Nuprl Lemma : natrec_wf
∀[T:ℕ ⟶ Type]. ∀[g:n:ℕ ⟶ (m:ℕn ⟶ T[m]) ⟶ T[n]]. (letrec f(n)=g[n;f] in f ∈ n:ℕ ⟶ T[n])
Proof
Definitions occuring in Statement :
natrec: natrec,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
natrec: natrec,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
ge: i ≥ j ,
guard: {T},
genrec: genrec,
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
true: True,
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
subtype_rel_self,
sq_stable__le,
not-le-2,
int_seg_subtype,
subtype_rel_dep_function,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-ge-2,
subtract_wf,
decidable__le,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lemma_by_obid,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
isect_memberEquality,
because_Cache,
cumulativity,
universeEquality,
intWeakElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
productElimination,
unionElimination,
addEquality,
voidEquality,
intEquality,
minusEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[T:\mBbbN{} {}\mrightarrow{} Type]. \mforall{}[g:n:\mBbbN{} {}\mrightarrow{} (m:\mBbbN{}n {}\mrightarrow{} T[m]) {}\mrightarrow{} T[n]]. (letrec f(n)=g[n;f] in f \mmember{} n:\mBbbN{} {}\mrightarrow{} T[n])
Date html generated:
2016_05_13-PM-04_03_11
Last ObjectModification:
2016_01_14-PM-07_24_29
Theory : int_1
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