Nuprl Lemma : find-xover-val_wf
∀[T:Type]
∀[test:T ⟶ 𝔹]. ∀[x:ℤ]. ∀[n:{x...}]. ∀[step:ℕ+]. ∀[f:{x...} ⟶ T].
find-xover-val(test;f;x;n;step) ∈ v:T
× n':{n':ℤ| (n ≤ n') ∧ (v = (f n') ∈ T) ∧ test v = tt}
× {x':ℤ|
((n' = n ∈ ℤ) ∧ (x' = x ∈ ℤ)) ∨ (((n ≤ x') ∧ test (f x') = ff) ∧ ((n' = (n + step) ∈ ℤ) ∨ ((n + step) ≤ x')))}
supposing ∃m:{n...}. ∀k:{m...}. test (f k) = tt
supposing value-type(T)
Proof
Definitions occuring in Statement :
find-xover-val: find-xover-val(test;f;m;n;step),
int_upper: {i...},
nat_plus: ℕ+,
value-type: value-type(T),
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],
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
add: n + m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
find-xover-val: find-xover-val(test;f;m;n;step),
has-value: (a)↓,
nat_plus: ℕ+,
int_upper: {i...},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
assert: ↑b,
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
sq_stable: SqStable(P),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
value-type-has-value,
int-value-type,
eqtt_to_assert,
nat_plus_properties,
int_upper_properties,
iff_imp_equal_bool,
btrue_wf,
true_wf,
bool_wf,
le_wf,
less_than_wf,
equal_wf,
equal-wf-T-base,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
istype-int_upper,
upper_subtype_upper,
sq_stable__le,
le_transitivity,
nat_plus_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
value-type_wf,
istype-universe,
itermMultiply_wf,
int_term_value_mul_lemma,
subtype_rel_function,
int_upper_wf,
squash_wf,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
iff_weakening_equal,
not_assert_elim,
btrue_neq_bfalse,
bfalse_wf,
assert_elim,
istype-assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
Error :lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
productElimination,
unionElimination,
applyEquality,
instantiate,
cumulativity,
intEquality,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
because_Cache,
Error :productIsType,
hypothesis_subsumption,
callbyvalueReduce,
multiplyEquality,
addEquality,
equalityElimination,
Error :dependent_pairEquality_alt,
Error :equalityIstype,
Error :inlFormation_alt,
baseClosed,
sqequalBase,
Error :unionIsType,
baseApply,
closedConclusion,
productEquality,
Error :setIsType,
imageMemberEquality,
imageElimination,
promote_hyp,
Error :functionIsType,
universeEquality,
minusEquality,
Error :inrFormation_alt
Latex:
\mforall{}[T:Type]
\mforall{}[test:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:\mBbbZ{}]. \mforall{}[n:\{x...\}]. \mforall{}[step:\mBbbN{}\msupplus{}]. \mforall{}[f:\{x...\} {}\mrightarrow{} T].
find-xover-val(test;f;x;n;step) \mmember{} v:T
\mtimes{} n':\{n':\mBbbZ{}| (n \mleq{} n') \mwedge{} (v = (f n')) \mwedge{} test v = tt\}
\mtimes{} \{x':\mBbbZ{}|
((n' = n) \mwedge{} (x' = x))
\mvee{} (((n \mleq{} x') \mwedge{} test (f x') = ff) \mwedge{} ((n' = (n + step)) \mvee{} ((n + step) \mleq{} x')))\}
supposing \mexists{}m:\{n...\}. \mforall{}k:\{m...\}. test (f k) = tt
supposing value-type(T)
Date html generated:
2019_06_20-PM-01_16_31
Last ObjectModification:
2019_01_09-PM-04_07_00
Theory : int_2
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