Nuprl Lemma : no_repeats_l_index-inj
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. Inj({x:T| (x ∈ L)} ;ℕ||L||;λx.index(L;x)) supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
l_index: index(L;x),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
length: ||as||,
list: T List,
deq: EqDecider(T),
inject: Inj(A;B;f),
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
lambda: λx.A[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
nat: ℕ,
cand: A c∧ B,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
subtype_rel: A ⊆r B,
squash: ↓T,
le: A ≤ B
Lemmas referenced :
no_repeats_wf,
list_wf,
deq_wf,
l_member_wf,
set_wf,
l_index_wf,
length_wf,
int_seg_wf,
equal_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
nat_properties,
select_wf,
less_than_wf,
nat_wf,
exists_wf,
le_wf,
squash_wf,
lelt_wf,
no_repeats_l_index
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :universeIsType,
extract_by_obid,
isectElimination,
universeEquality,
lambdaEquality,
because_Cache,
independent_isectElimination,
cumulativity,
natural_numberEquality,
rename,
setElimination,
lambdaFormation,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
unionElimination,
productEquality,
independent_pairFormation,
dependent_pairFormation,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
productElimination,
baseClosed,
imageMemberEquality,
imageElimination,
applyEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List].
Inj(\{x:T| (x \mmember{} L)\} ;\mBbbN{}||L||;\mlambda{}x.index(L;x)) supposing no\_repeats(T;L)
Date html generated:
2019_06_20-PM-01_57_00
Last ObjectModification:
2018_10_06-AM-11_23_40
Theory : decidable!equality
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