Nuprl Lemma : seq-add_wf
∀[T:Type]. ∀[n:ℕ]. ∀[s:ℕn ⟶ T]. ∀[x:T]. (s.x@n ∈ ℕn + 1 ⟶ T)
Proof
Definitions occuring in Statement :
seq-add: s.x@n,
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-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
false: False,
implies: P ⇒ Q,
not: ¬A,
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
less_than: a < b
Lemmas referenced :
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
less-iff-le,
condition-implies-le,
add-associates,
nat_wf,
minus-add,
minus-one-mul,
minus-one-mul-top,
zero-add,
le-add-cancel2,
and_wf,
le_wf,
less_than_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
int_eqEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
lemma_by_obid,
dependent_functionElimination,
unionElimination,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
addEquality,
natural_numberEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[s:\mBbbN{}n {}\mrightarrow{} T]. \mforall{}[x:T]. (s.x@n \mmember{} \mBbbN{}n + 1 {}\mrightarrow{} T)
Date html generated:
2016_05_13-PM-03_48_28
Last ObjectModification:
2015_12_26-AM-10_18_16
Theory : bar-induction
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