Nuprl Lemma : integer-sqrt-bin-search
∀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,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
cand: A c∧ B,
le: A ≤ B,
sq_exists: ∃x:{A| B[x]},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
int_upper: {i...},
int_seg: {i..j-},
nat_plus: ℕ+,
exp: i^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_seg_properties,
int_seg_subtype_nat,
mul_preserves_le,
exp_preserves_le,
not_wf,
all_wf,
assert_of_lt_int,
assert_wf,
lelt_wf,
primrec-unroll,
int_term_value_add_lemma,
itermAdd_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
primrec1_lemma,
iroot-property,
iroot_wf,
int_seg_wf,
lt_int_wf,
decidable__le,
binary-search_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
and_wf,
le_wf,
false_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
less_than_wf,
top_wf,
decidable__lt,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
cut,
lemma_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
unionElimination,
sqequalRule,
because_Cache,
lessCases,
isect_memberFormation,
isectElimination,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_functionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
dependent_set_memberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
computeAll,
applyEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
productEquality,
addLevel,
impliesFunctionality,
levelHypothesis,
andLevelFunctionality,
impliesLevelFunctionality,
setEquality
Latex:
\mforall{}x:\mBbbN{}. (\mexists{}r:\{\mBbbN{}| (((r * r) \mleq{} x) \mwedge{} x < (r + 1) * (r + 1))\})
Date html generated:
2016_05_15-PM-05_15_53
Last ObjectModification:
2016_01_16-AM-11_40_20
Theory : general
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