Nuprl Lemma : primrec-wf-nat-plus
∀[P:ℕ+ ⟶ ℙ]. ∀[b:P[1]]. ∀[s:∀n:ℕ+. (P[n] ⇒ P[n + 1])]. ∀[n:ℕ+]. (primrec(n - 1;b;λi,x. (s (i + 1) x)) ∈ P[n])
Proof
Definitions occuring in Statement :
primrec: primrec(n;b;c),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
int_upper: {i...},
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
top: Top,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
less_than: a < b,
squash: ↓T
Lemmas referenced :
all_wf,
le-add-cancel2,
add-swap,
not-le-2,
decidable__le,
add-zero,
add-associates,
minus-one-mul-top,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
subtype_rel_self,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
false_wf,
decidable__lt,
less_than_wf,
le_wf,
subtype_rel_sets,
int_upper_wf,
nat_plus_wf,
subtype_rel_dep_function,
primrec-wf-upper
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
natural_numberEquality,
isect_memberFormation,
hypothesis,
hypothesisEquality,
applyEquality,
instantiate,
cumulativity,
sqequalRule,
lambdaEquality,
universeEquality,
independent_isectElimination,
intEquality,
because_Cache,
setElimination,
rename,
setEquality,
lambdaFormation,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
isect_memberEquality,
voidEquality,
functionEquality,
dependent_set_memberEquality,
addEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
introduction,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[P:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[b:P[1]]. \mforall{}[s:\mforall{}n:\mBbbN{}\msupplus{}. (P[n] {}\mRightarrow{} P[n + 1])]. \mforall{}[n:\mBbbN{}\msupplus{}].
(primrec(n - 1;b;\mlambda{}i,x. (s (i + 1) x)) \mmember{} P[n])
Date html generated:
2016_05_13-PM-03_46_40
Last ObjectModification:
2016_01_14-PM-07_11_32
Theory : call!by!value_2
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