Nuprl Lemma : integer-sqrt-ext
∀x:ℕ. (∃r:ℕ [(((r * r) ≤ x) ∧ x < (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_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],
stable__from_decidable,
sq_stable__from_stable,
iff_preserves_decidability,
any: any x,
sq_stable_from_decidable,
squash_elim,
decidable_functionality,
decidable__int_equal,
decidable__less_than',
decidable__and,
decidable__squash,
decidable__equal_int,
decidable__lt,
rem_bounds_1,
div_nat_induction,
integer-nth-root,
integer-sqrt,
genrec-ap: genrec-ap,
natrec: natrec,
so_apply: x[s1;s2],
genrec: genrec,
efficient-exp-ext,
fastexp: i^n,
divide: n ÷ m,
subtract: n - m,
primtailrec: primtailrec(n;i;b;f),
primrec: primrec(n;b;c),
exp: i^n,
member: t ∈ T
Lemmas referenced :
lifting-strict-less,
lifting-strict-decide,
cbv_sqequal,
is-exception_wf,
istype-base,
has-value_wf_base,
lifting-strict-spread,
strict4-decide,
istype-void,
lifting-strict-int_eq,
integer-sqrt,
stable__from_decidable,
sq_stable__from_stable,
iff_preserves_decidability,
sq_stable_from_decidable,
squash_elim,
decidable_functionality,
decidable__int_equal,
decidable__less_than',
decidable__and,
decidable__squash,
decidable__equal_int,
decidable__lt,
rem_bounds_1,
div_nat_induction,
integer-nth-root,
efficient-exp-ext
Rules used in proof :
because_Cache,
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{}x:\mBbbN{}. (\mexists{}r:\mBbbN{} [(((r * r) \mleq{} x) \mwedge{} x < (r + 1) * (r + 1))])
Date html generated:
2019_06_20-PM-02_36_35
Last ObjectModification:
2019_06_12-PM-00_59_46
Theory : num_thy_1
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