Nuprl Lemma : isqrt-property
∀[x:ℕ]. (((isqrt(x) * isqrt(x)) ≤ x) ∧ x < (isqrt(x) + 1) * (isqrt(x) + 1))
Proof
Definitions occuring in Statement :
isqrt: isqrt(x),
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
isqrt: isqrt(x),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
less_than'_wf,
isqrt_wf,
nat_wf,
member-less_than,
integer-sqrt-ext,
subtype_rel_self,
sq_exists_wf,
le_wf,
less_than_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermMultiply_wf,
itermVar_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
sq_stable__le,
int_formula_prop_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
multiplyEquality,
applyEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
natural_numberEquality,
independent_isectElimination,
instantiate,
functionEquality,
productEquality,
lambdaFormation,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[x:\mBbbN{}]. (((isqrt(x) * isqrt(x)) \mleq{} x) \mwedge{} x < (isqrt(x) + 1) * (isqrt(x) + 1))
Date html generated:
2019_06_20-PM-02_36_53
Last ObjectModification:
2019_06_12-PM-00_25_48
Theory : num_thy_1
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