Nuprl Lemma : qsum_functionality_wrt_qle
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℚ].
Σn ≤ k < m. x[k] ≤ Σn ≤ k < m. y[k] supposing ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] ≤ y[k]))
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
rationals: ℚ,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
bfalse: ff,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
qle: r ≤ s,
grp_leq: a ≤ b,
infix_ap: x f y,
grp_le: ≤b,
pi1: fst(t),
pi2: snd(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
lt_int: i <z j,
qeq: qeq(r;s),
eq_int: (i =z j),
true: True,
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b
Lemmas referenced :
lt_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
less_than_wf,
eqtt_to_assert,
assert_of_lt_int,
le_int_wf,
le_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
qle_witness,
qsum_wf,
int_seg_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
all_wf,
qle_wf,
intformle_wf,
int_formula_prop_le_lemma,
rationals_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
qsum_unroll,
decidable__le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
nat_wf,
nat_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
ge_wf,
subtype_rel_dep_function,
int_seg_subtype,
subtype_rel_self,
qadd_wf,
qle_functionality_wrt_implies,
qadd_functionality_wrt_qle,
qle_weakening_eq_qorder,
qle_reflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_functionElimination,
lambdaEquality,
functionExtensionality,
addEquality,
natural_numberEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality,
promote_hyp,
instantiate,
cumulativity,
applyLambdaEquality,
intWeakElimination
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x,y:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbQ{}].
\mSigma{}n \mleq{} k < m. x[k] \mleq{} \mSigma{}n \mleq{} k < m. y[k] supposing \mforall{}k:\mBbbZ{}. ((n \mleq{} k) {}\mRightarrow{} (k \mleq{} m) {}\mRightarrow{} (x[k] \mleq{} y[k]))
Date html generated:
2018_05_22-AM-00_02_37
Last ObjectModification:
2017_07_26-PM-06_50_58
Theory : rationals
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