Nuprl Lemma : isqrt_newton_wf
∀n,x:ℕ+. isqrt_newton(n;x) ∈ ∃r:ℕ [(((r * r) ≤ n) ∧ n < (r + 1) * (r + 1))] supposing n < (x + 1) * (x + 1)
Proof
Definitions occuring in Statement :
isqrt_newton: isqrt_newton(n;x),
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
member: t ∈ T,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
nat: ℕ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
ge: i ≥ j ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
isqrt_newton: isqrt_newton(n;x),
nequal: a ≠ b ∈ T ,
has-value: (a)↓,
int_nzero: ℤ-o,
true: True,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
subtract_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
nat_plus_wf,
nat_properties,
ge_wf,
int_seg_wf,
int_seg_properties,
decidable__equal_int,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
lelt_wf,
subtype_rel_self,
equal-wf-base,
value-type-has-value,
int-value-type,
mul_preserves_eq,
equal_wf,
nat_wf,
div_rem_sum2,
subtype_rel_sets,
nequal_wf,
rem_bounds_1,
nat_plus_subtype_nat,
div_bounds_1,
true_wf,
mul-distributes,
mul-commutes,
add-commutes,
mul_preserves_le,
minus-one-mul,
mul-swap,
mul_cancel_in_lt,
top_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
square_non_neg,
multiply-is-int-iff,
add-is-int-iff,
false_wf,
mul-distributes-right,
add-associates,
mul-associates,
one-mul,
two-mul,
less_than_functionality,
le_weakening,
multiply_functionality_wrt_le,
mul_preserves_lt
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
extract_by_obid,
isectElimination,
multiplyEquality,
addEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
hypothesisEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
intWeakElimination,
productElimination,
applyEquality,
instantiate,
applyLambdaEquality,
hypothesis_subsumption,
divideEquality,
baseClosed,
callbyvalueReduce,
setEquality,
addLevel,
cumulativity,
imageElimination,
imageMemberEquality,
remainderEquality,
productEquality,
lessCases,
axiomSqEquality,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp,
int_eqReduceFalseSq,
pointwiseFunctionality,
baseApply,
closedConclusion,
minusEquality
Latex:
\mforall{}n,x:\mBbbN{}\msupplus{}.
isqrt\_newton(n;x) \mmember{} \mexists{}r:\mBbbN{} [(((r * r) \mleq{} n) \mwedge{} n < (r + 1) * (r + 1))] supposing n < (x + 1) * (x + 1)
Date html generated:
2019_06_20-PM-02_36_09
Last ObjectModification:
2019_06_12-PM-00_25_12
Theory : num_thy_1
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