Nuprl Lemma : next_wf_bound
∀[b:ℕ]. ∀[k:ℤ]. ∀[p:{i:ℤ| k < i} ⟶ 𝔹].
(next i > k s.t. ↑p[i]) ∈ {i:ℤ| (k < i ∧ (i ≤ (k + b))) ∧ (↑p[i]) ∧ (∀j:{k + 1..i-}. (¬↑p[j]))}
supposing ∃n:{i:ℤ| k < i ∧ (i ≤ (k + b))} . (↑p[n])
Proof
Definitions occuring in Statement :
next: (next i > k s.t. ↑p[i]),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
next: (next i > k s.t. ↑p[i]),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
cand: A c∧ B,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
sq_type: SQType(T),
sq_stable: SqStable(P),
squash: ↓T,
label: ...$L... t,
less_than: a < b
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
exists_wf,
le_wf,
assert_wf,
bool_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
value-type-has-value,
int-value-type,
int_seg_properties,
subtype_rel_sets,
lelt_wf,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
all_wf,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
subtype_rel_dep_function,
less-iff-le,
add-associates,
subtype_rel_self,
set_wf,
add-is-int-iff,
int_subtype_base,
decidable__equal_int,
assert_elim,
subtype_base_sq,
not_assert_elim,
btrue_neq_bfalse,
intformeq_wf,
int_formula_prop_eq_lemma,
sq_stable__and,
sq_stable__less_than,
sq_stable__le,
less_than'_wf,
squash_wf,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
productEquality,
addEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
productElimination,
functionEquality,
unionElimination,
because_Cache,
callbyvalueReduce,
baseClosed,
equalityElimination,
minusEquality,
baseApply,
closedConclusion,
instantiate,
cumulativity,
independent_pairEquality,
imageMemberEquality,
imageElimination,
addLevel,
impliesFunctionality,
levelHypothesis
Latex:
\mforall{}[b:\mBbbN{}]. \mforall{}[k:\mBbbZ{}]. \mforall{}[p:\{i:\mBbbZ{}| k < i\} {}\mrightarrow{} \mBbbB{}].
(next i > k s.t. \muparrow{}p[i]) \mmember{} \{i:\mBbbZ{}| (k < i \mwedge{} (i \mleq{} (k + b))) \mwedge{} (\muparrow{}p[i]) \mwedge{} (\mforall{}j:\{k + 1..i\msupminus{}\}. (\mneg{}\muparrow{}p[j]))\}
supposing \mexists{}n:\{i:\mBbbZ{}| k < i \mwedge{} (i \mleq{} (k + b))\} . (\muparrow{}p[n])
Date html generated:
2018_05_21-PM-06_54_54
Last ObjectModification:
2017_07_26-PM-04_59_16
Theory : general
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