Nuprl Lemma : genfact-step
∀[n:ℤ]. ∀[f:ℕ+ ⟶ ℤ]. ∀[b:ℤ]. (genfact(n;b;m.f[m]) = if n <z 1 then b else f[n] * genfact(n - 1;b;m.f[m]) fi ∈ ℤ)
Proof
Definitions occuring in Statement :
genfact: genfact(n;b;m.f[m]),
nat_plus: ℕ+,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
multiply: n * m,
subtract: 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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
genfact: genfact(n;b;m.f[m]),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtract: n - m,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
so_apply: x[s],
nat_plus: ℕ+,
decidable: Dec(P),
has-value: (a)↓,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
intformnot_wf,
int_formula_prop_not_lemma,
nat_plus_wf,
subtract-1-ge-0,
istype-top,
decidable__lt,
value-type-has-value,
int-value-type,
subtract_wf,
istype-nat,
decidable__le,
equal_wf,
istype-le,
genfact_wf,
iff_weakening_equal,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
squash_wf,
true_wf,
istype-universe,
genfact-base-linear,
subtype_rel_self
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
promote_hyp,
instantiate,
cumulativity,
Error :functionIsType,
lessCases,
axiomSqEquality,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
Error :dependent_set_memberEquality_alt,
callbyvalueReduce,
intEquality,
multiplyEquality,
closedConclusion,
universeEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[b:\mBbbZ{}].
(genfact(n;b;m.f[m]) = if n <z 1 then b else f[n] * genfact(n - 1;b;m.f[m]) fi )
Date html generated:
2019_06_20-PM-02_25_44
Last ObjectModification:
2019_02_08-AM-11_48_54
Theory : num_thy_1
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