Nuprl Lemma : record+_record
∀[T:Atom ⟶ 𝕌']. ∀[B:record(x.T[x]) ⟶ 𝕌']. ∀[z:Atom]. ∀[r:record(x.T[x])
z:B[self]].
(r ∈ record(x.if x =a z then B[r] else T[x] fi ))
Proof
Definitions occuring in Statement :
record+: record+,
record: record(x.T[x]),
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
record+: record+,
member: t ∈ T,
record: record(x.T[x]),
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀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,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
not: ¬A
Lemmas referenced :
record+_wf,
istype-atom,
record_wf,
istype-universe,
eq_atom_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_rel-equal,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
dependentIntersectionElimination,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
hypothesis,
functionIsType,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
functionExtensionality_alt,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
cumulativity,
equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
voidElimination
Latex:
\mforall{}[T:Atom {}\mrightarrow{} \mBbbU{}']. \mforall{}[B:record(x.T[x]) {}\mrightarrow{} \mBbbU{}']. \mforall{}[z:Atom]. \mforall{}[r:record(x.T[x])
z:B[self]].
(r \mmember{} record(x.if x =a z then B[r] else T[x] fi ))
Date html generated:
2019_10_15-AM-11_28_25
Last ObjectModification:
2018_10_16-PM-02_35_53
Theory : general
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