Nuprl Lemma : mul-list-insert-int
∀[ns:ℤ List]. ∀[x:ℤ]. (Π(insert-int(x;ns)) = (x * Π(ns) ) ∈ ℤ)
Proof
Definitions occuring in Statement :
mul-list: Π(ns) ,
insert-int: insert-int(x;l),
list: T List,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
implies: P ⇒ Q,
insert-int: insert-int(x;l),
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
mul-list: Π(ns) ,
reduce: reduce(f;k;as),
list_ind: list_ind,
cons: [a / b],
nil: [],
it: ⋅,
has-value: (a)↓,
prop: ℙ,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
uall_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
list_wf,
list_ind_nil_lemma,
mul_list_nil_lemma,
list_ind_cons_lemma,
reduce_cons_lemma,
value-type-has-value,
list-value-type,
insert-int_wf,
subtype_rel_self,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
mul-list_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
squash_wf,
true_wf,
iff_weakening_equal,
mul-swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
sqequalRule,
lambdaEquality,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
because_Cache,
independent_isectElimination,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
multiplyEquality,
natural_numberEquality,
lambdaFormation,
rename,
callbyvalueReduce,
axiomEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[ns:\mBbbZ{} List]. \mforall{}[x:\mBbbZ{}]. (\mPi{}(insert-int(x;ns)) = (x * \mPi{}(ns) ))
Date html generated:
2018_05_21-PM-06_57_24
Last ObjectModification:
2017_07_26-PM-04_59_41
Theory : general
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