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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