Nuprl Lemma : fun-path-no_repeats
∀[T:Type]. ∀[f:T ⟶ T]. ∀[L:T List]. ∀[x,y:T]. no_repeats(T;L) supposing x=f*(y) via L supposing retraction(T;f)
Proof
Definitions occuring in Statement :
retraction: retraction(T;f),
fun-path: y=f*(x) via L,
no_repeats: no_repeats(T;l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
retraction: retraction(T;f),
exists: ∃x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_lambda: λ2x.t[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
so_apply: x[s1;s2;s3],
uiff: uiff(P;Q),
sq_type: SQType(T),
select: L[n],
cons: [a / b],
ge: i ≥ j ,
le: A ≤ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
l_before: x before y ∈ l,
sublist: L1 ⊆ L2,
increasing: increasing(f;k)
Lemmas referenced :
no_repeats_witness,
fun-path_wf,
list_wf,
retraction_wf,
fun-path-induction,
all_wf,
int_seg_wf,
length_wf,
less_than_wf,
select_wf,
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,
nat_wf,
length_of_cons_lemma,
length_of_nil_lemma,
cons_wf,
nil_wf,
itermAdd_wf,
int_term_value_add_lemma,
add-is-int-iff,
false_wf,
member-less_than,
not_wf,
equal_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
squash_wf,
true_wf,
select_cons_tl,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
iff_weakening_equal,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
le_weakening2,
no_repeats_iff,
length_wf_nat,
nat_properties,
l_before_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
cumulativity,
functionExtensionality,
applyEquality,
sqequalRule,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
dependent_functionElimination,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
lambdaFormation,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
instantiate,
imageMemberEquality,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T].
\mforall{}[L:T List]. \mforall{}[x,y:T]. no\_repeats(T;L) supposing x=f*(y) via L supposing retraction(T;f)
Date html generated:
2018_05_21-PM-07_47_31
Last ObjectModification:
2017_07_26-PM-05_25_26
Theory : general
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