Nuprl Lemma : decidable__qless
∀a,b:ℚ. Dec(a < b)
Proof
Definitions occuring in Statement :
qless: r < s,
rationals: ℚ,
decidable: Dec(P),
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
qless: r < s,
grp_lt: a < b,
oset_of_ocmon: g↓oset,
set_lt: a <p b,
dset_of_mon: g↓set,
set_blt: a <b b,
set_le: ≤b,
pi2: snd(t),
qadd_grp: <ℚ+>,
grp_le: ≤b,
pi1: fst(t),
infix_ap: x f y,
q_le: q_le(r;s),
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False,
not: ¬A,
qsub: r - s,
bor: p ∨bq,
band: p ∧b q
Lemmas referenced :
valueall-type-has-valueall,
rationals_wf,
rationals-valueall-type,
evalall-reduce,
subtype_base_sq,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
qsub_wf,
qminus-qsub,
iff_weakening_equal,
band_wf,
bor_wf,
qpositive_wf,
qeq_wf2,
bnot_wf,
eqtt_to_assert,
assert-qeq,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
iff_imp_equal_bool,
btrue_wf,
qmul_wf,
bfalse_wf,
qless-int,
or_wf,
qless_wf,
not_wf,
false_wf,
iff_transitivity,
assert_wf,
iff_weakening_uiff,
assert_of_band,
assert_of_bor,
assert-qpositive,
int-subtype-rationals,
assert_of_bnot,
iff_wf,
qadd_wf,
qless_irreflexivity,
qmul_over_plus_qrng,
qinv_inv_q,
qadd_comm_q,
qinverse_q,
qadd_preserves_qless,
qadd_inv_assoc_q,
mon_ident_q,
qadd_ac_1_q,
qless_transitivity,
member-decide-assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
sqequalRule,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
hypothesisEquality,
callbyvalueReduce,
because_Cache,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
hyp_replacement,
applyLambdaEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
voidElimination,
minusEquality,
independent_pairFormation,
productEquality,
inlFormation,
addLevel,
impliesFunctionality,
orFunctionality
Latex:
\mforall{}a,b:\mBbbQ{}. Dec(a < b)
Date html generated:
2018_05_21-PM-11_53_30
Last ObjectModification:
2017_07_26-PM-06_45_39
Theory : rationals
Home
Index