Nuprl Lemma : Troelstra-lemma
¬(∀g:ℕ ⟶ ℕ. ∃n:ℕ. ∀f:ℕ ⟶ ℕ. ((f = g ∈ (ℕn ⟶ ℕ)) ⇒ (∀x:ℕ. ((g x) = 0 ∈ ℕ)) ⇒ (∀x:ℕ. ((f x) = 0 ∈ 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,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
less_than: a < b,
true: True,
squash: ↓T,
top: Top,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
false_wf,
le_wf,
nat_wf,
all_wf,
exists_wf,
equal_wf,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat,
subtype_rel_self,
equal-wf-T-base,
less_than_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
int_seg_properties,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_subtype_base,
zero-le-nat,
squash_wf,
true_wf,
iff_weakening_equal,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
productElimination,
functionEquality,
setElimination,
rename,
applyEquality,
because_Cache,
independent_isectElimination,
baseClosed,
lessCases,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
isect_memberFormation,
axiomSqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
independent_functionElimination,
functionExtensionality,
unionElimination,
equalityElimination,
approximateComputation,
dependent_pairFormation,
intEquality,
promote_hyp,
instantiate,
cumulativity,
int_eqEquality,
universeEquality,
addEquality,
applyLambdaEquality
Latex:
\mneg{}(\mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((f = g) {}\mRightarrow{} (\mforall{}x:\mBbbN{}. ((g x) = 0)) {}\mRightarrow{} (\mforall{}x:\mBbbN{}. ((f x) = 0))))
Date html generated:
2019_06_20-PM-02_49_34
Last ObjectModification:
2018_08_20-PM-09_32_57
Theory : fan-theorem
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