### Nuprl Lemma : finite-function-equipollent

`∀n:ℕ+. ∀[F:ℕn ⟶ Type]. i:ℕn ⟶ F[i] ~ i:ℕn - 1 ⟶ F[i] × F[n - 1]`

Proof

Definitions occuring in Statement :  equipollent: `A ~ B` int_seg: `{i..j-}` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` product: `x:A × B[x]` subtract: `n - m` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat_plus: `ℕ+` equipollent: `A ~ B` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` prop: `ℙ` le: `A ≤ B` less_than': `less_than'(a;b)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` true: `True` istype: `istype(T)` biject: `Bij(A;B;f)` inject: `Inj(A;B;f)` guard: `{T}` respects-equality: `respects-equality(S;T)` surject: `Surj(A;B;f)` pi1: `fst(t)` pi2: `snd(t)` sq_type: `SQType(T)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` squash: `↓T` sq_stable: `SqStable(P)` bfalse: `ff` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  int_seg_wf istype-universe nat_plus_wf subtype_rel_dep_function subtract_wf nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermVar_wf itermSubtract_wf itermConstant_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_wf istype-le istype-less_than int_seg_subtype istype-false decidable__le not-le-2 condition-implies-le add-associates minus-add minus-one-mul add-swap minus-one-mul-top add-mul-special zero-mul add-zero add-commutes le-add-cancel2 intformle_wf int_formula_prop_le_lemma biject_wf respects-equality-product respects-equality-function respects-equality-trivial istype-base decidable__equal_int subtype_base_sq int_subtype_base int_seg_properties intformeq_wf int_formula_prop_eq_lemma eq_int_wf subtype_rel-equal eqtt_to_assert assert_of_eq_int set_subtype_base less_than_wf sq_stable__and equal-wf-base sq_stable__equal eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot neg_assert_of_eq_int assert_wf equal_wf squash_wf true_wf eq_int_eq_true btrue_wf subtype_rel_self iff_weakening_equal btrue_neq_bfalse bnot_wf not_wf istype-assert uiff_transitivity iff_transitivity iff_weakening_uiff assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  Error :isect_memberFormation_alt,  Error :functionIsType,  Error :universeIsType,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis instantiate universeEquality Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  independent_pairEquality applyEquality sqequalRule because_Cache Error :dependent_set_memberEquality_alt,  productElimination independent_pairFormation dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination int_eqEquality Error :isect_memberEquality_alt,  voidElimination Error :productIsType,  addEquality minusEquality multiplyEquality functionEquality productEquality closedConclusion Error :equalityIstype,  Error :inhabitedIsType,  equalityTransitivity equalitySymmetry sqequalBase applyLambdaEquality Error :functionExtensionality_alt,  cumulativity intEquality equalityElimination baseApply baseClosed imageMemberEquality imageElimination axiomEquality Error :functionIsTypeImplies,  promote_hyp

Latex:
\mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}[F:\mBbbN{}n  {}\mrightarrow{}  Type].  i:\mBbbN{}n  {}\mrightarrow{}  F[i]  \msim{}  i:\mBbbN{}n  -  1  {}\mrightarrow{}  F[i]  \mtimes{}  F[n  -  1]

Date html generated: 2019_06_20-PM-02_19_19
Last ObjectModification: 2018_12_19-PM-05_13_29

Theory : equipollence!!cardinality!

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