Nuprl Lemma : mul-initial-seg-step
∀[f:ℕ ⟶ ℕ]. ∀[m:ℕ+]. ((mul-initial-seg(f) m) = ((mul-initial-seg(f) (m - 1)) * (f (m - 1))) ∈ ℤ)
Proof
Definitions occuring in Statement :
mul-initial-seg: mul-initial-seg(f)
,
nat_plus: ℕ+
,
nat: ℕ
,
uall: ∀[x:A]. B[x]
,
apply: f a
,
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
,
all: ∀x:A. B[x]
,
nat_plus: ℕ+
,
implies: P
⇒ Q
,
prop: ℙ
,
nat: ℕ
,
decidable: Dec(P)
,
or: P ∨ Q
,
uimplies: b supposing a
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
false: False
,
not: ¬A
,
top: Top
,
and: P ∧ Q
,
subtype_rel: A ⊆r B
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
mul-initial-seg: mul-initial-seg(f)
,
subtract: n - m
,
upto: upto(n)
,
from-upto: [n, m)
,
ifthenelse: if b then t else f fi
,
lt_int: i <z j
,
bfalse: ff
,
btrue: tt
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
uiff: uiff(P;Q)
,
true: True
,
append: as @ bs
,
so_lambda: so_lambda(x,y,z.t[x; y; z])
,
so_apply: x[s1;s2;s3]
,
ge: i ≥ j
,
squash: ↓T
,
guard: {T}
Lemmas referenced :
nat_plus_properties,
equal_wf,
mul-initial-seg_wf,
nat_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_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,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
nat_plus_wf,
map_nil_lemma,
reduce_nil_lemma,
map_cons_lemma,
reduce_cons_lemma,
decidable__equal_int,
false_wf,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
add-subtract-cancel,
upto_decomp1,
decidable__lt,
not-lt-2,
less-iff-le,
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,
less_than_wf,
map_append_sq,
map_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
subtype_rel_self,
upto_wf,
list_wf,
list_induction,
all_wf,
reduce_wf,
append_wf,
cons_wf,
nil_wf,
list_ind_nil_lemma,
nat_properties,
list_ind_cons_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
mul-associates
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
intEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
multiplyEquality,
axiomEquality,
functionEquality,
callbyvalueReduce,
sqleReflexivity,
addEquality,
productElimination,
independent_functionElimination,
minusEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[m:\mBbbN{}\msupplus{}]. ((mul-initial-seg(f) m) = ((mul-initial-seg(f) (m - 1)) * (f (m - 1))))
Date html generated:
2018_05_21-PM-08_37_33
Last ObjectModification:
2017_07_26-PM-06_01_51
Theory : general
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