Nuprl Lemma : iterated-conjugate
∀[T:Type]. ∀[f,g,h:T ⟶ T]. ∀[m:ℕ]. ((λf.(g o (f o h))^m f) = (g^m o (f o h^m)) ∈ (T ⟶ T))
Proof
Definitions occuring in Statement :
fun_exp: f^n,
compose: f o g,
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
fun_exp: f^n,
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
compose: f o g,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_wf,
nat_properties,
full-omega-unsat,
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,
primrec-unroll,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
compose_wf,
fun_exp_wf,
le_wf,
subtract-add-cancel,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
fun_exp_add1-sq2,
fun_exp_add1-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesisEquality,
because_Cache,
cut,
introduction,
extract_by_obid,
hypothesis,
functionEquality,
isect_memberFormation,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
sqequalRule,
independent_pairFormation,
axiomEquality,
functionExtensionality,
applyEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
promote_hyp,
instantiate,
cumulativity,
dependent_set_memberEquality,
addEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[f,g,h:T {}\mrightarrow{} T]. \mforall{}[m:\mBbbN{}]. ((\mlambda{}f.(g o (f o h))\^{}m f) = (g\^{}m o (f o h\^{}m)))
Date html generated:
2018_05_21-PM-08_17_19
Last ObjectModification:
2018_05_19-PM-04_55_41
Theory : general
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