Nuprl Lemma : identity-record+-update
∀[T:Atom ⟶ 𝕌']. ∀[B:record(x.T[x]) ⟶ 𝕌']. ∀[z:Atom]. ∀[r:record(x.T[x])
z:B[self]]. ∀[x:Atom].
(r[x := r.x] = r ∈ record(x.T[x])z:B[self])
Proof
Definitions occuring in Statement :
record-update: r[x := v],
record-select: r.x,
record+: record+,
record: record(x.T[x]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
atom: Atom,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
record+: record+,
so_lambda: λ2x.t[x],
member: t ∈ T,
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 ,
sq_type: SQType(T),
guard: {T},
record-update: r[x := v],
bfalse: ff,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
squash: ↓T,
label: ...$L... t,
top: Top,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
record-select: r.x,
nequal: a ≠ b ∈ T ,
not: ¬A,
prop: ℙ,
istype: istype(T)
Lemmas referenced :
identity-record-update,
eq_atom_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
equal_wf,
istype-universe,
top_wf,
istype-void,
iff_weakening_equal,
record+_wf,
istype-atom,
record_wf,
bool_wf,
eq_atom-reflexive,
subtype_rel-equal,
subtype_rel_wf,
squash_wf,
true_wf,
record-update_wf,
record-select_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
dependentIntersection_memberEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
inhabitedIsType,
dependentIntersectionElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
functionExtensionality_alt,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
instantiate,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
voidElimination,
imageElimination,
universeIsType,
isect_memberEquality_alt,
natural_numberEquality,
imageMemberEquality,
equalityIsType1,
functionIsType,
universeEquality,
cumulativity,
atomEquality
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]]. \mforall{}[x:Atom].
(r[x := r.x] = r)
Date html generated:
2019_10_15-AM-11_29_14
Last ObjectModification:
2018_10_16-PM-02_35_29
Theory : general
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