Nuprl Lemma : ball_char
∀s:DSet. ∀as:|s| List. ∀f:|s| ⟶ 𝔹. (↑(∀bx(:|s|) ∈ as. f[x])
⇐⇒ ∀x:|s|. ((↑(x ∈b as))
⇒ (↑f[x])))
Proof
Definitions occuring in Statement :
ball: ball,
mem: a ∈b as
,
list: T List
,
assert: ↑b
,
bool: 𝔹
,
so_apply: x[s]
,
all: ∀x:A. B[x]
,
iff: P
⇐⇒ Q
,
implies: P
⇒ Q
,
function: x:A ⟶ B[x]
,
dset: DSet
,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
nat: ℕ
,
implies: P
⇒ Q
,
false: False
,
ge: i ≥ j
,
uimplies: b supposing a
,
not: ¬A
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
top: Top
,
and: P ∧ Q
,
prop: ℙ
,
iff: P
⇐⇒ Q
,
guard: {T}
,
rev_implies: P
⇐ Q
,
dset: DSet
,
or: P ∨ Q
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
assert: ↑b
,
ifthenelse: if b then t else f fi
,
btrue: tt
,
bfalse: ff
,
true: True
,
cons: [a / b]
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
colength: colength(L)
,
nil: []
,
it: ⋅
,
sq_type: SQType(T)
,
less_than: a < b
,
squash: ↓T
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
,
decidable: Dec(P)
,
subtype_rel: A ⊆r B
,
uiff: uiff(P;Q)
,
rev_uimplies: rev_uimplies(P;Q)
,
ball: ball,
bool: 𝔹
,
unit: Unit
,
band: p ∧b q
,
bnot: ¬bb
,
infix_ap: x f y
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
assert_witness,
intformeq_wf,
int_formula_prop_eq_lemma,
set_car_wf,
list-cases,
ball_nil_lemma,
mem_nil_lemma,
true_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
ball_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
ball_cons_lemma,
mem_cons_lemma,
nat_wf,
bool_wf,
list_wf,
dset_wf,
assert_functionality_wrt_uiff,
assert_wf,
mem_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
bfalse_wf,
bor_wf,
set_eq_wf,
or_wf,
equal_wf,
iff_weakening_uiff,
assert_of_band,
iff_transitivity,
assert_of_bor,
assert_of_dset_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
unionElimination,
functionIsType,
promote_hyp,
hypothesis_subsumption,
equalityIsType1,
dependent_set_memberEquality_alt,
applyEquality,
instantiate,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
unionIsType,
productIsType,
inlFormation_alt,
inrFormation_alt,
equalityElimination,
productEquality
Latex:
\mforall{}s:DSet. \mforall{}as:|s| List. \mforall{}f:|s| {}\mrightarrow{} \mBbbB{}. (\muparrow{}(\mforall{}\msubb{}x(:|s|) \mmember{} as. f[x]) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} as)) {}\mRightarrow{} (\muparrow{}f[x])))
Date html generated:
2019_10_16-PM-01_02_54
Last ObjectModification:
2018_10_08-AM-11_23_26
Theory : list_2
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