Nuprl Lemma : frs-non-dec-sorted-by
∀[p:ℝ List]. (frs-non-dec(p) ⇐⇒ sorted-by(λx,y. (x ≤ y);p))
Proof
Definitions occuring in Statement :
frs-non-dec: frs-non-dec(L),
rleq: x ≤ y,
real: ℝ,
sorted-by: sorted-by(R;L),
list: T List,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
lambda: λx.A[x]
Definitions unfolded in proof :
sorted-by: sorted-by(R;L),
frs-non-dec: frs-non-dec(L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
sq_type: SQType(T),
rleq: x ≤ y,
rnonneg: rnonneg(x),
nat_plus: ℕ+,
subtype_rel: A ⊆r B
Lemmas referenced :
rleq_weakening_equal,
list_wf,
nat_plus_wf,
nat_plus_properties,
rsub_wf,
less_than'_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
int_term_value_constant_lemma,
itermConstant_wf,
select_wf,
rleq_wf,
le_wf,
all_wf,
int_seg_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
real_wf,
length_wf,
int_seg_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality,
hypothesisEquality,
productElimination,
lemma_by_obid,
isectElimination,
natural_numberEquality,
unionElimination,
imageElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
because_Cache,
functionEquality,
instantiate,
cumulativity,
independent_pairEquality,
applyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[p:\mBbbR{} List]. (frs-non-dec(p) \mLeftarrow{}{}\mRightarrow{} sorted-by(\mlambda{}x,y. (x \mleq{} y);p))
Date html generated:
2016_05_18-AM-08_53_24
Last ObjectModification:
2016_01_17-AM-02_32_24
Theory : reals
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