Nuprl Lemma : qmin-eq-iff
∀[q,r,s:ℚ]. uiff(qmin(q;r) = s ∈ ℚ;((r ≤ q) ⇒ (s = r ∈ ℚ)) ∧ ((q ≤ r) ⇒ (s = q ∈ ℚ)))
Proof
Definitions occuring in Statement :
qmin: qmin(x;y),
qle: r ≤ s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
guard: {T},
prop: ℙ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
qmin: qmin(x;y),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
trivial-equal,
qle_weakening_lt_qorder,
rationals_wf,
equal_wf,
qless_trichot_qorder,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
qle_wf,
qle_weakening_eq_qorder,
qle_transitivity_qorder,
qle_antisymmetry,
iff_weakening_equal,
assert-q_le-eq,
eqtt_to_assert,
q_le_wf
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
applyEquality,
voidElimination,
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation_alt,
functionIsType,
productIsType,
equalityIstype,
functionIsTypeImplies,
axiomEquality,
dependent_functionElimination,
lambdaEquality_alt,
independent_pairEquality,
universeIsType,
because_Cache,
independent_pairFormation,
sqequalRule,
independent_functionElimination,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
lambdaFormation_alt,
inhabitedIsType,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[q,r,s:\mBbbQ{}]. uiff(qmin(q;r) = s;((r \mleq{} q) {}\mRightarrow{} (s = r)) \mwedge{} ((q \mleq{} r) {}\mRightarrow{} (s = q)))
Date html generated:
2019_10_29-AM-07_44_08
Last ObjectModification:
2019_10_21-PM-06_18_57
Theory : rationals
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