Nuprl Lemma : isqrrrt-program
∀n:ℕ. (∃r:ℕ [(((r * r) ≤ n) ∧ n < (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 :
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
or: P ∨ Q,
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
strict4: strict4(F),
uimplies: b supposing a,
so_apply: x[s],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
any: any x,
decidable__false,
decidable__implies,
decidable__less_than',
decidable__not,
decidable__and,
decidable__le,
sq_stable__le,
sq_stable__and,
div_nat_induction-ext,
isqrrrt,
genrec-ap: genrec-ap,
genrec: genrec,
natrec: natrec,
divide: n ÷ m,
member: t ∈ T
Lemmas referenced :
lifting-strict-less,
strict4-decide,
lifting-strict-decide,
is-exception_wf,
istype-base,
has-value_wf_base,
istype-void,
lifting-strict-spread,
isqrrrt,
decidable__false,
decidable__implies,
decidable__less_than',
decidable__not,
decidable__and,
decidable__le,
sq_stable__le,
sq_stable__and,
div_nat_induction-ext
Rules used in proof :
Error :inlFormation_alt,
exceptionSqequal,
imageElimination,
imageMemberEquality,
Error :inrFormation_alt,
callbyvalueExceptionCases,
hypothesisEquality,
closedConclusion,
baseApply,
Error :universeIsType,
callbyvalueReduce,
callbyvalueCallbyvalue,
Error :lambdaFormation_alt,
independent_pairFormation,
independent_isectElimination,
voidElimination,
Error :isect_memberEquality_alt,
baseClosed,
isectElimination,
equalitySymmetry,
equalityTransitivity,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}n:\mBbbN{}. (\mexists{}r:\mBbbN{} [(((r * r) \mleq{} n) \mwedge{} n < (r + 1) * (r + 1))])
Date html generated:
2019_06_20-PM-02_35_20
Last ObjectModification:
2019_06_12-PM-00_58_36
Theory : num_thy_1
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