Nuprl Lemma : div-search-lemma
∀a:ℤ. ∀b:{a + 1...}. ∀f:ℤ ⟶ 𝔹.
∃x:{a..b-} [((∀y:{a..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..b + 1-}. (↑(f z))))]
supposing ∃x:{a..b-} [((∀y:{a..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..b + 1-}. (↑(f z))))]
Proof
Definitions occuring in Statement :
int_upper: {i...},
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
cand: A c∧ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
true: True,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_upper: {i...},
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
int_seg: {i..j-},
so_apply: x[s],
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
sq_stable__and,
sq_stable__all,
sq_stable__not,
sq_stable_from_decidable,
decidable__assert,
assert_witness,
istype-assert,
int_seg_properties,
int_upper_properties,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
itermConstant_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
divide-and-conquer,
isect_wf,
istype-false,
istype-le,
member-less_than,
istype-less_than,
upper_subtype_upper,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
sq_exists_wf,
int_seg_wf,
all_wf,
not_wf,
assert_wf,
bool_wf,
istype-int_upper,
istype-int
Rules used in proof :
Error :inrFormation_alt,
promote_hyp,
Error :inlFormation_alt,
Error :isectIsType,
Error :dependent_set_memberFormation_alt,
Error :functionIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
independent_functionElimination,
productElimination,
independent_pairEquality,
independent_isectElimination,
Error :productIsType,
unionElimination,
voidElimination,
Error :isect_memberEquality_alt,
minusEquality,
multiplyEquality,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
sqequalRule,
Error :lambdaEquality_alt,
productEquality,
addEquality,
because_Cache,
closedConclusion,
natural_numberEquality,
applyEquality,
Error :functionIsType,
Error :inhabitedIsType
Latex:
\mforall{}a:\mBbbZ{}. \mforall{}b:\{a + 1...\}. \mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbB{}.
\mexists{}x:\{a..b\msupminus{}\} [((\mforall{}y:\{a..x + 1\msupminus{}\}. (\mneg{}\muparrow{}(f y))) \mwedge{} (\mforall{}z:\{x + 1..b + 1\msupminus{}\}. (\muparrow{}(f z))))]
supposing \mexists{}x:\{a..b\msupminus{}\} [((\mforall{}y:\{a..x + 1\msupminus{}\}. (\mneg{}\muparrow{}(f y))) \mwedge{} (\mforall{}z:\{x + 1..b + 1\msupminus{}\}. (\muparrow{}(f z))))]
Date html generated:
2019_06_20-PM-02_12_35
Last ObjectModification:
2019_06_20-PM-02_08_48
Theory : int_2
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