Nuprl Lemma : iterated-conjugate2
∀[T:Type]. ∀[f,g,h:T ⟶ T].
(∀[n:ℕ]. (g o (f o h)^n = (g o (f^n o h)) ∈ (T ⟶ T))) supposing
((∀b:T. ((h (g b)) = b ∈ T)) and
(∀a:T. ((g (h a)) = a ∈ T)))
Proof
Definitions occuring in Statement :
fun_exp: f^n,
compose: f o g,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
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],
true: True,
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
ge: i ≥ j ,
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,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
istype-void,
compose_wf,
fun_exp_wf,
decidable__le,
subtract_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
istype-nat,
istype-universe,
nat_properties,
ge_wf,
istype-less_than,
primrec-unroll,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
subtract-1-ge-0,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
primrec_wf,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
natural_numberEquality,
isect_memberEquality_alt,
voidElimination,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
functionEquality,
hypothesisEquality,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsType,
equalityIsType1,
inhabitedIsType,
applyEquality,
instantiate,
isect_memberFormation_alt,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
axiomEquality,
functionIsTypeImplies,
functionExtensionality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
equalityElimination,
equalityIsType4,
baseApply,
closedConclusion,
promote_hyp,
cumulativity,
addEquality,
minusEquality,
isectIsTypeImplies
Latex:
\mforall{}[T:Type]. \mforall{}[f,g,h:T {}\mrightarrow{} T].
(\mforall{}[n:\mBbbN{}]. (g o (f o h)\^{}n = (g o (f\^{}n o h)))) supposing
((\mforall{}b:T. ((h (g b)) = b)) and
(\mforall{}a:T. ((g (h a)) = a)))
Date html generated:
2019_10_15-AM-11_19_13
Last ObjectModification:
2018_10_19-PM-01_30_20
Theory : general
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