Nuprl Lemma : bag-map-equal
∀[T,A:Type].
∀f,g:T ⟶ A. ∀P:T ⟶ 𝔹.
((∀x:T. ((¬↑(P x)) ⇒ ((f x) = (g x) ∈ A)))
⇒ (∀as:bag(T). ((↑null([x∈as|P x])) ⇒ (bag-map(f;as) = bag-map(g;as) ∈ bag(A)))))
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
bag: bag(T),
null: null(as),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bag-null: bag-null(bs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
guard: {T},
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
assert_wf,
bag-null_wf,
bag-filter_wf,
bag_wf,
all_wf,
not_wf,
equal_wf,
bool_wf,
bag-map_wf,
list_wf,
permutation_wf,
equal-wf-base,
list-subtype-bag,
map_equal,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
less_than_wf,
length_wf,
nat_wf,
assert_of_null,
filter_wf5,
l_member_wf,
member_filter,
select_member,
lelt_wf,
assert_functionality_wrt_uiff,
eta_conv,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
and_wf,
null_wf,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
functionEquality,
axiomEquality,
because_Cache,
isect_memberEquality,
independent_functionElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
rename,
productEquality,
independent_isectElimination,
setElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}[T,A:Type].
\mforall{}f,g:T {}\mrightarrow{} A. \mforall{}P:T {}\mrightarrow{} \mBbbB{}.
((\mforall{}x:T. ((\mneg{}\muparrow{}(P x)) {}\mRightarrow{} ((f x) = (g x))))
{}\mRightarrow{} (\mforall{}as:bag(T). ((\muparrow{}null([x\mmember{}as|P x])) {}\mRightarrow{} (bag-map(f;as) = bag-map(g;as)))))
Date html generated:
2017_10_01-AM-08_45_41
Last ObjectModification:
2017_07_26-PM-04_30_51
Theory : bags
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