Nuprl Lemma : swap_adjacent_decomp
∀[A:Type]
∀i:ℕ. ∀L:A List.
∃X,Y:A List
((L = (X @ [L[i]; L[i + 1]] @ Y) ∈ (A List)) ∧ (swap(L;i;i + 1) = (X @ [L[i + 1]; L[i]] @ Y) ∈ (A List)))
supposing i + 1 < ||L||
Proof
Definitions occuring in Statement :
swap: swap(L;i;j),
select: L[n],
length: ||as||,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ge: i ≥ j ,
so_apply: x[s],
nat: ℕ,
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
listp: A List+,
sq_type: SQType(T),
select: L[n],
cons: [a / b],
subtract: n - m,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
all_wf,
list_wf,
isect_wf,
less_than_wf,
subtract_wf,
length_wf,
exists_wf,
equal_wf,
append_wf,
cons_wf,
select_wf,
subtract-add-cancel,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
nil_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
length-append,
length_of_cons_lemma,
non_neg_length,
set_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
member-less_than,
tl_wf,
false_wf,
list_extensionality,
listp_properties,
length_tl,
iff_weakening_equal,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
squash_wf,
true_wf,
subtype_rel_self,
subtype_base_sq,
int_subtype_base,
select_cons_tl,
select_tl,
lelt_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
swap_wf,
swap_length,
swapped_select,
and_wf,
list_decomp,
hd_wf,
length_wf_nat,
swap_cons,
add-subtract-cancel,
int_seg_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
rename,
setElimination,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
addEquality,
natural_numberEquality,
because_Cache,
productEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
productElimination,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
applyEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
instantiate,
minusEquality,
hyp_replacement
Latex:
\mforall{}[A:Type]
\mforall{}i:\mBbbN{}. \mforall{}L:A List.
\mexists{}X,Y:A List
((L = (X @ [L[i]; L[i + 1]] @ Y)) \mwedge{} (swap(L;i;i + 1) = (X @ [L[i + 1]; L[i]] @ Y)))
supposing i + 1 < ||L||
Date html generated:
2018_05_21-PM-06_21_09
Last ObjectModification:
2018_05_19-PM-05_34_38
Theory : list!
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