Nuprl Lemma : integer-sqrt
∀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_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
sq_exists: ∃x:A [B[x]],
exp: i^n,
top: Top,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A
Lemmas referenced :
integer-nth-root,
less_than_wf,
and_wf,
le_wf,
primrec-unroll,
primrec1_lemma,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
isectElimination,
lambdaFormation,
setElimination,
rename,
hyp_replacement,
equalitySymmetry,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
productEquality,
multiplyEquality,
addEquality
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_27
Last ObjectModification:
2019_06_12-PM-00_24_47
Theory : num_thy_1
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