Nuprl Lemma : uniform-continuity-pi-search-prop2
∀[n:ℕ]. ∀[P:ℕ ⟶ ℙ]. ∀[G:∀m:ℕn. Dec(P[m])]. ∀[x:ℕ].
(uniform-continuity-pi-search(
G;
n;x) ∈ {k:{x..n + 1-}| P[k] ∧ (∀m:{x..k-}. (¬P[m])) ∧ (∀m:{x..n + 1-}. (P[m] ⇒ (k ≤ m)))} ) supposing
((x ≤ n) and
(∃n:{x..n + 1-}. P[n]))
Proof
Definitions occuring in Statement :
uniform-continuity-pi-search: uniform-continuity-pi-search,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
sq_type: SQType(T),
uniform-continuity-pi-search: uniform-continuity-pi-search,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
isl: isl(x),
subtract: n - m,
squash: ↓T
Lemmas referenced :
le_wf,
exists_wf,
int_seg_wf,
nat_wf,
int_seg_subtype_nat,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
all_wf,
decidable_wf,
false_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
equal_wf,
subtype_base_sq,
int_subtype_base,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
less_than_transitivity1,
less_than_irreflexivity,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
add-zero,
decidable__lt,
lelt_wf,
not_wf,
add-commutes,
add-associates,
add-swap,
zero-add,
int_seg_subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
addEquality,
natural_numberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
applyLambdaEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
lambdaFormation,
functionEquality,
cumulativity,
universeEquality,
isect_memberFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
instantiate,
independent_functionElimination,
intWeakElimination,
equalityElimination,
promote_hyp,
productEquality,
imageMemberEquality,
baseClosed,
imageElimination,
hyp_replacement
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[G:\mforall{}m:\mBbbN{}n. Dec(P[m])]. \mforall{}[x:\mBbbN{}].
(uniform-continuity-pi-search(
G;
n;x) \mmember{} \{k:\{x..n + 1\msupminus{}\}| P[k] \mwedge{} (\mforall{}m:\{x..k\msupminus{}\}. (\mneg{}P[m])) \mwedge{} (\mforall{}m:\{x..n + 1\msupminus{}\}. (P[m] {}\mRightarrow{} (k \mleq{} m)))\} ) supp\000Cosing
((x \mleq{} n) and
(\mexists{}n:\{x..n + 1\msupminus{}\}. P[n]))
Date html generated:
2017_04_17-AM-09_59_01
Last ObjectModification:
2017_02_27-PM-05_53_01
Theory : continuity
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