Nuprl Lemma : isqrtn_wf
∀x:ℕ. (isqrtn(x) ∈ {r:ℕ| ((r * r) ≤ x) ∧ x < (r + 1) * (r + 1)} )
Proof
Definitions occuring in Statement :
isqrtn: isqrtn(x),
nat: ℕ,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
isqrtn: isqrtn(x),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
nat: ℕ,
so_apply: x[s],
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
integer-sqrt-newton-ext,
subtype_rel_self,
nat_wf,
sq_exists_wf,
le_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
lambdaEquality,
productEquality,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
addEquality,
natural_numberEquality,
setEquality
Latex:
\mforall{}x:\mBbbN{}. (isqrtn(x) \mmember{} \{r:\mBbbN{}| ((r * r) \mleq{} x) \mwedge{} x < (r + 1) * (r + 1)\} )
Date html generated:
2019_06_20-PM-02_36_29
Last ObjectModification:
2019_06_12-PM-00_25_29
Theory : num_thy_1
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