Nuprl Lemma : int-sq-root
∀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,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
implies: P ⇒ Q,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
sq_type: SQType(T),
false: False,
sq_exists: ∃x:A [B[x]],
le: A ≤ B,
cand: A c∧ B,
int_nzero: ℤ-o,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
div_nat_induction-ext,
less_than_wf,
sq_exists_wf,
nat_wf,
le_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nat_plus_wf,
false_wf,
div_rem_sum,
nequal_wf,
rem_bounds_1,
nat_plus_subtype_nat,
equal_wf,
set-value-type,
int-value-type,
decidable__lt,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermMultiply_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_formula_prop_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,
lambdaEquality,
productEquality,
multiplyEquality,
setElimination,
rename,
because_Cache,
addEquality,
independent_functionElimination,
lambdaFormation,
divideEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
dependent_set_memberFormation,
productElimination,
applyEquality,
cutEval,
unionElimination,
imageElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
promote_hyp
Latex:
\mforall{}x:\mBbbN{}. (\mexists{}r:\mBbbN{} [(((r * r) \mleq{} x) \mwedge{} x < (r + 1) * (r + 1))])
Date html generated:
2018_05_21-PM-07_49_44
Last ObjectModification:
2017_11_20-PM-00_43_58
Theory : general
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