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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