Nuprl Lemma : integer-sqrt-xover
∀x:ℕ. (∃r:ℕ [(((r * r) ≤ x) ∧ x < (r + 1) * (r + 1))])
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_upper: {i...},
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
prop: ℙ,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
btrue: tt,
sq_type: SQType(T),
guard: {T},
lt_int: i <z j,
lelt: i ≤ j < k,
less_than: a < b,
sq_exists: ∃x:A [B[x]],
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
nat_wf,
exact-xover_wf,
lt_int_wf,
int_upper_wf,
iff_imp_equal_bool,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
less_than_wf,
false_wf,
assert_of_lt_int,
assert_wf,
iff_wf,
exists_wf,
all_wf,
int_seg_wf,
equal-wf-T-base,
bool_wf,
subtract_wf,
set_wf,
primrec-wf2,
le_wf,
btrue_wf,
mul_bounds_1b,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
true_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
bfalse_wf,
int_seg_subtype,
int_seg_cases,
eqff_to_assert,
eqtt_to_assert,
bnot_wf,
not_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
int_upper_properties,
itermMultiply_wf,
intformnot_wf,
itermSubtract_wf,
int_term_value_mul_lemma,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
mul_bounds_1a,
decidable__le,
itermAdd_wf,
int_term_value_add_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
lelt_wf,
mul-distributes,
mul-distributes-right,
add-associates,
mul-commutes,
one-mul,
add-swap,
two-mul,
sq_stable__and,
sq_stable__le,
sq_stable__equal,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
cut,
extract_by_obid,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
lambdaEquality,
setElimination,
rename,
because_Cache,
multiplyEquality,
independent_isectElimination,
independent_pairFormation,
hypothesisEquality,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addLevel,
impliesFunctionality,
applyEquality,
productEquality,
baseClosed,
dependent_set_memberEquality,
unionElimination,
independent_functionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
hypothesis_subsumption,
addEquality,
allFunctionality,
levelHypothesis,
promote_hyp,
andLevelFunctionality,
allLevelFunctionality,
existsFunctionality,
setEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}x:\mBbbN{}. (\mexists{}r:\mBbbN{} [(((r * r) \mleq{} x) \mwedge{} x < (r + 1) * (r + 1))])
Date html generated:
2019_06_20-PM-02_35_38
Last ObjectModification:
2019_06_12-PM-00_24_52
Theory : num_thy_1
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