Nuprl Lemma : square-iff-isqrt
∀x:ℕ. (∃y:ℕ. ((y * y) = x ∈ ℤ) ⇐⇒ (isqrt(x) * isqrt(x)) = x ∈ ℤ)
Proof
Definitions occuring in Statement :
isqrt: isqrt(x),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
isqrt: isqrt(x),
integer-sqrt-ext,
genrec-ap: genrec-ap,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
nat_plus: ℕ+,
le: A ≤ B,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
subtract: n - m
Lemmas referenced :
integer-sqrt-ext,
int_entire,
int_formula_prop_or_lemma,
intformor_wf,
less_than_wf,
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,
false_wf,
multiply-is-int-iff,
mul_preserves_lt,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
le_wf,
decidable__le,
nat_properties,
mul_preserves_le,
decidable__lt,
decidable__equal_int,
isqrt-property,
isqrt_wf,
equal_wf,
nat_wf,
exists_wf,
int_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
instantiate,
lemma_by_obid,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalRule,
lambdaEquality,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
dependent_pairFormation,
applyEquality,
because_Cache,
natural_numberEquality,
unionElimination,
addEquality,
dependent_set_memberEquality,
imageElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
introduction,
baseApply,
closedConclusion,
baseClosed,
minusEquality,
pointwiseFunctionality,
promote_hyp
Latex:
\mforall{}x:\mBbbN{}. (\mexists{}y:\mBbbN{}. ((y * y) = x) \mLeftarrow{}{}\mRightarrow{} (isqrt(x) * isqrt(x)) = x)
Date html generated:
2019_06_20-PM-02_37_05
Last ObjectModification:
2019_06_12-PM-00_25_57
Theory : num_thy_1
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