Nuprl Lemma : q-floor-property
∀[r:ℚ]. (([r] ≤ r) ∧ r < [r] + 1)
Proof
Definitions occuring in Statement :
q-floor: [r],
qle: r ≤ s,
qless: r < s,
qadd: r + s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
q-floor: [r],
all: ∀x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
pi1: fst(t),
cand: A c∧ B,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
true: True,
sq_stable: SqStable(P),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rat-int-part_wf2,
set_wf,
rationals_wf,
qle_wf,
qless_wf,
equal_wf,
qadd_wf,
qle_witness,
q-floor_wf,
int-subtype-rationals,
qless_witness,
squash_wf,
qadd_preserves_qle,
qadd_preserves_qless,
qmul_wf,
sq_stable__and,
sq_stable_from_decidable,
decidable__qle,
decidable__qless,
true_wf,
qadd_comm_q,
qadd_ac_1_q,
qinverse_q,
qadd_inv_assoc_q,
iff_weakening_equal,
mon_ident_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
productEquality,
intEquality,
setEquality,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
spreadEquality,
productElimination,
independent_pairEquality,
setElimination,
rename,
dependent_set_memberEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
isect_memberEquality,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
independent_pairFormation,
minusEquality,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality
Latex:
\mforall{}[r:\mBbbQ{}]. (([r] \mleq{} r) \mwedge{} r < [r] + 1)
Date html generated:
2018_05_22-AM-00_28_01
Last ObjectModification:
2017_07_26-PM-06_56_58
Theory : rationals
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