Nuprl Lemma : bag-eq-subtype1
∀[A:Type]. ∀[B:A ⟶ ℙ]. ∀[d1,d2:bag({a:A| B[a]} )]. d1 = d2 ∈ bag({a:A| B[a]} ) supposing d1 = d2 ∈ bag(A)
Proof
Definitions occuring in Statement :
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
equal_wf,
bag_wf,
subtype_rel_bag,
bag_to_squash_list,
equal_functionality_wrt_subtype_rel2,
subtype_rel_wf,
squash_wf,
true_wf,
iff_weakening_equal,
list-subtype-bag,
subtype_rel_self,
permutation-strong-subtype,
strong-subtype-set2,
quotient-member-eq,
list_wf,
permutation_wf,
permutation-equiv,
member_wf,
subtype_rel_list
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
applyEquality,
setEquality,
functionExtensionality,
lambdaEquality,
sqequalRule,
universeEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
functionEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
productElimination,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
pertypeElimination,
productEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[d1,d2:bag(\{a:A| B[a]\} )]. d1 = d2 supposing d1 = d2
Date html generated:
2017_10_01-AM-08_57_42
Last ObjectModification:
2017_07_26-PM-04_39_47
Theory : bags
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