Nuprl Lemma : rminimum_functionality_wrt_rleq
∀[n,m:ℤ].
  ∀[x,y:{n..m + 1-} ⟶ ℝ].
    rminimum(n;m;k.x[k]) ≤ rminimum(n;m;k.y[k]) supposing ∀k:ℤ. ((n ≤ k) 
⇒ (k ≤ m) 
⇒ (x[k] ≤ y[k])) 
  supposing n ≤ m
Proof
Definitions occuring in Statement : 
rminimum: rminimum(n;m;k.x[k])
, 
rleq: x ≤ y
, 
real: ℝ
, 
int_seg: {i..j-}
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
le: A ≤ B
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
add: n + m
, 
natural_number: $n
, 
int: ℤ
Definitions unfolded in proof : 
true: True
, 
less_than': less_than'(a;b)
, 
subtract: n - m
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
squash: ↓T
, 
less_than: a < b
, 
cand: A c∧ B
, 
bfalse: ff
, 
ifthenelse: if b then t else f fi 
, 
subtype_rel: A ⊆r B
, 
btrue: tt
, 
it: ⋅
, 
unit: Unit
, 
bool: 𝔹
, 
req_int_terms: t1 ≡ t2
, 
uiff: uiff(P;Q)
, 
rge: x ≥ y
, 
rev_uimplies: rev_uimplies(P;Q)
, 
lelt: i ≤ j < k
, 
int_seg: {i..j-}
, 
so_apply: x[s]
, 
le: A ≤ B
, 
rnonneg: rnonneg(x)
, 
rleq: x ≤ y
, 
sq_type: SQType(T)
, 
ge: i ≥ j 
, 
guard: {T}
, 
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
, 
rminimum: rminimum(n;m;k.x[k])
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
Lemmas referenced : 
subtract-add-cancel, 
int_seg_properties, 
rmin_wf, 
primrec_wf, 
rmin_functionality_wrt_rleq, 
subtype_rel_self, 
le-add-cancel, 
add-commutes, 
add-zero, 
zero-add, 
zero-mul, 
add-mul-special, 
minus-one-mul-top, 
add-swap, 
minus-one-mul, 
minus-add, 
add-associates, 
condition-implies-le, 
not-le-2, 
istype-false, 
le_reflexive, 
int_seg_subtype, 
subtype_rel_function, 
primrec-unroll, 
assert_of_le_int, 
bnot_of_lt_int, 
assert_functionality_wrt_uiff, 
eqff_to_assert, 
bnot_wf, 
le_wf, 
le_int_wf, 
assert_of_lt_int, 
eqtt_to_assert, 
less_than_wf, 
assert_wf, 
bool_wf, 
equal-wf-base, 
uiff_transitivity, 
lt_int_wf, 
real_term_value_const_lemma, 
real_term_value_var_lemma, 
real_term_value_sub_lemma, 
int-to-real_wf, 
real_polynomial_null, 
req-iff-rsub-is-0, 
rleq_weakening, 
rleq_weakening_equal, 
rleq_functionality_wrt_implies, 
real_wf, 
int_seg_wf, 
decidable__lt, 
rleq_wf, 
subtract-1-ge-0, 
primrec0_lemma, 
le_witness_for_triv, 
istype-less_than, 
ge_wf, 
int_formula_prop_less_lemma, 
intformless_wf, 
int_term_value_add_lemma, 
int_formula_prop_eq_lemma, 
itermAdd_wf, 
intformeq_wf, 
decidable__equal_int, 
nat_properties, 
int_subtype_base, 
subtype_base_sq, 
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
Rules used in proof : 
multiplyEquality, 
minusEquality, 
imageElimination, 
baseClosed, 
closedConclusion, 
baseApply, 
equalityElimination, 
isectIsTypeImplies, 
productIsType, 
addEquality, 
applyEquality, 
functionIsType, 
equalityIstype, 
functionIsTypeImplies, 
productElimination, 
intWeakElimination, 
because_Cache, 
rename, 
setElimination, 
applyLambdaEquality, 
equalitySymmetry, 
equalityTransitivity, 
intEquality, 
cumulativity, 
instantiate, 
lambdaFormation_alt, 
inhabitedIsType, 
universeIsType, 
independent_pairFormation, 
voidElimination, 
isect_memberEquality_alt, 
int_eqEquality, 
lambdaEquality_alt, 
dependent_pairFormation_alt, 
independent_functionElimination, 
approximateComputation, 
independent_isectElimination, 
unionElimination, 
natural_numberEquality, 
dependent_functionElimination, 
hypothesis, 
hypothesisEquality, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
dependent_set_memberEquality_alt, 
sqequalRule, 
thin, 
cut, 
introduction, 
isect_memberFormation_alt, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}[n,m:\mBbbZ{}].
    \mforall{}[x,y:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].
        rminimum(n;m;k.x[k])  \mleq{}  rminimum(n;m;k.y[k]) 
        supposing  \mforall{}k:\mBbbZ{}.  ((n  \mleq{}  k)  {}\mRightarrow{}  (k  \mleq{}  m)  {}\mRightarrow{}  (x[k]  \mleq{}  y[k])) 
    supposing  n  \mleq{}  m
Date html generated:
2019_11_06-PM-00_29_51
Last ObjectModification:
2019_11_05-PM-00_10_36
Theory : reals
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