g))))
Date html generated:
2017_09_29-PM-06_05_02
Last ObjectModification:
2017_09_22-PM-04_45_41
Theory : fan-theorem
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