Nuprl Lemma : bag-equality
∀[A,B:Type]. ∀[f,g:bag(A) ⟶ bag(B)]. ∀[b:bag(A)]. (f[b] = g[b] ∈ bag(B)) supposing ∀[b:A List]. (f[b] = g[b] ∈ bag(B))
Proof
Definitions occuring in Statement :
bag: bag(T),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
so_apply: x[s],
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x]
Lemmas referenced :
bag_wf,
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
equal_wf,
list-subtype-bag,
iff_weakening_equal,
equal-wf-base,
uall_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
imageElimination,
functionExtensionality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f,g:bag(A) {}\mrightarrow{} bag(B)].
\mforall{}[b:bag(A)]. (f[b] = g[b]) supposing \mforall{}[b:A List]. (f[b] = g[b])
Date html generated:
2017_10_01-AM-08_44_53
Last ObjectModification:
2017_07_26-PM-04_30_24
Theory : bags
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