Nuprl Lemma : test-recursion-extract
∀k:ℕ. ∀f:ℕ ⟶ ℚ. ((∃n:ℕk. ((f n) = 0 ∈ ℚ)) ∨ True)
Proof
Definitions occuring in Statement :
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
or: P ∨ Q,
exists: ∃x:A. B[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
decidable: Dec(P),
sq_type: SQType(T),
guard: {T},
true: True,
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
nequal: a ≠ b ∈ T ,
int_upper: {i...},
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
int_seg_wf,
rationals_wf,
int_seg_subtype_nat,
istype-false,
true_wf,
istype-nat,
natrec_wf,
all_wf,
nat_wf,
or_wf,
exists_wf,
equal-wf-T-base,
subtype_rel_function,
subtype_rel_self,
function-valueall-type,
rationals-value-type,
evalall-reduce,
set-value-type,
equal_wf,
valueall-type-value-type,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
decidable__equal_rationals,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
int-subtype-rationals,
decidable__lt,
less_than_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
upper_subtype_nat,
nequal-le-implies,
zero-add,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
qexp_wf,
bnot_wf,
not_wf,
bool_cases,
iff_transitivity,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
sqequalRule,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
inhabitedIsType,
unionIsType,
productIsType,
equalityIsType3,
applyEquality,
independent_isectElimination,
independent_pairFormation,
baseClosed,
lambdaEquality_alt,
functionEquality,
because_Cache,
functionExtensionality,
dependent_functionElimination,
imageMemberEquality,
equalityTransitivity,
equalitySymmetry,
cutEval,
dependent_set_memberEquality_alt,
equalityIsType1,
hyp_replacement,
applyLambdaEquality,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_functionElimination,
inrFormation_alt,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
inlFormation_alt,
equalityElimination,
hypothesis_subsumption,
equalityIsType2,
baseApply,
closedConclusion,
promote_hyp
Latex:
\mforall{}k:\mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbQ{}. ((\mexists{}n:\mBbbN{}k. ((f n) = 0)) \mvee{} True)
Date html generated:
2019_10_16-PM-00_34_09
Last ObjectModification:
2018_10_10-AM-11_04_51
Theory : rationals
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