Nuprl Lemma : rat-int-bound_wf
∀[q:ℚ]. (rat-int-bound(q) ∈ {n:ℤ| n - 1 < q ∧ (q ≤ n)} )
Proof
Definitions occuring in Statement :
rat-int-bound: rat-int-bound(q),
qle: r ≤ s,
qless: r < s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rat-int-bound: rat-int-bound(q),
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,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
true: True,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qmul: r * s,
lt_int: i <z j,
qeq: qeq(r;s),
eq_int: (i =z j),
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,
int-subtype-rationals,
qeq_wf2,
bool_wf,
eqtt_to_assert,
assert-qeq,
subtract_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
qsub-sub,
qadd_preserves_qless,
qsub_wf,
qmul_wf,
squash_wf,
true_wf,
qmul_one_qrng,
mon_assoc_q,
qadd_ac_1_q,
qadd_comm_q,
qinverse_q,
mon_ident_q,
iff_weakening_equal,
qle_reflexivity,
add-subtract-cancel,
qadd_inv_assoc_q,
qle-iff,
qless-int,
qadd-add,
qadd_preserves_qle,
qle_weakening_lt_qorder
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,
productElimination,
setElimination,
rename,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
baseClosed,
addEquality,
minusEquality,
imageElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[q:\mBbbQ{}]. (rat-int-bound(q) \mmember{} \{n:\mBbbZ{}| n - 1 < q \mwedge{} (q \mleq{} n)\} )
Date html generated:
2018_05_22-AM-00_27_55
Last ObjectModification:
2017_07_26-PM-06_56_53
Theory : rationals
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