Nuprl Lemma : apply-alist-cases
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[L:(T × Top) List].
(apply-alist(eq;L;x) ~ ff supposing ¬(x ∈ map(λp.(fst(p));L))
∧ (∀[i:ℕ||L||]
(apply-alist(eq;L;x) ~ inl (snd(L[i]))) supposing (((fst(L[i])) = x ∈ T) and (∀j:ℕi. (¬((fst(L[j])) = x ∈ T))))))
Proof
Definitions occuring in Statement :
apply-alist: apply-alist(eq;L;x),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
deq: EqDecider(T),
int_seg: {i..j-},
bfalse: ff,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
lambda: λx.A[x],
product: x:A × B[x],
inl: inl x,
natural_number: $n,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
le: A ≤ B,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
lelt: i ≤ j < k,
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
bool: 𝔹,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
apply-alist: apply-alist(eq;L;x),
list_ind: list_ind,
nat_plus: ℕ+,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
deq: EqDecider(T),
exists: ∃x:A. B[x],
unit: Unit,
btrue: tt,
eqof: eqof(d),
select: L[n]
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
not_wf,
l_member_wf,
map_wf,
top_wf,
pi1_wf,
equal_wf,
select_wf,
all_wf,
int_seg_wf,
length_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le_wf,
less_than_transitivity2,
le_weakening2,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
list_wf,
deq_wf,
map_nil_lemma,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
apply-alist_wf,
nil_wf,
subtype_rel_union,
unit_wf2,
bfalse_wf,
assert_wf,
length_of_nil_lemma,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
minus-zero,
omega-shadow,
int_seg_properties,
map_cons_lemma,
list_ind_cons_lemma,
length_of_cons_lemma,
cons_wf,
non_neg_length,
length_wf_nat,
not-lt-2,
mul-distributes,
mul-associates,
mul-commutes,
bnot_wf,
eqof_wf,
uiff_transitivity,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
decidable__lt,
cons_member,
decidable__equal_int,
not-equal-2,
lelt_wf,
subtract-add-cancel,
select_cons_tl_sq,
le-add-cancel2,
squash_wf,
true_wf,
select_cons_tl,
iff_weakening_equal,
add-member-int_seg2,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
sqequalAxiom,
cumulativity,
productEquality,
voidEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
applyEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
intEquality,
instantiate,
universeEquality,
multiplyEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
equalityElimination,
addLevel,
impliesFunctionality,
levelHypothesis,
inlFormation,
hyp_replacement,
inrFormation,
functionExtensionality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[L:(T \mtimes{} Top) List].
(apply-alist(eq;L;x) \msim{} ff supposing \mneg{}(x \mmember{} map(\mlambda{}p.(fst(p));L))
\mwedge{} (\mforall{}[i:\mBbbN{}||L||]
(apply-alist(eq;L;x) \msim{} inl (snd(L[i]))) supposing
(((fst(L[i])) = x) and
(\mforall{}j:\mBbbN{}i. (\mneg{}((fst(L[j])) = x))))))
Date html generated:
2017_04_14-AM-08_47_21
Last ObjectModification:
2017_02_27-PM-03_35_51
Theory : list_0
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