Nuprl Lemma : assert-bdd-all
∀n:ℕ. ∀P:ℕn ⟶ 𝔹. (↑bdd-all(n;i.P[i]) ⇐⇒ ∀i:ℕn. (↑P[i]))
Proof
Definitions occuring in Statement :
bdd-all: bdd-all(n;i.P[i]),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_apply: x[s],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
bdd-all: bdd-all(n;i.P[i]),
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
subtype_rel: A ⊆r B,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
istype: istype(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
band: p ∧b q,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
label: ...$L... t,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
assert_witness,
bdd-all_wf,
int_seg_wf,
primrec0_lemma,
istype-void,
true_wf,
assert_wf,
bool_wf,
le_wf,
subtract-1-ge-0,
subtype_rel_dep_function,
subtract_wf,
subtype_rel_sets,
and_wf,
decidable__lt,
istype-false,
not-lt-2,
less-iff-le,
condition-implies-le,
add-associates,
istype-int,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
primrec-unroll,
nat_wf,
lt_int_wf,
equal-wf-base,
int_subtype_base,
add-zero,
btrue_wf,
le_int_wf,
bnot_wf,
iff_weakening_uiff,
decidable__le,
not-le-2,
zero-add,
minus-minus,
le-add-cancel,
add-mul-special,
zero-mul,
le-add-cancel-alt,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert_of_band,
uiff_transitivity,
assert_of_lt_int,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
decidable__int_equal,
le_antisymmetry_iff,
not-equal-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
sqequalRule,
Error :lambdaEquality_alt,
dependent_functionElimination,
productElimination,
independent_pairEquality,
applyEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
independent_pairFormation,
Error :functionIsType,
Error :dependent_set_memberEquality_alt,
because_Cache,
intEquality,
Error :setIsType,
Error :productIsType,
unionElimination,
addEquality,
minusEquality,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
equalityElimination,
Error :dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType1,
instantiate,
productEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}P:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. (\muparrow{}bdd-all(n;i.P[i]) \mLeftarrow{}{}\mRightarrow{} \mforall{}i:\mBbbN{}n. (\muparrow{}P[i]))
Date html generated:
2019_06_20-AM-11_32_42
Last ObjectModification:
2018_10_06-AM-09_00_33
Theory : bool_1
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