Nuprl Lemma : implies-eq-upto-baire2cantor

`∀a,b:ℕ ⟶ ℕ. ∀n:ℕ.  ((a = b ∈ (ℕn ⟶ ℕ)) `` (baire2cantor(a) = baire2cantor(b) ∈ (ℕn ⟶ 𝔹)))`

Proof

Definitions occuring in Statement :  baire2cantor: `baire2cantor(a)` int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` baire2cantor: `baire2cantor(a)` nat-pred: `n-1` member: `t ∈ T` uall: `∀[x:A]. B[x]` int_seg: `{i..j-}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` guard: `{T}` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` prop: `ℙ` bfalse: `ff` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T ` squash: `↓T` true: `True`
Lemmas referenced :  eq_int_wf eqtt_to_assert assert_of_eq_int istype-false int_seg_properties nat_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformeq_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf istype-le istype-less_than eqff_to_assert set_subtype_base lelt_wf int_subtype_base bool_subtype_base bool_cases_sqequal subtype_base_sq bool_wf assert-bnot neg_assert_of_eq_int subtract_wf decidable__le intformle_wf itermSubtract_wf int_formula_prop_le_lemma int_term_value_subtract_lemma ifthenelse_wf squash_wf true_wf istype-universe bfalse_wf btrue_wf int_seg_wf subtype_rel_function nat_wf int_seg_subtype_nat subtype_rel_self istype-nat
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut Error :functionExtensionality_alt,  sqequalRule thin introduction extract_by_obid sqequalHypSubstitution isectElimination setElimination rename hypothesisEquality hypothesis natural_numberEquality Error :inhabitedIsType,  unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination int_eqReduceTrueSq Error :dependent_set_memberEquality_alt,  independent_pairFormation dependent_functionElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination Error :universeIsType,  Error :productIsType,  because_Cache Error :equalityIsType4,  baseApply closedConclusion baseClosed applyEquality intEquality promote_hyp instantiate cumulativity int_eqReduceFalseSq Error :equalityIsType1,  imageElimination universeEquality applyLambdaEquality imageMemberEquality Error :functionIsType

Latex:
\mforall{}a,b:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \mforall{}n:\mBbbN{}.    ((a  =  b)  {}\mRightarrow{}  (baire2cantor(a)  =  baire2cantor(b)))

Date html generated: 2019_06_20-PM-03_07_42
Last ObjectModification: 2018_10_30-PM-02_07_53

Theory : continuity

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