Nuprl Lemma : swap_select
∀[T:Type]. ∀[L:T List]. ∀[i,j,x:ℕ||L||]. (swap(L;i;j)[x] = L[(i, j) x] ∈ T)
Proof
Definitions occuring in Statement :
swap: swap(L;i;j),
flip: (i, j),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
swap: swap(L;i;j),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ
Lemmas referenced :
permute_list_select,
flip_wf,
int_seg_properties,
decidable__le,
length_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
istype-le,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
productElimination,
imageElimination,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
because_Cache
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[i,j,x:\mBbbN{}||L||]. (swap(L;i;j)[x] = L[(i, j) x])
Date html generated:
2020_05_20-AM-07_48_56
Last ObjectModification:
2019_12_26-PM-04_47_15
Theory : list!
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