Nuprl Lemma : fun_exp-mul
∀[T:Type]. ∀[f:T ⟶ T]. ∀[n,m:ℕ]. ∀[x:T]. ((f^n * m x) = (λx.(f^m x)^n x) ∈ T)
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
multiply: n * m,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fun_exp: f^n,
primrec: primrec(n;b;c),
compose: f o g,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
fun_exp0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
zero-mul,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
fun_exp_add-sq,
mul_bounds_1a,
le_wf,
equal_wf,
squash_wf,
true_wf,
fun_exp_wf,
iff_weakening_equal,
fun_exp_unroll,
eq_int_wf,
bool_wf,
equal-wf-base,
assert_wf,
bnot_wf,
not_wf,
compose_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
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,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
because_Cache,
unionElimination,
functionEquality,
cumulativity,
universeEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
multiplyEquality,
applyEquality,
imageElimination,
functionExtensionality,
imageMemberEquality,
baseClosed,
productElimination,
baseApply,
closedConclusion,
equalityElimination,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[x:T]. ((f\^{}n * m x) = (\mlambda{}x.(f\^{}m x)\^{}n x))
Date html generated:
2017_04_14-AM-09_13_12
Last ObjectModification:
2017_02_27-PM-03_50_17
Theory : int_2
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