Nuprl Lemma : gensearch_wf
∀[A,B:Type]. ∀[r:A ⟶ ℕ]. ∀[f:A ⟶ (B + Top)]. ∀[g:A ⟶ (A + Top)].
∀[a:A]. (gensearch(f;g;a) ∈ B + Top) supposing ∀a:A. ((↑isl(g a)) ⇒ r outl(g a) < r a)
Proof
Definitions occuring in Statement :
gensearch: gensearch(f;g;x),
nat: ℕ,
outl: outl(x),
assert: ↑b,
isl: isl(x),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
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},
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
gensearch: gensearch(f;g;x),
outl: outl(x),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
less_than: a < b,
squash: ↓T
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
le_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
subtype_rel_self,
top_wf,
equal_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
all_wf,
assert_wf,
isl_wf,
assert_elim,
bfalse_wf,
and_wf,
btrue_neq_bfalse,
btrue_wf,
bool_wf,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
productElimination,
unionElimination,
instantiate,
cumulativity,
applyLambdaEquality,
dependent_set_memberEquality,
hypothesis_subsumption,
unionEquality,
inlEquality,
inrEquality,
addEquality,
functionEquality,
universeEquality,
hyp_replacement,
functionExtensionality,
imageElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[r:A {}\mrightarrow{} \mBbbN{}]. \mforall{}[f:A {}\mrightarrow{} (B + Top)]. \mforall{}[g:A {}\mrightarrow{} (A + Top)].
\mforall{}[a:A]. (gensearch(f;g;a) \mmember{} B + Top) supposing \mforall{}a:A. ((\muparrow{}isl(g a)) {}\mRightarrow{} r outl(g a) < r a)
Date html generated:
2019_10_15-AM-11_06_57
Last ObjectModification:
2018_08_21-PM-03_44_11
Theory : general
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