Nuprl Lemma : list-injection
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀L:T List. ∀f:{x:T| (x ∈ L)} ⟶ {x:T| (x ∈ L)} .
∀x:{x:T| (x ∈ L)} . ∃m:{1..||L|| + 1-}. ((f^m x) = x ∈ {x:T| (x ∈ L)} ) supposing Inj({x:T| (x ∈ L)} ;{x:T| (x ∈\000C L)} ;f)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
length: ||as||,
list: T List,
fun_exp: f^n,
inject: Inj(A;B;f),
int_seg: {i..j-},
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,
set: {x:A| B[x]} ,
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),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
surject: Surj(A;B;f),
sq_stable: SqStable(P),
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
cand: A c∧ B,
ge: i ≥ j
Lemmas referenced :
equal_wf,
l_member_wf,
finite-injection,
decidable__equal_set,
length_wf_nat,
select_wf,
list-subtype,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
length_wf,
set_wf,
inject_wf,
list_wf,
all_wf,
decidable_wf,
sq_stable__l_member,
lelt_wf,
select_member,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
setEquality,
cumulativity,
applyEquality,
functionExtensionality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
functionEquality,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}L:T List. \mforall{}f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \{x:T| (x \mmember{} L)\} .
\mforall{}x:\{x:T| (x \mmember{} L)\} . \mexists{}m:\{1..||L|| + 1\msupminus{}\}. ((f\^{}m x) = x) supposing Inj(\{x:T| (x \mmember{} L)\} ;\{x:T| (x\000C \mmember{} L)\} ;f)))
Date html generated:
2017_04_17-AM-07_47_32
Last ObjectModification:
2017_02_27-PM-04_18_28
Theory : list_1
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