Nuprl Lemma : concat-lifting-list-member
∀[B:Type]. ∀[n:ℕ]. ∀[m:ℕn + 1]. ∀[A:ℕn ⟶ Type]. ∀[bags:k:ℕn ⟶ bag(A k)]. ∀[f:funtype(n - m;λx.(A (x + m));bag(B))].
∀[b:B].
(b ↓∈ concat-lifting-list(n;bags) m f
⇐⇒ ↓∃lst:k:{m..n-} ⟶ (A k). ((∀[k:{m..n-}]. lst k ↓∈ bags k) ∧ b ↓∈ uncurry-gen(n) m (λx.f) lst))
Proof
Definitions occuring in Statement :
concat-lifting-list: concat-lifting-list(n;bags),
uncurry-gen: uncurry-gen(n),
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,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
bag-member: x ↓∈ bs,
rev_implies: P ⇐ Q,
squash: ↓T,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
nat: ℕ,
guard: {T},
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_type: SQType(T),
subtype_rel: A ⊆r B,
cand: A c∧ B,
uncurry-gen: uncurry-gen(n),
concat: concat(ll),
bag-union: bag-union(bbs),
single-bag: {x},
subtract: n - m,
lt_int: i <z j,
funtype: funtype(n;A;T),
btrue: tt,
ifthenelse: if b then t else f fi ,
true: True,
lifting-gen-list-rev: lifting-gen-list-rev(n;bags),
concat-lifting-list: concat-lifting-list(n;bags),
bfalse: ff,
nequal: a ≠ b ∈ T ,
le: A ≤ B,
less_than: a < b,
assert: ↑b,
bnot: ¬bb,
it: ⋅,
unit: Unit,
bool: 𝔹,
gt: i > j,
sq_stable: SqStable(P),
istype: istype(T),
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_wf,
add-member-int_seg1,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
funtype_wf,
istype-universe,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
intformless_wf,
decidable__lt,
bag_wf,
uncurry-gen_wf2,
int_seg_wf,
uall_wf,
less_than_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
exists_wf,
squash_wf,
concat-lifting-list_wf,
bag-member_wf,
subtract-1-ge-0,
add-member-int_seg2,
ge_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
bag-subtype-list,
append_nil_sq,
reduce_nil_lemma,
reduce_cons_lemma,
primrec-unroll,
iff_weakening_equal,
subtype_rel_self,
btrue_wf,
eq_int_eq_true,
true_wf,
equal_wf,
bool_subtype_base,
bool_wf,
bag-combine_wf,
bag-member-union,
bfalse_wf,
eq_int_eq_false,
lifting-gen-list-rev_wf,
istype-le,
istype-less_than,
funtype-unroll,
eq_int_wf,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
istype-assert,
subtype_rel-equal,
bool_cases,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
bag-member-combine,
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
le_reflexive,
add-is-int-iff,
int_seg_subtype,
btrue_neq_bfalse,
int_formula_prop_or_lemma,
intformor_wf,
decidable__or,
subtract-add-cancel,
easy-member-int_seg,
apply_uncurry,
eq_int_eq_false_intro,
apply_larger_list,
eq_int_eq_true_intro,
sq_stable__bag-member,
bag-union_wf,
single-bag_wf,
assert_elim,
add-commutes,
add-swap,
minus-one-mul,
add-associates,
le-add-cancel,
add-zero,
zero-add,
zero-mul,
add-mul-special,
minus-one-mul-top,
minus-minus,
minus-add,
condition-implies-le,
not-le-2,
istype-false,
subtype_rel_dep_function,
int_term_value_mul_lemma,
itermMultiply_wf,
apply_gen_wf
Rules used in proof :
functionIsTypeImplies,
independent_pairEquality,
baseClosed,
imageMemberEquality,
imageElimination,
lambdaFormation_alt,
isect_memberFormation_alt,
universeEquality,
functionIsType,
inhabitedIsType,
addEquality,
productEquality,
productIsType,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
productElimination,
dependent_set_memberEquality_alt,
rename,
setElimination,
applyEquality,
because_Cache,
functionEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
universeIsType,
intWeakElimination,
equalitySymmetry,
equalityTransitivity,
cumulativity,
instantiate,
intEquality,
closedConclusion,
baseApply,
equalityIsType4,
isectIsType,
Error :memTop,
equalityIstype,
sqequalBase,
functionExtensionality_alt,
promote_hyp,
equalityElimination,
applyLambdaEquality,
hyp_replacement,
minusEquality,
multiplyEquality
Latex:
\mforall{}[B:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[m:\mBbbN{}n + 1]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type]. \mforall{}[bags:k:\mBbbN{}n {}\mrightarrow{} bag(A k)].
\mforall{}[f:funtype(n - m;\mlambda{}x.(A (x + m));bag(B))]. \mforall{}[b:B].
(b \mdownarrow{}\mmember{} concat-lifting-list(n;bags) m f
\mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mexists{}lst:k:\{m..n\msupminus{}\} {}\mrightarrow{} (A k)
((\mforall{}[k:\{m..n\msupminus{}\}]. lst k \mdownarrow{}\mmember{} bags k) \mwedge{} b \mdownarrow{}\mmember{} uncurry-gen(n) m (\mlambda{}x.f) lst))
Date html generated:
2020_05_20-AM-08_03_32
Last ObjectModification:
2020_02_04-PM-02_19_59
Theory : bags
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