### Nuprl Lemma : CCC-finite

`∀[T:Type]. (finite(T) `` CCC(T))`

Proof

Definitions occuring in Statement :  contra-cc: `CCC(T)` finite: `finite(T)` uall: `∀[x:A]. B[x]` implies: `P `` Q` universe: `Type`
Definitions unfolded in proof :  subtract: `n - m` primtailrec: `primtailrec(n;i;b;f)` primrec: `primrec(n;b;c)` exp: `i^n` true: `True` less_than': `less_than'(a;b)` so_apply: `x[s]` so_lambda: `λ2x.t[x]` surject: `Surj(A;B;f)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` assert: `↑b` bnot: `¬bb` bfalse: `ff` squash: `↓T` less_than: `a < b` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` biject: `Bij(A;B;f)` equipollent: `A ~ B` subtype_rel: `A ⊆r B` contra-cc: `CCC(T)` guard: `{T}` sq_type: `SQType(T)` le: `A ≤ B` and: `P ∧ Q` lelt: `i ≤ j < k` int_seg: `{i..j-}` false: `False` top: `Top` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` or: `P ∨ Q` decidable: `Dec(P)` ge: `i ≥ j ` int_upper: `{i...}` uimplies: `b supposing a` nat: `ℕ` all: `∀x:A. B[x]` prop: `ℙ` member: `t ∈ T` exists: `∃x:A. B[x]` finite: `finite(T)` implies: `P `` Q` uall: `∀[x:A]. B[x]`
Lemmas referenced :  equipollent_inversion istype-false CCC-product int_formula_prop_eq_lemma intformeq_wf subtract-add-cancel iff_weakening_equal exp_add true_wf squash_wf equal_wf exp_wf4 equipollent-multiply exp0_lemma primrec-wf2 int_term_value_subtract_lemma itermSubtract_wf subtract_wf contra-cc_wf lelt_wf le_wf set_subtype_base less_than_wf assert_wf iff_weakening_uiff assert-bnot bool_subtype_base bool_cases_sqequal eqff_to_assert assert_of_lt_int eqtt_to_assert lt_int_wf CCC-bool surject_wf equipollent-two bool_wf CCC-surjection istype-less_than decidable__lt int_seg_cases int_seg_subtype_special subtype_rel_self int_seg_wf int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_and_lemma itermVar_wf intformless_wf intformand_wf int_seg_properties int_subtype_base subtype_base_sq decidable__equal_int istype-le int_formula_prop_wf int_term_value_constant_lemma int_formula_prop_le_lemma istype-void int_formula_prop_not_lemma istype-int itermConstant_wf intformle_wf intformnot_wf full-omega-unsat decidable__le nat_properties exp-greater exp_wf2 le_weakening2 istype-universe finite_wf
Rules used in proof :  imageMemberEquality productEquality Error :setIsType,  sqequalBase baseClosed closedConclusion baseApply promote_hyp Error :equalityIstype,  imageElimination equalityElimination hypothesis_subsumption applyEquality Error :productIsType,  Error :inhabitedIsType,  Error :functionIsType,  independent_pairFormation int_eqEquality equalitySymmetry equalityTransitivity intEquality cumulativity because_Cache sqequalRule voidElimination Error :isect_memberEquality_alt,  Error :lambdaEquality_alt,  independent_functionElimination approximateComputation unionElimination Error :dependent_set_memberEquality_alt,  independent_isectElimination natural_numberEquality rename setElimination dependent_functionElimination Error :dependent_pairFormation_alt,  universeEquality instantiate hypothesis hypothesisEquality isectElimination extract_by_obid introduction Error :universeIsType,  cut thin productElimination sqequalHypSubstitution Error :lambdaFormation_alt,  Error :isect_memberFormation_alt,  sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}[T:Type].  (finite(T)  {}\mRightarrow{}  CCC(T))

Date html generated: 2019_06_20-PM-03_01_10
Last ObjectModification: 2019_06_12-PM-09_48_12

Theory : continuity

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