Nuprl Lemma : fresh-nat-exists
∀L:ℕ List. ∃n:ℕ. ∀i:ℕ||L||. L[i] < n
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
select: L[n],
uimplies: b supposing a,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
cons: [a / b],
exists: ∃x:A. B[x],
nat: ℕ,
decidable: Dec(P),
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
guard: {T},
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_exists: (∃x∈L. P[x]),
uiff: uiff(P;Q)
Lemmas referenced :
nat_wf,
list-cases,
length_of_nil_lemma,
stuck-spread,
istype-base,
istype-void,
product_subtype_list,
length_of_cons_lemma,
list_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
int_seg_properties,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_seg_wf,
nat_properties,
imax-list-cons-is-nat,
subtype_rel_list,
add_nat_wf,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
length_wf,
istype-less_than,
select_wf,
cons_wf,
non_neg_length,
decidable__lt,
imax-list-ub,
add-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
sqequalRule,
baseClosed,
independent_isectElimination,
Error :isect_memberEquality_alt,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :universeIsType,
Error :dependent_pairFormation_alt,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
Error :lambdaEquality_alt,
setElimination,
rename,
int_eqEquality,
independent_pairFormation,
Error :functionIsType,
applyEquality,
intEquality,
Error :inhabitedIsType,
addEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
closedConclusion,
applyLambdaEquality,
Error :equalityIstype,
imageElimination,
Error :productIsType,
pointwiseFunctionality,
baseApply
Latex:
\mforall{}L:\mBbbN{} List. \mexists{}n:\mBbbN{}. \mforall{}i:\mBbbN{}||L||. L[i] < n
Date html generated:
2019_06_20-PM-01_19_34
Last ObjectModification:
2019_03_26-PM-00_51_22
Theory : list_1
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