`(∀P:(ℕ ⟶ ℕ) ⟶ ℙ. ∀f:ℕ ⟶ ℕ.  ((P f) `` ⇃(∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) `` (P g)))))`
` (¬(∀A:ℕ ⟶ ℙ. ((∀n:ℕ. ((A n) ∨ (¬(A n)))) `` (¬¬(∃n:ℕ. (A n))) `` (∃n:ℕ. (A n)))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` or: `P ∨ Q` true: `True` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  implies: `P `` Q` not: `¬A` false: `False` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` or: `P ∨ Q` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` nat: `ℕ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` uimplies: `b supposing a` and: `P ∧ Q` ge: `i ≥ j ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` is-absolutely-free: `is-absolutely-free{i:l}(f)` increasing-sequence: `increasing-sequence(a)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` guard: `{T}` sq_stable: `SqStable(P)` squash: `↓T` cand: `A c∧ B`
Lemmas referenced :  istype-nat subtype_rel_self istype-void quotient_wf nat_wf equal_wf int_seg_wf true_wf equiv_rel_true init0_wf increasing-sequence_wf ge_wf nat_properties decidable__or le_wf decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int 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 istype-le not_wf decidable__not intformor_wf int_formula_prop_or_lemma Kripke2a sq_stable_from_decidable equal-wf-base itermAdd_wf int_term_value_add_lemma set_subtype_base int_subtype_base decidable__equal_nat add_nat_wf intformeq_wf int_formula_prop_eq_lemma false_wf Kripke2b init0-zero-seq increasing-zero-seq zero-seq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut thin because_Cache hypothesis sqequalHypSubstitution independent_functionElimination voidElimination sqequalRule functionIsType introduction extract_by_obid universeIsType universeEquality unionIsType applyEquality hypothesisEquality instantiate isectElimination productIsType inhabitedIsType productEquality functionEquality natural_numberEquality setElimination rename lambdaEquality_alt equalityIstype independent_isectElimination setIsType dependent_functionElimination dependent_set_memberEquality_alt unionElimination approximateComputation dependent_pairFormation_alt int_eqEquality Error :memTop,  independent_pairFormation unionEquality productElimination addEquality intEquality equalityTransitivity equalitySymmetry applyLambdaEquality pointwiseFunctionality promote_hyp imageMemberEquality baseClosed imageElimination

Latex:
(\mforall{}P:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.    ((P  f)  {}\mRightarrow{}  \00D9(\mexists{}k:\mBbbN{}.  \mforall{}g:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  ((f  =  g)  {}\mRightarrow{}  (P  g)))))
{}\mRightarrow{}  (\mneg{}(\mforall{}A:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  ((\mforall{}n:\mBbbN{}.  ((A  n)  \mvee{}  (\mneg{}(A  n))))  {}\mRightarrow{}  (\mneg{}\mneg{}(\mexists{}n:\mBbbN{}.  (A  n)))  {}\mRightarrow{}  (\mexists{}n:\mBbbN{}.  (A  n)))))

Date html generated: 2020_05_19-PM-10_06_04
Last ObjectModification: 2020_01_04-PM-08_04_11

Theory : continuity

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