Nuprl Lemma : radd-list_functionality_wrt_rleq
∀[L1,L2:ℝ List]. radd-list(L1) ≤ radd-list(L2) supposing (||L1|| = ||L2|| ∈ ℤ) ∧ (∀i:ℕ||L1||. (L1[i] ≤ L2[i]))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
radd-list: radd-list(L),
real: ℝ,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
so_apply: x[s],
subtype_rel: A ⊆r B,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
real: ℝ,
ge: i ≥ j ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
subtract: n - m,
less_than: a < b,
true: True,
squash: ↓T,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rge: x ≥ y,
less_than': less_than'(a;b),
nat_plus: ℕ+,
cons: [a / b]
Lemmas referenced :
list_induction,
real_wf,
uall_wf,
list_wf,
isect_wf,
equal_wf,
length_wf,
all_wf,
int_seg_wf,
rleq_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
radd-list_wf-bag,
list-subtype-bag,
subtype_rel_self,
equal-wf-base-T,
nil_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
radd_list_nil_lemma,
rleq_weakening_equal,
int-to-real_wf,
less_than'_wf,
rsub_wf,
nat_plus_wf,
equal-wf-base,
length_of_cons_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
cons_wf,
add-is-int-iff,
false_wf,
equal-wf-T-base,
rleq_functionality,
radd_wf,
radd-list-cons,
decidable__equal_int,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
squash_wf,
le_wf,
less_than_wf,
add-subtract-cancel,
subtype_base_sq,
int_subtype_base,
true_wf,
select_cons_tl,
iff_weakening_equal,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
add_nat_plus,
length_wf_nat,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
intEquality,
because_Cache,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
independent_functionElimination,
baseClosed,
lambdaFormation,
independent_pairEquality,
minusEquality,
axiomEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addEquality,
dependent_set_memberEquality,
hyp_replacement,
imageElimination,
cumulativity,
universeEquality,
imageMemberEquality,
instantiate,
applyLambdaEquality
Latex:
\mforall{}[L1,L2:\mBbbR{} List].
radd-list(L1) \mleq{} radd-list(L2) supposing (||L1|| = ||L2||) \mwedge{} (\mforall{}i:\mBbbN{}||L1||. (L1[i] \mleq{} L2[i]))
Date html generated:
2017_10_03-AM-08_26_13
Last ObjectModification:
2017_07_28-AM-07_24_12
Theory : reals
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