Nuprl Lemma : qeq_wf
∀[r,s:ℤ ⋃ (ℤ × ℤ-o)]. (qeq(r;s) ∈ 𝔹)
Proof
Definitions occuring in Statement :
qeq: qeq(r;s),
int_nzero: ℤ-o,
b-union: A ⋃ B,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
qeq: qeq(r;s),
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
btrue: tt,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_nzero: ℤ-o,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
assert: ↑b,
bfalse: ff,
false: False,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb
Lemmas referenced :
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
eq_int_wf,
int_nzero_wf,
product-valueall-type,
set-valueall-type,
nequal_wf,
bfalse_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
b-union_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
unionElimination,
equalityElimination,
sqequalRule,
extract_by_obid,
isectElimination,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
callbyvalueReduce,
because_Cache,
isintReduceTrue,
productEquality,
lambdaEquality,
independent_functionElimination,
lambdaFormation,
natural_numberEquality,
voidElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
multiplyEquality,
setElimination,
rename,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[r,s:\mBbbZ{} \mcup{} (\mBbbZ{} \mtimes{} \mBbbZ{}\msupminus{}\msupzero{})]. (qeq(r;s) \mmember{} \mBbbB{})
Date html generated:
2018_05_21-PM-11_43_34
Last ObjectModification:
2017_07_26-PM-06_42_53
Theory : rationals
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