Nuprl Lemma : primrec_wf
∀[T:Type]. ∀[n:ℕ]. ∀[b:T]. ∀[c:ℕn ⟶ T ⟶ T]. (primrec(n;b;c) ∈ T)
Proof
Definitions occuring in Statement :
primrec: primrec(n;b;c),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
top: Top,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
primrec-unroll,
istype-void,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-assert,
not-lt-2,
int_seg_wf,
subtract-1-ge-0,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
add-mul-special,
zero-mul,
le-add-cancel-alt,
istype-le,
subtype_rel_dep_function,
subtype_rel_sets_simple,
and_wf,
le_wf,
le-add-cancel2,
subtype_rel_self,
istype-nat,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
Error :lambdaFormation_alt,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
Error :isect_memberEquality_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
unionElimination,
equalityElimination,
because_Cache,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
instantiate,
cumulativity,
Error :functionIsType,
applyEquality,
Error :dependent_set_memberEquality_alt,
addEquality,
minusEquality,
Error :productIsType,
functionEquality,
intEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[b:T]. \mforall{}[c:\mBbbN{}n {}\mrightarrow{} T {}\mrightarrow{} T]. (primrec(n;b;c) \mmember{} T)
Date html generated:
2019_06_20-AM-11_27_41
Last ObjectModification:
2019_01_28-PM-05_28_01
Theory : call!by!value_2
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