Nuprl Lemma : natrec_wf_intseg
∀[k:ℤ]. ∀[T:{k...} ⟶ Type]. ∀[g:n:{k...} ⟶ (m:{k..n-} ⟶ T[m]) ⟶ T[n]]. (letrec f(n)=g[n;f] in f ∈ n:{k...} ⟶ T[n])
Proof
Definitions occuring in Statement :
natrec: natrec,
int_upper: {i...},
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
natrec: natrec,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_upper: {i...},
so_apply: x[s],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
uimplies: b supposing a,
lelt: i ≤ j < k,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
genrec: genrec,
so_apply: x[s1;s2],
sq_stable: SqStable(P),
uiff: uiff(P;Q),
subtract: n - m,
squash: ↓T,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
sq_type: SQType(T),
nat_plus: ℕ+,
less_than: a < b
Lemmas referenced :
int_upper_properties,
mul-associates,
mul-distributes,
omega-shadow,
mul-distributes-right,
two-mul,
one-mul,
minus-zero,
zero-mul,
add-mul-special,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
nat_wf,
subtype_rel_self,
not-le-2,
le_reflexive,
int_seg_subtype,
subtype_rel_dep_function,
add-zero,
minus-minus,
not-ge-2,
false_wf,
subtract_wf,
decidable__le,
le-add-cancel,
add-commutes,
add_functionality_wrt_le,
zero-add,
add-associates,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
sq_stable__le,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
le_wf,
lelt_wf,
subtype_rel_sets,
int_seg_wf,
int_upper_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,
hypothesisEquality,
because_Cache,
setElimination,
rename,
applyEquality,
intEquality,
lambdaEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
productElimination,
isect_memberEquality,
cumulativity,
universeEquality,
intWeakElimination,
natural_numberEquality,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
addEquality,
minusEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
independent_pairFormation,
voidEquality,
dependent_set_memberEquality,
instantiate,
multiplyEquality
Latex:
\mforall{}[k:\mBbbZ{}]. \mforall{}[T:\{k...\} {}\mrightarrow{} Type]. \mforall{}[g:n:\{k...\} {}\mrightarrow{} (m:\{k..n\msupminus{}\} {}\mrightarrow{} T[m]) {}\mrightarrow{} T[n]].
(letrec f(n)=g[n;f] in
f \mmember{} n:\{k...\} {}\mrightarrow{} T[n])
Date html generated:
2016_05_13-PM-04_03_14
Last ObjectModification:
2016_01_14-PM-07_24_45
Theory : int_1
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