Nuprl Lemma : pair_support
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[m,k:ℕn].
(Σ(f[x] | x < n) = (f[m] + f[k]) ∈ ℤ) supposing
((∀x:ℕn. ((¬(x = m ∈ ℤ)) ⇒ (¬(x = k ∈ ℤ)) ⇒ (f[x] = 0 ∈ ℤ))) and
(¬(m = k ∈ ℤ)))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int_seg: {i..j-},
prop: ℙ,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
squash: ↓T,
nequal: a ≠ b ∈ T ,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P)
Lemmas referenced :
not_wf,
equal_wf,
equal-wf-T-base,
all_wf,
int_seg_wf,
nat_wf,
isolate_summand,
singleton_support_sum,
ifthenelse_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
eq_int_eq_false,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
bfalse_wf,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
satisfiable-full-omega-tt,
add-is-int-iff,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
hypothesis,
sqequalRule,
Error :functionIsType,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
setElimination,
rename,
because_Cache,
applyEquality,
baseClosed,
isect_memberEquality,
axiomEquality,
natural_numberEquality,
lambdaEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
imageElimination,
universeEquality,
imageMemberEquality,
approximateComputation,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll,
closedConclusion,
baseApply,
pointwiseFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m,k:\mBbbN{}n].
(\mSigma{}(f[x] | x < n) = (f[m] + f[k])) supposing
((\mforall{}x:\mBbbN{}n. ((\mneg{}(x = m)) {}\mRightarrow{} (\mneg{}(x = k)) {}\mRightarrow{} (f[x] = 0))) and
(\mneg{}(m = k)))
Date html generated:
2019_06_20-PM-02_29_27
Last ObjectModification:
2018_09_26-PM-06_03_38
Theory : num_thy_1
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