### Nuprl Lemma : fan-implies-bar-sep

`∀[T:Type]. (Fan_d(T) `` (∃size:ℕ. T ~ ℕsize) `` BarSep(T;T))`

Proof

Definitions occuring in Statement :  bar-separation: `BarSep(T;S)` dfan: `Fan_d(T)` equipollent: `A ~ B` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` bar-separation: `BarSep(T;S)` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` prop: `ℙ` tbar: `tbar(T;X)` iff: `P `⇐⇒` Q` and: `P ∧ Q` squash: `↓T` true: `True` rev_implies: `P `` Q` not: `¬A` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` false: `False` equipollent: `A ~ B` biject: `Bij(A;B;f)` surject: `Surj(A;B;f)` int_seg: `{i..j-}` lelt: `i ≤ j < k` dfan: `Fan_d(T)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` ubar: `ubar(T;X)` dbar: `dbar(T;X)` dec-predicate: `Decidable(X)` cand: `A c∧ B` jbar: `jbar(T;S;X;Y)` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` nat_plus: `ℕ+` compose: `f o g` pi1: `fst(t)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` pi2: `snd(t)` sq_stable: `SqStable(P)` less_than: `a < b` int_iseg: `{i...j}` iseg: `l1 ≤ l2` select: `L[n]`
Lemmas referenced :  decidable__equal_int subtype_base_sq int_subtype_base jbar_wf dec-predicate_wf list_wf istype-nat equipollent_wf int_seg_wf dfan_wf istype-universe equipollent-zero squash_wf true_wf istype-int iff_weakening_equal nat_properties decidable__le full-omega-unsat intformnot_wf intformle_wf itermConstant_wf int_formula_prop_not_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_wf istype-le tbar_wf decidable__lt intformand_wf intformless_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_eq_lemma istype-less_than length_wf unshuffle_wf firstn_wf map_wf pi1_wf pi2_wf decidable__or decidable__exists_int_seg itermMultiply_wf int_term_value_mul_lemma itermAdd_wf int_term_value_add_lemma length-unshuffle upto_wf map-length length_upto subtype_rel_function nat_wf int_seg_subtype_nat istype-false subtype_rel_self nequal_wf less_than_wf div_mul_cancel divide_wfa mul_com subtype_rel_list top_wf list_subtype_base set_subtype_base lelt_wf firstn_upto le_int_wf eqtt_to_assert assert_of_le_int int_seg_subtype eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf le_wf firstn_map unshuffle-map map-map sq_stable__le div_rem_sum rem_bounds_1 length_wf_nat lt_int_wf assert_of_lt_int select_wf list_extensionality int_seg_properties length_firstn subtype_rel_sets_simple select-map select-upto select-firstn unshuffle-iseg firstn-iseg iseg_length firstn_append equal_wf decidable__all_length shuffle_wf length-shuffle unshuffle-shuffle eta_conv bnot_wf not_wf istype-assert bool_cases iff_transitivity assert_of_bnot decidable__exists_length decidable__all_int_seg decidable__not map_length_nat select-unshuffle not_over_exists equal-wf-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  sqequalHypSubstitution productElimination thin cut introduction extract_by_obid dependent_functionElimination setElimination rename because_Cache hypothesis natural_numberEquality unionElimination instantiate isectElimination cumulativity intEquality independent_isectElimination independent_functionElimination Error :universeIsType,  hypothesisEquality Error :inhabitedIsType,  Error :functionIsType,  universeEquality sqequalRule Error :productIsType,  Error :inlFormation_alt,  applyEquality Error :lambdaEquality_alt,  imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed Error :dependent_set_memberEquality_alt,  approximateComputation Error :dependent_pairFormation_alt,  Error :isect_memberEquality_alt,  voidElimination independent_pairFormation int_eqEquality promote_hyp unionEquality productEquality multiplyEquality addEquality Error :unionIsType,  Error :equalityIstype,  sqequalBase closedConclusion functionExtensionality equalityElimination Error :inrFormation_alt,  Error :setIsType,  hyp_replacement applyLambdaEquality independent_pairEquality functionEquality baseApply

Latex:
\mforall{}[T:Type].  (Fan\_d(T)  {}\mRightarrow{}  (\mexists{}size:\mBbbN{}.  T  \msim{}  \mBbbN{}size)  {}\mRightarrow{}  BarSep(T;T))

Date html generated: 2019_06_20-PM-02_47_21
Last ObjectModification: 2019_03_06-AM-11_06_01

Theory : fan-theorem

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