### Nuprl Lemma : permutation_transitivity

`∀[A:Type]. ∀as,bs,cs:A List.  (permutation(A;as;bs) `` permutation(A;bs;cs) `` permutation(A;as;cs))`

Proof

Definitions occuring in Statement :  permutation: `permutation(T;L1;L2)` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` universe: `Type`
Definitions unfolded in proof :  permutation: `permutation(T;L1;L2)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` exists: `∃x:A. B[x]` and: `P ∧ Q` member: `t ∈ T` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` cand: `A c∧ B` compose: `f o g` inject: `Inj(A;B;f)` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b`
Lemmas referenced :  permute_list_length equal_wf length_wf list_wf compose_wf int_seg_wf subtype_rel_dep_function int_seg_subtype false_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf subtype_rel_self int_seg_properties decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf squash_wf true_wf permute_permute_list permute_list_wf iff_weakening_equal inject_wf exists_wf and_wf le_wf subtype_rel_weakening ext-eq_weakening
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin cut hypothesis introduction extract_by_obid isectElimination hypothesisEquality isect_memberEquality voidElimination voidEquality because_Cache hyp_replacement equalitySymmetry applyLambdaEquality intEquality cumulativity promote_hyp dependent_set_memberEquality equalityTransitivity dependent_pairFormation natural_numberEquality applyEquality lambdaEquality independent_isectElimination independent_pairFormation dependent_functionElimination unionElimination int_eqEquality computeAll functionExtensionality setElimination rename imageElimination equalityUniverse levelHypothesis imageMemberEquality baseClosed universeEquality independent_functionElimination productEquality functionEquality instantiate

Latex:
\mforall{}[A:Type].  \mforall{}as,bs,cs:A  List.    (permutation(A;as;bs)  {}\mRightarrow{}  permutation(A;bs;cs)  {}\mRightarrow{}  permutation(A;as;cs))

Date html generated: 2017_04_17-AM-08_11_11
Last ObjectModification: 2017_02_27-PM-04_38_18

Theory : list_1

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