Nuprl Lemma : finite-injection
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ T. (Surj(ℕn;T;s) ⇒ (∀f:T ⟶ T. ∀x:T. ∃m:ℕ+n + 1. ((f^m x) = x ∈ T) supposing Inj(T;T;f)))))
Proof
Definitions occuring in Statement :
fun_exp: f^n,
surject: Surj(A;B;f),
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
inject: Inj(A;B;f),
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
so_apply: x[s],
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
guard: {T},
true: True,
rev_implies: P ⇐ Q,
cand: A c∧ B,
subtract: n - m,
sq_type: SQType(T)
Lemmas referenced :
surject-inverse,
inject_wf,
surject_wf,
int_seg_wf,
istype-nat,
decidable_wf,
equal_wf,
istype-universe,
decidable__exists_int_seg,
fun_exp_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_var_lemma,
int_formula_prop_wf,
istype-le,
int_seg_subtype_nat,
istype-false,
set_subtype_base,
lelt_wf,
int_subtype_base,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
decidable__lt,
subtract_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
int_term_value_add_lemma,
istype-less_than,
fun_exp-injection,
fun_exp_add_apply,
minus-one-mul,
add-commutes,
add-associates,
add-mul-special,
zero-mul,
zero-add,
subtype_base_sq,
add-zero,
add-swap,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
pigeon-hole
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
rename,
extract_by_obid,
isectElimination,
because_Cache,
productElimination,
independent_functionElimination,
universeIsType,
functionIsType,
natural_numberEquality,
setElimination,
instantiate,
universeEquality,
addEquality,
applyEquality,
dependent_set_memberEquality_alt,
imageElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
equalityIstype,
intEquality,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
productIsType,
cumulativity
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T.
(Surj(\mBbbN{}n;T;s) {}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T. \mforall{}x:T. \mexists{}m:\mBbbN{}\msupplus{}n + 1. ((f\^{}m x) = x) supposing Inj(T;T;f)))))
Date html generated:
2020_05_19-PM-09_43_47
Last ObjectModification:
2020_01_04-PM-08_18_24
Theory : list_1
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