Nuprl Lemma : primrec-wf-nsub
∀[b:ℕ+]. ∀[P:ℕb ⟶ ℙ]. ∀[init:P[0]]. ∀[s:∀n:ℕb - 1. (P[n] ⇒ P[n + 1])]. ∀[n:ℕb]. (primrec(n;init;s) ∈ P[n])
Proof
Definitions occuring in Statement :
primrec: primrec(n;b;c),
int_seg: {i..j-},
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,
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,
nat_plus: ℕ+,
uimplies: b supposing a,
int_seg: {i..j-},
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P)
Lemmas referenced :
primrec-wf-int_seg,
subtype_rel-equal,
all_wf,
int_seg_wf,
subtract_wf,
decidable__lt,
istype-false,
not-lt-2,
less-iff-le,
condition-implies-le,
add-associates,
istype-void,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add_functionality_wrt_le,
add-commutes,
le-add-cancel2,
istype-le,
istype-less_than,
add-member-int_seg2,
decidable__le,
not-le-2,
zero-add,
add-zero,
add-member-int_seg1,
minus-zero,
sq_stable__le,
le-add-cancel,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
setElimination,
rename,
hypothesis,
independent_isectElimination,
independent_pairFormation,
imageElimination,
productElimination,
hypothesisEquality,
applyEquality,
closedConclusion,
natural_numberEquality,
sqequalRule,
Error :lambdaEquality_alt,
functionEquality,
Error :dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
Error :lambdaFormation_alt,
voidElimination,
independent_functionElimination,
addEquality,
minusEquality,
Error :isect_memberEquality_alt,
Error :productIsType,
Error :universeIsType,
imageMemberEquality,
baseClosed,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsType,
universeEquality
Latex:
\mforall{}[b:\mBbbN{}\msupplus{}]. \mforall{}[P:\mBbbN{}b {}\mrightarrow{} \mBbbP{}]. \mforall{}[init:P[0]]. \mforall{}[s:\mforall{}n:\mBbbN{}b - 1. (P[n] {}\mRightarrow{} P[n + 1])]. \mforall{}[n:\mBbbN{}b].
(primrec(n;init;s) \mmember{} P[n])
Date html generated:
2019_06_20-AM-11_27_37
Last ObjectModification:
2019_01_28-PM-05_24_18
Theory : call!by!value_2
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