Nuprl Lemma : simple-primrec-add
∀[b,F:Top]. ∀[n,m:ℕ]. (primrec(n + m;b;λi.F) ~ primrec(n;primrec(m;b;λi.F);λi.F))
Proof
Definitions occuring in Statement :
primrec: primrec(n;b;c),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
lambda: λx.A[x],
add: n + m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
top: Top,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
primrec0_lemma,
zero-add,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
nat_wf,
top_wf,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
le_weakening,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
iff_weakening_uiff,
assert_of_bnot,
minus-zero,
add-mul-special,
zero-mul,
not-equal-implies-less,
subtype_rel_self,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
omega-shadow,
mul-associates,
general_arith_equation1,
primrec-unroll,
uiff_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
because_Cache,
voidEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
applyEquality,
intEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
multiplyEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseApply,
closedConclusion
Latex:
\mforall{}[b,F:Top]. \mforall{}[n,m:\mBbbN{}]. (primrec(n + m;b;\mlambda{}i.F) \msim{} primrec(n;primrec(m;b;\mlambda{}i.F);\mlambda{}i.F))
Date html generated:
2017_04_14-AM-07_25_50
Last ObjectModification:
2017_02_27-PM-02_55_37
Theory : call!by!value_2
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