### Nuprl Lemma : permutation-iff-count1

`∀[T:Type]`
`  ∀eq:T ⟶ T ⟶ 𝔹`
`    ((∀x,y:T.  (↑(eq x y) `⇐⇒` x = y ∈ T))`
`    `` (∀a1,b1:T List.  (∀x:T. (||filter(eq x;a1)|| = ||filter(eq x;b1)|| ∈ ℤ) `⇐⇒` permutation(T;a1;b1))))`

Proof

Definitions occuring in Statement :  permutation: `permutation(T;L1;L2)` length: `||as||` filter: `filter(P;l)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  rev_implies: `P `` Q` and: `P ∧ Q` iff: `P `⇐⇒` Q` nat: `ℕ` istype: `istype(T)` uimplies: `b supposing a` so_apply: `x[s]` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x.t[x]` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` top: `Top` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` ge: `i ≥ j ` false: `False` le: `A ≤ B` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bfalse: `ff` it: `⋅` unit: `Unit` bool: `𝔹` decidable: `Dec(P)` squash: `↓T` true: `True` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` deq: `EqDecider(T)` cand: `A c∧ B` permutation: `permutation(T;L1;L2)`
Lemmas referenced :  istype-universe istype-assert cons_wf permutation-nil nil_wf permutation_wf int_subtype_base istype-int le_wf set_subtype_base l_member_wf bool_wf subtype_rel_dep_function filter_wf5 length_wf_nat equal-wf-base list_wf list_induction filter_nil_lemma istype-void filter_cons_lemma length_of_nil_lemma bool_cases subtype_base_sq bool_subtype_base eqtt_to_assert length_of_cons_lemma non_neg_length full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_formula_prop_wf eqff_to_assert assert_of_bnot not_wf bnot_wf assert_wf equal-wf-T-base uiff_transitivity permutation-cons2 decidable__equal_int add-is-int-iff intformnot_wf int_formula_prop_not_lemma false_wf member-exists2 decidable__lt intformless_wf int_formula_prop_less_lemma member_filter squash_wf true_wf subtype_rel_self iff_weakening_equal l_member_decomp append_wf istype-nat list_ind_cons_lemma list_ind_nil_lemma length-append add-associates length_wf filter_append_sq equal_wf add_functionality_wrt_eq ifthenelse_wf ite_rw_false iff_weakening_uiff assert-deq permutation-swap-first2 permutation_inversion permutation_transitivity permutation-rotate iff_wf set_wf all_wf permute_list_wf int_seg_wf inject_wf nat_wf subtype_rel_list permutation-filter permutation-length
Rules used in proof :  cut universeEquality instantiate productIsType dependent_functionElimination equalitySymmetry sqequalBase equalityIstype functionIsType independent_functionElimination natural_numberEquality rename setElimination independent_isectElimination universeIsType setIsType setEquality inhabitedIsType because_Cache applyEquality intEquality hypothesis functionEquality lambdaEquality_alt sqequalRule hypothesisEquality isectElimination sqequalHypSubstitution extract_by_obid introduction thin lambdaFormation_alt isect_memberFormation_alt sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution isect_memberEquality_alt voidElimination equalityTransitivity unionElimination cumulativity productElimination closedConclusion approximateComputation dependent_pairFormation_alt int_eqEquality independent_pairFormation baseClosed equalityElimination hyp_replacement applyLambdaEquality pointwiseFunctionality promote_hyp baseApply equalityIsType1 imageElimination imageMemberEquality dependent_set_memberEquality_alt addEquality functionExtensionality lambdaEquality lambdaFormation isect_memberFormation productEquality dependent_set_memberEquality dependent_pairFormation

Latex:
\mforall{}[T:Type]
\mforall{}eq:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbB{}
((\mforall{}x,y:T.    (\muparrow{}(eq  x  y)  \mLeftarrow{}{}\mRightarrow{}  x  =  y))
{}\mRightarrow{}  (\mforall{}a1,b1:T  List.
(\mforall{}x:T.  (||filter(eq  x;a1)||  =  ||filter(eq  x;b1)||)  \mLeftarrow{}{}\mRightarrow{}  permutation(T;a1;b1))))

Date html generated: 2019_10_15-AM-10_24_11
Last ObjectModification: 2019_08_05-PM-02_09_03

Theory : decidable!equality

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