### Nuprl Lemma : gen-continuity-is-false

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

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` true: `True` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  not: `¬A` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` nat: `ℕ` subtype_rel: `A ⊆r B` so_apply: `x[s]` exists: `∃x:A. B[x]` int_upper: `{i...}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` prop: `ℙ` zero-seq: `0s` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` squash: `↓T` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` int_seg: `{i..j-}` lelt: `i ≤ j < k` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  all_wf nat_wf exists_wf int_upper_wf equal-wf-T-base int_upper_subtype_nat zero-seq_wf nat_properties decidable__le full-omega-unsat intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf le_wf quotient_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self true_wf equiv_rel_true squash-from-quotient equal-wf-base-T lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than_wf int_seg_properties decidable__equal_int intformand_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_upper_properties int_subtype_base bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis sqequalHypSubstitution dependent_functionElimination thin lambdaEquality introduction extract_by_obid isectElimination sqequalRule setElimination rename hypothesisEquality intEquality applyEquality functionExtensionality baseClosed because_Cache functionEquality independent_functionElimination dependent_pairFormation dependent_set_memberEquality unionElimination natural_numberEquality independent_isectElimination approximateComputation int_eqEquality isect_memberEquality voidElimination voidEquality instantiate cumulativity universeEquality independent_pairFormation imageElimination productElimination equalityElimination equalityTransitivity equalitySymmetry promote_hyp impliesFunctionality

Latex:
\mneg{}(\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)))))

Date html generated: 2017_09_29-PM-06_10_10
Last ObjectModification: 2017_07_11-PM-05_33_38

Theory : continuity

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