Nuprl Lemma : permutation-strong-subtype

`∀[A,B:Type].`
`  ∀L1:B List. ∀L2:A List.  (permutation(A;L1;L2) `` {(L2 ∈ B List) ∧ permutation(B;L1;L2)}) `
`  supposing strong-subtype(B;A)`

Proof

Definitions occuring in Statement :  permutation: `permutation(T;L1;L2)` list: `T List` strong-subtype: `strong-subtype(A;B)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` guard: `{T}` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` member: `t ∈ T` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` strong-subtype: `strong-subtype(A;B)` cand: `A c∧ B` guard: `{T}` prop: `ℙ` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` permutation: `permutation(T;L1;L2)`
Lemmas referenced :  strong-subtype_witness member-permutation subtype_rel_list permutation_wf list_wf strong-subtype_wf strong-subtype-implies list-subtype l_member_wf select_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf 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_wf inject_wf int_seg_wf length_wf equal_wf permute_list_wf strong-subtype-list
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_functionElimination hypothesis rename lambdaFormation because_Cache dependent_functionElimination applyEquality independent_isectElimination productElimination sqequalRule cumulativity universeEquality equalityTransitivity equalitySymmetry setEquality lambdaEquality setElimination natural_numberEquality unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll promote_hyp productEquality functionExtensionality

Latex:
\mforall{}[A,B:Type].
\mforall{}L1:B  List.  \mforall{}L2:A  List.    (permutation(A;L1;L2)  {}\mRightarrow{}  \{(L2  \mmember{}  B  List)  \mwedge{}  permutation(B;L1;L2)\})
supposing  strong-subtype(B;A)

Date html generated: 2017_04_17-AM-08_13_34
Last ObjectModification: 2017_02_27-PM-04_39_13

Theory : list_1

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