Nuprl Lemma : apply_alist_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[V:Type]. ∀[L:(T × V) List].
apply_alist(eq;L;x) ∈ {v:V| (<x, v> ∈ L)} supposing (x ∈ map(λa.(fst(a));L))
Proof
Definitions occuring in Statement :
apply_alist: apply_alist(eq;L;x),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
top: Top,
not: ¬A,
cons: [a / b],
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
subtract: n - m,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
pi1: fst(t),
apply_alist: apply_alist(eq;L;x),
deq: EqDecider(T),
bool: 𝔹,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
true: True,
assert: ↑b,
ifthenelse: if b then t else f fi
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
list-cases,
map_nil_lemma,
istype-void,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
l_member_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
spread_cons_lemma,
sq_stable__le,
add-associates,
add-commutes,
add-swap,
zero-add,
map_cons_lemma,
cons_member,
map_wf,
pi1_wf,
deq_property,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
cons_wf,
istype-assert,
le_weakening2,
istype-nat,
list_wf,
deq_wf,
istype-universe,
subtype_rel_sets_simple
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
Error :isect_memberEquality_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
productEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
Error :equalityIstype,
because_Cache,
Error :dependent_set_memberEquality_alt,
instantiate,
cumulativity,
intEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
baseApply,
closedConclusion,
applyEquality,
sqequalBase,
independent_pairEquality,
Error :inlFormation_alt,
universeEquality,
Error :productIsType,
Error :isectIsType,
Error :inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[V:Type]. \mforall{}[L:(T \mtimes{} V) List].
apply\_alist(eq;L;x) \mmember{} \{v:V| (<x, v> \mmember{} L)\} supposing (x \mmember{} map(\mlambda{}a.(fst(a));L))
Date html generated:
2019_06_20-PM-00_42_44
Last ObjectModification:
2019_02_28-PM-00_01_09
Theory : list_0
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