Nuprl Lemma : next_wf
∀[k:ℤ]. ∀[p:{i:ℤ| k < i} ⟶ 𝔹]. (next i > k s.t. ↑p[i]) ∈ {i:ℤ| k < i ∧ (↑p[i]) ∧ (∀j:{k + 1..i-}. (¬↑p[j]))} supposi\000Cng ∃n:{i:ℤ| k < i} . (↑p[n])
Proof
Definitions occuring in Statement :
next: (next i > k s.t. ↑p[i]),
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
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,
uimplies: b supposing a,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
subtract: n - m,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
le-add-cancel,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
false_wf,
decidable__lt,
int_subtype_base,
add-is-int-iff,
lelt_wf,
not_wf,
int_seg_wf,
all_wf,
subtype_rel_sets,
and_wf,
int_term_value_add_lemma,
itermAdd_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
subtract_wf,
decidable__le,
next_wf_bound,
bool_wf,
assert_wf,
less_than_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
setEquality,
intEquality,
hypothesisEquality,
lambdaEquality,
applyEquality,
isect_memberEquality,
because_Cache,
functionEquality,
setElimination,
rename,
dependent_set_memberEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addEquality,
lambdaFormation,
productEquality,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
minusEquality
Latex:
\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{} (\muparrow{}p[i]) \mwedge{} (\mforall{}j:\{k + 1..i\msupminus{}\}. (\mneg{}\muparrow{}p[j]))\} supposing \mexists{}n:\{i:\mBbbZ{}| \000Ck < i\} . (\muparrow{}p[n])
Date html generated:
2016_05_15-PM-04_00_12
Last ObjectModification:
2016_01_16-AM-11_00_15
Theory : general
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