Nuprl Lemma : super-fact-int-prod-exp
∀[k:ℕ]. ((k)!! = Π((k - i)^(i + 1) | i < k) ∈ ℤ)
Proof
Definitions occuring in Statement :
super-fact: (n)!!,
exp: i^n,
int-prod: Π(f[x] | x < k),
nat: ℕ,
uall: ∀[x:A]. B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
super-fact: (n)!!,
primrec: primrec(n;b;c),
int-prod: Π(f[x] | x < k),
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
sq_type: SQType(T),
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
exp: i^n
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int-prod_wf,
false_wf,
le_wf,
exp_wf2,
int_seg_properties,
subtract_wf,
int_seg_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
super-fact-unroll,
subtype_base_sq,
int_subtype_base,
nat_wf,
equal_wf,
squash_wf,
true_wf,
itermAdd_wf,
int_term_value_add_lemma,
int-prod-factor,
int_seg_subtype_nat,
iff_weakening_equal,
exp_step,
decidable__lt,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
add-subtract-cancel,
fact0_redex_lemma,
int_prod0_lemma,
int-prod-split,
lelt_wf,
fact_unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
primrec1_lemma,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma,
add-swap,
one-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
dependent_set_memberEquality,
because_Cache,
productElimination,
unionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
applyEquality,
imageElimination,
universeEquality,
addEquality,
imageMemberEquality,
baseClosed,
minusEquality,
multiplyEquality,
equalityElimination,
promote_hyp,
functionEquality
Latex:
\mforall{}[k:\mBbbN{}]. ((k)!! = \mPi{}((k - i)\^{}(i + 1) | i < k))
Date html generated:
2018_05_21-PM-01_04_52
Last ObjectModification:
2018_01_28-PM-02_13_18
Theory : num_thy_1
Home
Index