Nuprl Lemma : Legendre-roots-rless
∀n:ℕ. ∀i,j:ℕn. (i < j ⇒ (Legendre-root(n;i) < Legendre-root(n;j)))
Proof
Definitions occuring in Statement :
Legendre-root: Legendre-root(n;i),
rless: x < y,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
Legendre-root: Legendre-root(n;i),
Legendre-roots-ext,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
uimplies: b supposing a,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtract: n - m,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
sq_stable: SqStable(P),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sequence: sequence(T),
trans: Trans(T;x,y.E[x; y]),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
seq-item: s[i],
seq-len: ||s||,
pi1: fst(t),
pi2: snd(t),
sorted-seq: sorted-seq(x,y.R[x; y];s),
squash: ↓T
Lemmas referenced :
Legendre-roots-ext,
subtype_rel_self,
sq_exists_wf,
rless_wf,
rmul_wf,
int-to-real_wf,
exp_wf2,
Legendre_wf,
all_wf,
add-member-int_seg2,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
subtract_wf,
istype-less_than,
int_seg_wf,
real_wf,
i-member_wf,
rooint_wf,
req_wf,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
itermAdd_wf,
int_term_value_add_lemma,
sq_stable__rless,
member_rooint_lemma,
sorted-seq-iff,
rless_transitivity2,
rleq_weakening_rless,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
because_Cache,
setEquality,
productEquality,
lambdaEquality_alt,
setElimination,
rename,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
independent_isectElimination,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
functionIsType,
setIsType,
minusEquality,
addEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_pairEquality_alt,
functionExtensionality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i,j:\mBbbN{}n. (i < j {}\mRightarrow{} (Legendre-root(n;i) < Legendre-root(n;j)))
Date html generated:
2019_10_31-AM-06_20_14
Last ObjectModification:
2019_01_19-PM-00_54_20
Theory : reals_2
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