Nuprl Lemma : lifting-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));B)]. ∀[b:B].
(b ↓∈ lifting-gen-list-rev(n;bags) m f
⇐⇒ ↓∃lst:k:{m..n-} ⟶ (A k). ((∀[k:{m..n-}]. lst k ↓∈ bags k) ∧ ((uncurry-gen(n) m (λx.f) lst) = b ∈ B)))
Proof
Definitions occuring in Statement :
uncurry-gen: uncurry-gen(n),
lifting-gen-list-rev: lifting-gen-list-rev(n;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,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
squash: ↓T,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
lifting-gen-list-rev: lifting-gen-list-rev(n;bags),
true: True,
ifthenelse: if b then t else f fi ,
btrue: tt,
funtype: funtype(n;A;T),
lt_int: i <z j,
subtract: n - m,
uncurry-gen: uncurry-gen(n),
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
bfalse: ff,
assert: ↑b,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
gt: i > j,
sq_stable: SqStable(P),
le: A ≤ B,
less_than': less_than'(a;b),
istype: istype(T)
Lemmas referenced :
uncurry-gen_wf2,
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
bag-member_wf,
lifting-gen-list-rev_wf,
squash_wf,
exists_wf,
uall_wf,
equal_wf,
istype-universe,
funtype_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-member-int_seg1,
bag_wf,
nat_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
ge_wf,
add-member-int_seg2,
subtract-1-ge-0,
bool_wf,
bool_subtype_base,
true_wf,
eq_int_eq_true,
btrue_wf,
subtype_rel_self,
iff_weakening_equal,
primrec-unroll,
bag-member-single,
eq_int_eq_false,
bfalse_wf,
bag-member-combine,
iff_imp_equal_bool,
lt_int_wf,
iff_functionality_wrt_iff,
assert_wf,
false_wf,
iff_weakening_uiff,
assert_of_lt_int,
subtype_rel-equal,
primrec_wf,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
zero-mul,
add-zero,
itermMultiply_wf,
int_term_value_mul_lemma,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
bool_cases,
decidable__or,
equal-wf-base,
intformor_wf,
int_formula_prop_or_lemma,
btrue_neq_bfalse,
apply_uncurry,
minus-add,
minus-minus,
one-mul,
apply_larger_list,
sq_stable__bag-member,
subtype_rel_dep_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-one-mul-top,
zero-add,
le-add-cancel,
le_reflexive
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
hypothesis,
dependent_functionElimination,
addEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
productIsType,
functionIsType,
because_Cache,
applyEquality,
lambdaFormation_alt,
imageElimination,
imageMemberEquality,
baseClosed,
functionEquality,
productEquality,
inhabitedIsType,
universeEquality,
isect_memberFormation_alt,
independent_pairEquality,
functionIsTypeImplies,
equalityIsType4,
baseApply,
closedConclusion,
intEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
intWeakElimination,
isectIsType,
equalityIsType1,
multiplyEquality,
minusEquality,
promote_hyp,
equalityElimination,
equalityIsType2,
hyp_replacement,
applyLambdaEquality
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));B)]. \mforall{}[b:B].
(b \mdownarrow{}\mmember{} lifting-gen-list-rev(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{} ((uncurry-gen(n) m (\mlambda{}x.f) lst) = b)))
Date html generated:
2019_10_15-AM-11_04_47
Last ObjectModification:
2018_10_09-AM-10_53_21
Theory : bags
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