Nuprl Lemma : length-insert-int
∀[T:Type]. ∀[x:T]. ∀[l:T List]. (||insert-int(x;l)|| = (||l|| + 1) ∈ ℤ) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
length: ||as||,
insert-int: insert-int(x;l),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
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],
has-value: (a)↓,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
list_induction,
equal_wf,
length_wf,
insert-int_wf,
list_wf,
list_ind_nil_lemma,
length_of_nil_lemma,
length_of_cons_lemma,
list_ind_cons_lemma,
value-type-has-value,
list-value-type,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
intEquality,
cumulativity,
hypothesisEquality,
independent_isectElimination,
hypothesis,
addEquality,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
rename,
callbyvalueReduce,
applyEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[l:T List]. (||insert-int(x;l)|| = (||l|| + 1)) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2017_04_14-AM-08_36_08
Last ObjectModification:
2017_02_27-PM-03_28_33
Theory : list_0
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