Nuprl Lemma : record+_extensionality
∀[T:Atom ⟶ 𝕌']. ∀[B:record(x.T[x]) ⟶ 𝕌']. ∀[z:Atom]. ∀[r1,r2:record(x.T[x])
z:B[self]].
uiff(r1 = r2 ∈ record(x.T[x])z:B[self];(r1 = r2 ∈ record(x.T[x])) ∧ (r1.z = r2.z ∈ B[r1]))
Proof
Definitions occuring in Statement :
record-select: r.x,
record+: record+,
record: record(x.T[x]),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
function: x:A ⟶ B[x],
atom: Atom,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
record+: record+,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
record-select: r.x,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top
Lemmas referenced :
record+_wf,
istype-atom,
record_wf,
istype-universe,
subtype_rel-equal,
eq_atom_wf,
eqtt_to_assert,
assert_of_eq_atom,
top_wf,
eqff_to_assert,
atom_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_atom,
equal_wf,
equal-wf-base,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
istype-void,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
record-select_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
independent_pairFormation,
sqequalRule,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
isect_memberEquality_alt,
isectElimination,
hypothesisEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
because_Cache,
extract_by_obid,
lambdaEquality_alt,
applyEquality,
functionIsType,
instantiate,
universeEquality,
dependentIntersectionEqElimination,
applyLambdaEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
independent_isectElimination,
cumulativity,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
voidElimination,
dependentIntersection_memberEquality,
dependentIntersectionElimination,
functionExtensionality_alt,
atomEquality,
productIsType
Latex:
\mforall{}[T:Atom {}\mrightarrow{} \mBbbU{}']. \mforall{}[B:record(x.T[x]) {}\mrightarrow{} \mBbbU{}']. \mforall{}[z:Atom]. \mforall{}[r1,r2:record(x.T[x])
z:B[self]].
uiff(r1 = r2;(r1 = r2) \mwedge{} (r1.z = r2.z))
Date html generated:
2019_10_15-AM-11_28_58
Last ObjectModification:
2018_10_16-PM-02_30_02
Theory : general
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