Nuprl Lemma : C-comb_wf_funtype
∀[T:Type]. ∀[n:ℕ]. ∀[A:ℕn ⟶ Type].
C-comb() ∈ funtype(n;A;T) ⟶ funtype(n;λk.if (k =z 0) then A 1 if (k =z 1) then A 0 else A k fi ;T) supposing 2 ≤ n
Proof
Definitions occuring in Statement :
C-comb: C-comb(),
funtype: funtype(n;A;T),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
C-comb: C-comb(),
funtype: funtype(n;A;T),
top: Top,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
nequal: a ≠ b ∈ T ,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
subtype_rel: A ⊆r B,
subtract: n - m,
squash: ↓T,
true: True
Lemmas referenced :
primrec-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
le_wf,
nequal-le-implies,
zero-add,
subtract_wf,
int_upper_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
int_seg_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
subtype_rel-equal,
decidable__equal_int,
add-associates,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
zero-mul,
add-zero,
primrec_wf,
int_seg_properties,
funtype_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
setElimination,
rename,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
hypothesisEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
independent_pairFormation,
computeAll,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
applyEquality,
functionExtensionality,
functionEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type].
C-comb() \mmember{} funtype(n;A;T) {}\mrightarrow{} funtype(n;\mlambda{}k.if (k =\msubz{} 0) then A 1
if (k =\msubz{} 1) then A 0
else A k
fi ;T)
supposing 2 \mleq{} n
Date html generated:
2018_05_21-PM-08_02_33
Last ObjectModification:
2017_07_26-PM-05_39_08
Theory : general
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