Nuprl Lemma : exact-xover_wf
∀[n:ℤ]. ∀[f:{n...} ⟶ 𝔹].
exact-xover(f;n) ∈ {x:ℤ| (n ≤ x) ∧ f x = ff ∧ f (x + 1) = tt}
supposing (∃m:{n...}. ((∀k:{n..m-}. f k = ff) ∧ (∀k:{m...}. f k = tt))) ∧ f n = ff
Proof
Definitions occuring in Statement :
exact-xover: exact-xover(f;n),
int_upper: {i...},
int_seg: {i..j-},
bfalse: ff,
btrue: tt,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
nat: ℕ,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
guard: {T},
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
exact-xover: exact-xover(f;n),
less_than: a < b,
nat_plus: ℕ+,
squash: ↓T,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
label: ...$L... t,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P)
Lemmas referenced :
subtract_wf,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
all_wf,
int_seg_wf,
equal-wf-T-base,
subtype_rel_sets,
int_upper_wf,
bool_wf,
int_upper_subtype_int_upper,
int_seg_properties,
exists_wf,
nat_properties,
ge_wf,
less_than_wf,
less_than_transitivity1,
less_than_irreflexivity,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_wf,
find-xover_wf,
or_wf,
equal-wf-base,
int_subtype_base,
equal_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
squash_wf,
true_wf,
iff_weakening_equal,
not_wf,
subtype_rel_dep_function,
subtype_rel_self,
sq_stable__and,
sq_stable__le,
sq_stable__equal,
btrue_neq_bfalse,
intformor_wf,
int_formula_prop_or_lemma
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
dependent_functionElimination,
dependent_set_memberEquality,
addEquality,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesisEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
productEquality,
applyEquality,
setEquality,
lambdaFormation,
baseClosed,
functionExtensionality,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality,
intWeakElimination,
independent_functionElimination,
hypothesis_subsumption,
imageMemberEquality,
baseApply,
closedConclusion,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
independent_pairEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[f:\{n...\} {}\mrightarrow{} \mBbbB{}].
exact-xover(f;n) \mmember{} \{x:\mBbbZ{}| (n \mleq{} x) \mwedge{} f x = ff \mwedge{} f (x + 1) = tt\}
supposing (\mexists{}m:\{n...\}. ((\mforall{}k:\{n..m\msupminus{}\}. f k = ff) \mwedge{} (\mforall{}k:\{m...\}. f k = tt))) \mwedge{} f n = ff
Date html generated:
2017_04_14-AM-09_18_02
Last ObjectModification:
2017_02_27-PM-03_55_16
Theory : int_2
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