### Nuprl Lemma : unsquashed-continuity-false-troelstra

`¬(∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀a:ℕ ⟶ ℕ.  ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((a = b ∈ (ℕn ⟶ ℕ)) `` ((F a) = (F b) ∈ ℕ)))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` 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` exists: `∃x:A. B[x]` uall: `∀[x:A]. B[x]` nat: `ℕ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` pi1: `fst(t)` phi-star: `Phi*` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` prop: `ℙ` cand: `A c∧ B` isl: `isl(x)` outl: `outl(x)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` guard: `{T}` gamma-neighbourhood: `gamma-neighbourhood(beta;n0)` exposed-bfalse: `exposed-bfalse` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` assert: `↑b` bfalse: `ff` bnot: `¬bb` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` finite-nat-seq: `finite-nat-seq()` mk-finite-nat-seq: `f^(n)` pi2: `snd(t)` ext-finite-nat-seq: `ext-finite-nat-seq(f;x)` append-finite-nat-seq: `f**g` int_seg: `{i..j-}` less_than: `a < b` ge: `i ≥ j ` lelt: `i ≤ j < k`
Lemmas referenced :  istype-nat zero-seq_wf int_seg_wf subtype_rel_function nat_wf int_seg_subtype_nat istype-false subtype_rel_self gamma-neighbourhood-prop1 finite-nat-seq_wf phi-star_wf decidable__le full-omega-unsat intformnot_wf intformle_wf itermConstant_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_wf istype-le istype-assert gamma-neighbourhood_wf mk-finite-nat-seq_wf btrue_wf bfalse_wf assert_elim union_subtype_base unit_wf2 set_subtype_base le_wf int_subtype_base unit_subtype_base btrue_neq_bfalse append-finite-nat-seq_wf assert_wf equal-wf-base subtype_base_sq init-seg-nat-seq_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot extend-seq1-all-dec decidable_wf not_wf true_wf equal_wf squash_wf iff_weakening_equal gamma-neighbourhood-prop2 decidable__equal_int ext-finite-nat-seq_wf lt_int_wf assert_of_lt_int istype-top iff_weakening_uiff less_than_wf istype-less_than int_seg_properties nat_properties intformand_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma itermAdd_wf int_term_value_add_lemma gamma-neighbourhood-prop3 gamma-neighbourhood-prop4 intformeq_wf int_formula_prop_eq_lemma gamma-neighbourhood-prop5 gamma-neighbourhood-prop6 eq-seg-nat-seq_wf assert-eq-seg-nat-seq Troelstra-lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality Error :functionIsType,  Error :inhabitedIsType,  introduction extract_by_obid because_Cache promote_hyp productElimination sqequalRule Error :productIsType,  Error :equalityIstype,  Error :universeIsType,  isectElimination natural_numberEquality setElimination rename applyEquality independent_isectElimination independent_pairFormation Error :lambdaEquality_alt,  Error :dependent_pairFormation_alt,  functionExtensionality functionEquality equalityTransitivity equalitySymmetry independent_functionElimination Error :dependent_set_memberEquality_alt,  unionElimination approximateComputation Error :isect_memberEquality_alt,  voidElimination baseApply closedConclusion baseClosed intEquality sqequalBase applyLambdaEquality Error :unionIsType,  productEquality Error :functionExtensionality_alt,  instantiate cumulativity equalityElimination imageElimination universeEquality imageMemberEquality Error :equalityIsType4,  Error :equalityIsType1,  addEquality lessCases Error :isect_memberFormation_alt,  axiomSqEquality Error :isectIsTypeImplies,  int_eqEquality unionEquality

Latex:
\mneg{}(\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}.  \mforall{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.    \mexists{}n:\mBbbN{}.  \mforall{}b:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  ((a  =  b)  {}\mRightarrow{}  ((F  a)  =  (F  b))))

Date html generated: 2019_06_20-PM-03_05_14
Last ObjectModification: 2018_12_06-PM-11_57_48

Theory : continuity

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