Nuprl Lemma : comparison-sort-permutation
∀T:Type. (valueall-type(T) ⇒ (∀cmp:comparison(T). ∀L:T List. permutation(T;comparison-sort(cmp;L);L)))
Proof
Definitions occuring in Statement :
comparison-sort: comparison-sort(cmp;L),
comparison: comparison(T),
permutation: permutation(T;L1;L2),
list: T List,
valueall-type: valueall-type(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
comparison-sort: comparison-sort(cmp;L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
top: Top,
eager-accum: eager-accum(x,a.f[x; a];y;l),
list_ind: list_ind,
nil: [],
it: ⋅,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
append: as @ bs,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
all_wf,
list_wf,
permutation_wf,
eager-accum_wf,
insert-no-combine_wf,
list-valueall-type,
append_wf,
append-nil,
subtype_rel_list,
top_wf,
list_ind_cons_lemma,
valueall-type-has-valueall,
evalall-reduce,
comparison_wf,
valueall-type_wf,
cons_wf,
nil_wf,
insert-no-combine-permutation,
permutation-nil,
permutation_functionality_wrt_permutation,
permutation_weakening,
permutation-rotate,
append_functionality_wrt_permutation,
append_assoc,
list_ind_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
functionEquality,
independent_isectElimination,
independent_functionElimination,
applyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
dependent_functionElimination,
callbyvalueReduce,
universeEquality,
productElimination
Latex:
\mforall{}T:Type
(valueall-type(T) {}\mRightarrow{} (\mforall{}cmp:comparison(T). \mforall{}L:T List. permutation(T;comparison-sort(cmp;L);L)))
Date html generated:
2016_05_14-PM-02_44_11
Last ObjectModification:
2015_12_26-PM-02_41_17
Theory : list_1
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