Nuprl Lemma : lifting-member-simple
∀[B:Type]. ∀[n:ℕ]. ∀[A:ℕn ⟶ Type]. ∀[bags:k:ℕn ⟶ bag(A k)]. ∀[f:funtype(n;A;B)]. ∀[b:B].
(b ↓∈ lifting-gen-rev(n;f;bags)
⇐⇒ ↓∃lst:k:ℕn ⟶ (A k). ((∀[k:ℕn]. lst k ↓∈ bags k) ∧ ((uncurry-rev(n;f) lst) = b ∈ B)))
Proof
Definitions occuring in Statement :
uncurry-rev: uncurry-rev(n;f),
lifting-gen-rev: lifting-gen-rev(n;f;bags),
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,
equal: s = t ∈ T
Definitions unfolded in proof :
lifting-gen-rev: lifting-gen-rev(n;f;bags),
uncurry-rev: uncurry-rev(n;f),
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs
Lemmas referenced :
lifting-member,
false_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
subtype_rel-equal,
funtype_wf,
int_seg_wf,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
add-member-int_seg2,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
add-zero,
bag-member_wf,
lifting-gen-rev_wf,
squash_wf,
exists_wf,
uall_wf,
equal_wf,
uncurry-rev_wf,
bag_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_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,
functionExtensionality,
productElimination,
imageElimination,
imageMemberEquality,
baseClosed,
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;B)]. \mforall{}[b:B].
(b \mdownarrow{}\mmember{} lifting-gen-rev(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{} ((uncurry-rev(n;f) lst) = b)))
Date html generated:
2017_10_01-AM-09_04_21
Last ObjectModification:
2017_07_26-PM-04_44_48
Theory : bags
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