Nuprl Lemma : integer-sqrt-newton
∀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],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
top: Top,
subtract: n - m,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
two-mul,
add-swap,
one-mul,
mul-commutes,
mul-distributes-right,
mul-distributes,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermMultiply_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
square_non_neg,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
zero-add,
add-zero,
add-associates,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
decidable__lt,
isqrt_newton_wf,
less_than_wf,
and_wf,
le_wf,
false_wf,
nat_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
hypothesis,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
introduction,
sqequalRule,
independent_pairFormation,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
productElimination,
voidElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
minusEquality,
dependent_pairFormation,
int_eqEquality,
computeAll
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_36_14
Last ObjectModification:
2019_06_12-PM-00_25_16
Theory : num_thy_1
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