Nuprl Lemma : seq-append_wf
∀[T:Type]. ∀[n,m:ℕ]. ∀[s1:ℕn ⟶ T]. ∀[s2:ℕm ⟶ T]. (seq-append(n;m;s1;s2) ∈ ℕn + m ⟶ T)
Proof
Definitions occuring in Statement :
seq-append: seq-append(n;m;s1;s2),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
seq-append: seq-append(n;m;s1;s2),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
lelt: i ≤ j < k,
sq_stable: SqStable(P),
uimplies: b supposing a,
subtract: n - m,
le: A ≤ B,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
less_than_wf,
int_seg_wf,
and_wf,
le_wf,
top_wf,
subtract_wf,
nat_wf,
sq_stable__and,
sq_stable__le,
sq_stable__less_than,
member-less_than,
squash_wf,
add-commutes,
minus-one-mul,
not-le-2,
add_functionality_wrt_le,
le_reflexive,
minus-one-mul-top,
add-associates,
minus-zero,
add-zero,
one-mul,
zero-add,
add-swap,
add-mul-special,
zero-mul,
not-lt-2,
two-mul,
mul-distributes-right,
omega-shadow,
mul-distributes,
mul-associates,
le-add-cancel,
less-iff-le,
minus-add,
mul-swap,
nat_properties,
decidable__le,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
because_Cache,
hypothesis,
lessCases,
independent_pairFormation,
isectElimination,
baseClosed,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
hypothesisEquality,
axiomSqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
imageElimination,
productElimination,
extract_by_obid,
independent_functionElimination,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
axiomEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_isectElimination,
multiplyEquality,
minusEquality,
intEquality,
unionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[s1:\mBbbN{}n {}\mrightarrow{} T]. \mforall{}[s2:\mBbbN{}m {}\mrightarrow{} T]. (seq-append(n;m;s1;s2) \mmember{} \mBbbN{}n + m {}\mrightarrow{} T)
Date html generated:
2019_06_20-AM-11_28_33
Last ObjectModification:
2018_08_20-PM-09_29_12
Theory : bar-induction
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