Nuprl Lemma : funinv-compose
∀[n:ℕ]. ∀[f,g:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ]. (inv(f o g) = (inv(g) o inv(f)) ∈ {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} )
Proof
Definitions occuring in Statement :
funinv: inv(f),
inject: Inj(A;B;f),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
prop: ℙ,
squash: ↓T,
compose: f o g,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
and: P ∧ Q,
label: ...$L... t,
guard: {T},
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T)
Lemmas referenced :
compose-injections,
int_seg_wf,
inject_wf,
nat_wf,
funinv_wf2,
equal_wf,
squash_wf,
true_wf,
istype-universe,
funinv-property,
int_seg_properties,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
le_wf,
less_than_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
subtype_rel_self,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
Error :inhabitedIsType,
sqequalRule,
Error :isect_memberEquality_alt,
axiomEquality,
Error :setIsType,
Error :functionIsType,
Error :universeIsType,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
Error :dependent_set_memberEquality_alt,
Error :functionExtensionality_alt,
dependent_functionElimination,
applyEquality,
Error :lambdaEquality_alt,
independent_functionElimination,
universeEquality,
productElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
Error :productIsType,
instantiate,
cumulativity,
intEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f,g:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} ]. (inv(f o g) = (inv(g) o inv(f)))
Date html generated:
2019_06_20-PM-01_17_42
Last ObjectModification:
2018_10_07-AM-00_37_00
Theory : int_2
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