Nuprl Lemma : decidable__equal_rationals
∀r,s:ℚ. Dec(r = s ∈ ℚ)
Proof
Definitions occuring in Statement :
rationals: ℚ,
decidable: Dec(P),
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exposed-btrue: exposed-btrue,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top
Lemmas referenced :
rationals_wf,
qeq_wf2,
bool_wf,
eqtt_to_assert,
assert-qeq,
it_wf,
subtype_rel_union,
unit_wf2,
equal_wf,
not_wf,
equal_subtype,
equal-wf-base,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
top_wf,
subtype_rel_dep_function,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
cut,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
inlEquality,
voidEquality,
applyEquality,
intEquality,
natural_numberEquality,
because_Cache,
baseClosed,
lambdaEquality,
voidElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
inrEquality,
functionEquality,
isect_memberEquality
Latex:
\mforall{}r,s:\mBbbQ{}. Dec(r = s)
Date html generated:
2018_05_21-PM-11_52_09
Last ObjectModification:
2017_07_26-PM-06_44_53
Theory : rationals
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