Nuprl Lemma : apply-updated-alist
∀[A,T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:T]. ∀[L:(T × A) List]. ∀[z:A]. ∀[f:A ⟶ A].
(apply-alist(eq;update-alist(eq;L;x;z;v.f[v]);y)
= case apply-alist(eq;L;y)
of inl(v) =>
inl if eq x y then f[v] else v fi
| inr(any) =>
if eq x y then inl z else inr ⋅ fi
∈ (A?))
Proof
Definitions occuring in Statement :
apply-alist: apply-alist(eq;L;x),
update-alist: update-alist(eq;L;x;z;v.f[v]),
list: T List,
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
it: ⋅,
uall: ∀[x:A]. B[x],
so_apply: x[s],
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
deq: EqDecider(T),
prop: ℙ,
apply-alist: apply-alist(eq;L;x),
update-alist: update-alist(eq;L;x;z;v.f[v]),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
list_ind: list_ind,
nil: [],
it: ⋅,
pi1: fst(t),
pi2: snd(t),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
btrue: tt,
eqof: eqof(d),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
list_induction,
uall_wf,
equal_wf,
unit_wf2,
apply-alist_wf,
update-alist_wf,
ifthenelse_wf,
it_wf,
list_wf,
deq_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqof_wf,
uiff_transitivity,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
apply_alist_cons_lemma,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-unit,
bool_cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
unionEquality,
hypothesis,
applyEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
unionElimination,
inlEquality,
setElimination,
rename,
inrEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
isect_memberEquality,
axiomEquality,
universeEquality,
voidElimination,
voidEquality,
baseClosed,
equalityElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
addLevel,
impliesFunctionality,
levelHypothesis,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[A,T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:T]. \mforall{}[L:(T \mtimes{} A) List]. \mforall{}[z:A]. \mforall{}[f:A {}\mrightarrow{} A].
(apply-alist(eq;update-alist(eq;L;x;z;v.f[v]);y)
= case apply-alist(eq;L;y)
of inl(v) =>
inl if eq x y then f[v] else v fi
| inr(any) =>
if eq x y then inl z else inr \mcdot{} fi )
Date html generated:
2019_06_20-PM-00_43_06
Last ObjectModification:
2018_08_21-PM-01_53_58
Theory : list_0
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