Nuprl Lemma : concat-lifting-member
∀[B:Type]. ∀[n:ℕ]. ∀[A:ℕn ⟶ Type]. ∀[bags:k:ℕn ⟶ bag(A k)]. ∀[f:funtype(n;A;bag(B))]. ∀[b:B].
(b ↓∈ concat-lifting(n;f;bags) ⇐⇒ ↓∃lst:k:ℕn ⟶ (A k). ((∀[k:ℕn]. lst k ↓∈ bags k) ∧ b ↓∈ uncurry-rev(n;f) lst))
Proof
Definitions occuring in Statement :
concat-lifting: concat-lifting(n;f;bags),
uncurry-rev: uncurry-rev(n;f),
bag-member: x ↓∈ bs,
bag: bag(T),
funtype: funtype(n;A;T),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
concat-lifting: concat-lifting(n;f;bags),
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
subtract: n - m,
less_than: a < b,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
uncurry-rev: uncurry-rev(n;f),
uncurry-gen: uncurry-gen(n),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-member: x ↓∈ bs
Lemmas referenced :
nat_wf,
concat-lifting_wf,
uncurry-rev_wf,
uall_wf,
exists_wf,
squash_wf,
concat-lifting-list_wf,
bag-member_wf,
add-zero,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_seg_wf,
add-member-int_seg2,
le_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__le,
subtract_wf,
bag_wf,
funtype_wf,
subtype_rel-equal,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
false_wf,
concat-lifting-list-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
lambdaFormation,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality,
cumulativity,
productElimination,
imageElimination,
functionExtensionality,
imageMemberEquality,
baseClosed,
introduction,
independent_functionElimination,
functionEquality,
productEquality,
universeEquality,
isect_memberFormation,
independent_pairEquality
Latex:
\mforall{}[B:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type]. \mforall{}[bags:k:\mBbbN{}n {}\mrightarrow{} bag(A k)]. \mforall{}[f:funtype(n;A;bag(B))]. \mforall{}[b:B].
(b \mdownarrow{}\mmember{} concat-lifting(n;f;bags)
\mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mexists{}lst:k:\mBbbN{}n {}\mrightarrow{} (A k). ((\mforall{}[k:\mBbbN{}n]. lst k \mdownarrow{}\mmember{} bags k) \mwedge{} b \mdownarrow{}\mmember{} uncurry-rev(n;f) lst))
Date html generated:
2016_05_15-PM-03_06_42
Last ObjectModification:
2016_01_16-AM-08_35_07
Theory : bags
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