Nuprl Lemma : qmin-eq-iff-cases
∀q,r,s:ℚ. uiff(qmin(q;r) = s ∈ ℚ;((q ≤ r) ∧ (s = q ∈ ℚ)) ∨ ((r ≤ q) ∧ (s = r ∈ ℚ)))
Proof
Definitions occuring in Statement :
qmin: qmin(x;y),
qle: r ≤ s,
rationals: ℚ,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
qmin: qmin(x;y),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
cand: A c∧ B,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
not: ¬A,
rev_implies: P ⇐ Q
Lemmas referenced :
q_le_wf,
eqtt_to_assert,
assert-q_le-eq,
iff_weakening_equal,
qless_trichot_qorder,
qless_transitivity_2_qorder,
qless_irreflexivity,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
qle_wf,
qle_weakening_lt_qorder,
qle_weakening_eq_qorder,
rationals_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
dependent_functionElimination,
independent_pairFormation,
isect_memberFormation_alt,
axiomEquality,
rename,
inlFormation_alt,
because_Cache,
voidElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
universeIsType,
inrFormation_alt
Latex:
\mforall{}q,r,s:\mBbbQ{}. uiff(qmin(q;r) = s;((q \mleq{} r) \mwedge{} (s = q)) \mvee{} ((r \mleq{} q) \mwedge{} (s = r)))
Date html generated:
2020_05_20-AM-09_16_47
Last ObjectModification:
2019_11_02-PM-00_43_22
Theory : rationals
Home
Index