Nuprl Lemma : mu-property2
∀[P:ℕ ⟶ ℙ]. ∀d:∀n:ℕ. Dec(P[n]). {P[mu(d)] ∧ (∀i:ℕ. ¬P[i] supposing i < mu(d))} supposing ∃n:ℕ. P[n]
Proof
Definitions occuring in Statement :
mu: mu(f),
nat: ℕ,
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
mu: mu(f),
nat: ℕ,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
top: Top,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
mu-ge-property2,
subtype_rel_function,
nat_wf,
int_upper_wf,
upper_subtype_nat,
istype-false,
subtype_rel_self,
all_wf,
decidable_wf,
mu-ge_wf2,
subtype_rel_dep_function,
top_wf,
subtype_rel_union,
not_wf,
istype-void,
assert_wf,
btrue_wf,
bfalse_wf,
less_than_wf,
nat_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
le_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
natural_numberEquality,
Error :isect_memberFormation_alt,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
Error :lambdaFormation_alt,
Error :lambdaEquality_alt,
Error :universeIsType,
productElimination,
Error :dependent_pairFormation_alt,
promote_hyp,
Error :productIsType,
Error :functionIsType,
unionEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :unionIsType,
functionExtensionality,
Error :inhabitedIsType,
unionElimination,
Error :equalityIsType1,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
Error :functionIsTypeImplies,
setElimination,
rename,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
approximateComputation,
int_eqEquality
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. \mforall{}d:\mforall{}n:\mBbbN{}. Dec(P[n]). \{P[mu(d)] \mwedge{} (\mforall{}i:\mBbbN{}. \mneg{}P[i] supposing i < mu(d))\} supposing \mexists{}n:\mBbbN{}. P[n]
Date html generated:
2019_06_20-PM-01_17_23
Last ObjectModification:
2018_10_06-AM-11_21_44
Theory : int_2
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