Nuprl Lemma : rminimum_wf
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (rminimum(n;m;k.x[k]) ∈ ℝ) supposing n ≤ m
Proof
Definitions occuring in Statement :
rminimum: rminimum(n;m;k.x[k]),
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
nat: ℕ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rminimum: rminimum(n;m;k.x[k])
Lemmas referenced :
int_seg_wf,
int_seg_properties,
rmin_wf,
istype-less_than,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
intformless_wf,
decidable__lt,
istype-le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
subtract_wf,
real_wf,
primrec_wf
Rules used in proof :
inhabitedIsType,
isectIsTypeImplies,
functionIsType,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
imageElimination,
productElimination,
rename,
setElimination,
productIsType,
addEquality,
applyEquality,
universeIsType,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
dependent_functionElimination,
hypothesisEquality,
dependent_set_memberEquality_alt,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. (rminimum(n;m;k.x[k]) \mmember{} \mBbbR{}) supposing n \mleq{} m
Date html generated:
2019_11_06-PM-00_29_20
Last ObjectModification:
2019_11_05-AM-11_55_32
Theory : reals
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