Nuprl Lemma : sum-as-primrec
∀[k:ℕ]. ∀[f:ℕk ⟶ ℤ]. (Σ(f[x] | x < k) ~ primrec(k;0;λj,x. (x + f[j])))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
primrec: primrec(n;b;c),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sum: Σ(f[x] | x < k),
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
so_apply: x[s],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
lelt: i ≤ j < k,
true: True,
so_lambda: λ2x.t[x]
Lemmas referenced :
sum_aux-as-primrec,
lelt_wf,
int_seg_properties,
false_wf,
int_seg_subtype,
zero-add,
le_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
itermSubtract_wf,
intformeq_wf,
subtract_wf,
decidable__equal_int,
true_wf,
squash_wf,
primrec_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
nat_wf,
int_seg_wf,
int_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
functionEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
intEquality,
sqequalRule,
isect_memberEquality,
lambdaEquality,
applyEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
universeEquality,
dependent_set_memberEquality,
functionExtensionality,
addEquality,
lambdaFormation,
setEquality,
productElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}(f[x] | x < k) \msim{} primrec(k;0;\mlambda{}j,x. (x + f[j])))
Date html generated:
2016_05_14-AM-07_31_20
Last ObjectModification:
2016_01_14-PM-09_56_40
Theory : int_2
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