Nuprl Lemma : mk_lambdas_fun-unroll-ite
∀[F:Top]. ∀[m:ℕ].
(mk_lambdas_fun(F;m) ~ if (m =z 0) then F (λx.x) else λx.mk_lambdas_fun(λg.(F (λf.(g (f x))));m - 1) fi )
Proof
Definitions occuring in Statement :
mk_lambdas_fun: mk_lambdas_fun(F;m),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
lambda: λx.A[x],
subtract: n - m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
mk_lambdas_fun: mk_lambdas_fun(F;m),
mk_lambdas-fun: mk_lambdas-fun(F;G;n;m),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
assert: ↑b,
false: False,
not: ¬A,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
nat_plus: ℕ+,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
true: True
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
btrue_wf,
assert_of_le_int,
eqff_to_assert,
le_int_wf,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
le_wf,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
nequal-le-implies,
zero-add,
mk_lambdas_fun-unroll-first,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
le-add-cancel,
less_than_wf,
nat_wf,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
sqequalRule,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
voidElimination,
lambdaEquality,
isect_memberEquality,
voidEquality,
computeAll,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
applyEquality,
sqequalAxiom
Latex:
\mforall{}[F:Top]. \mforall{}[m:\mBbbN{}].
(mk\_lambdas\_fun(F;m) \msim{} if (m =\msubz{} 0)
then F (\mlambda{}x.x)
else \mlambda{}x.mk\_lambdas\_fun(\mlambda{}g.(F (\mlambda{}f.(g (f x))));m - 1)
fi )
Date html generated:
2017_10_01-AM-08_40_48
Last ObjectModification:
2017_07_26-PM-04_28_16
Theory : untyped!computation
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