Nuprl Lemma : q-archimedean-ext
∀a:ℚ. ∃n:ℕ. ((-(n) ≤ a) ∧ (a ≤ n))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qmul: r * s,
rationals: ℚ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
experimental: experimental{impliesFunctionality}(possibleextract),
q-archimedean,
better-q-elim,
decidable__le,
rem_bounds_1,
qmul_preserves_qle,
rem_bounds_2,
q-elim,
decidable__and,
decidable__not,
decidable__less_than',
any: any x,
decidable__implies,
decidable__false,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
q-archimedean,
lifting-strict-spread,
istype-void,
strict4-spread,
lifting-strict-decide,
strict4-decide,
lifting-strict-less,
better-q-elim,
decidable__le,
rem_bounds_1,
qmul_preserves_qle,
rem_bounds_2,
q-elim,
decidable__and,
decidable__not,
decidable__less_than',
decidable__implies,
decidable__false
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}a:\mBbbQ{}. \mexists{}n:\mBbbN{}. ((-(n) \mleq{} a) \mwedge{} (a \mleq{} n))
Date html generated:
2019_10_16-PM-00_32_08
Last ObjectModification:
2019_06_26-PM-04_08_18
Theory : rationals
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