Nuprl Lemma : seq-min-upper-le
∀[k:ℕ]. ∀[n:ℕ+]. ∀[f:ℕ+ ⟶ ℤ]. (seq-min-upper(k;n;f) ≤ n)
Proof
Definitions occuring in Statement :
seq-min-upper: seq-min-upper(k;n;f),
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
seq-min-upper: seq-min-upper(k;n;f),
all: ∀x:A. B[x],
top: Top,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
le: A ≤ B,
false: False,
not: ¬A,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
primrec1_lemma,
le_int_wf,
nat_plus_wf,
less_than_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
nat_plus_properties,
seq-min-upper_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
primrec-wf-nat-plus,
less_than'_wf,
nat_wf,
primrec-unroll,
squash_wf,
true_wf,
eq_int_eq_false,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
equal-wf-base,
int_subtype_base,
bfalse_wf,
iff_weakening_equal,
subtract_wf,
add-subtract-cancel,
decidable__lt,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
minus-minus,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_weakening2,
le_reflexive,
add-is-int-iff,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
addEquality,
multiplyEquality,
natural_numberEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
setElimination,
rename,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairEquality,
axiomEquality,
functionEquality,
imageElimination,
universeEquality,
baseApply,
closedConclusion,
minusEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. (seq-min-upper(k;n;f) \mleq{} n)
Date html generated:
2017_10_03-AM-08_43_31
Last ObjectModification:
2017_09_09-AM-10_43_35
Theory : reals
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