Nuprl Lemma : assert-b-exists
∀n:ℕ. ∀P:ℕn ⟶ 𝔹. (↑(∃i<n.P[i])_b ⇐⇒ ∃i:ℕn. (↑P[i]))
Proof
Definitions occuring in Statement :
b-exists: (∃i<n.P[i])_b,
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
b-exists: (∃i<n.P[i])_b,
member: t ∈ T,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
istype: istype(T),
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
sq_type: SQType(T)
Lemmas referenced :
primrec0_lemma,
istype-void,
false_wf,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
assert_wf,
bool_wf,
subtype_rel_dep_function,
subtype_rel_sets,
and_wf,
le_wf,
less_than_wf,
subtract_wf,
decidable__lt,
istype-false,
not-lt-2,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
primrec-unroll,
b-exists_wf,
decidable__le,
not-le-2,
zero-add,
minus-minus,
add-zero,
le-add-cancel,
primrec-wf2,
all_wf,
iff_wf,
exists_wf,
nat_wf,
lt_int_wf,
equal-wf-base,
int_subtype_base,
bfalse_wf,
assert_witness,
le_int_wf,
bnot_wf,
iff_weakening_uiff,
bor_wf,
add-mul-special,
zero-mul,
le-add-cancel-alt,
or_wf,
assert_of_bor,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
decidable__int_equal,
le_weakening,
subtype_base_sq,
not-equal-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
hypothesis,
independent_pairFormation,
Error :universeIsType,
productElimination,
setElimination,
rename,
isectElimination,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
Error :productIsType,
applyEquality,
Error :functionIsType,
Error :lambdaEquality_alt,
because_Cache,
intEquality,
Error :inhabitedIsType,
Error :setIsType,
unionElimination,
addEquality,
minusEquality,
Error :dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
functionEquality,
baseApply,
closedConclusion,
baseClosed,
Error :inlFormation_alt,
Error :inrFormation_alt,
Error :unionIsType,
promote_hyp,
equalityElimination,
Error :equalityIsType1,
Error :dependent_pairFormation_alt,
instantiate
Latex:
\mforall{}n:\mBbbN{}. \mforall{}P:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. (\muparrow{}(\mexists{}i<n.P[i])\_b \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. (\muparrow{}P[i]))
Date html generated:
2019_06_20-AM-11_32_34
Last ObjectModification:
2018_09_28-PM-10_56_45
Theory : bool_1
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