### Nuprl Lemma : eq-finite-seqs-iff-eq-upto

`∀a,b:ℕ ⟶ ℕ. ∀x:ℕ.  (↑eq-finite-seqs(a;b;x) `⇐⇒` a = b ∈ (ℕx ⟶ ℕ))`

Proof

Definitions occuring in Statement :  eq-finite-seqs: `eq-finite-seqs(a;b;x)` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` lelt: `i ≤ j < k` int_seg: `{i..j-}` guard: `{T}` true: `True` btrue: `tt` ifthenelse: `if b then t else f fi ` assert: `↑b` eq-finite-seqs: `eq-finite-seqs(a;b;x)` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` so_lambda: `λ2x.t[x]` so_apply: `x[s]` band: `p ∧b q` subtract: `n - m`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than assert_witness istype-assert eq-finite-seqs_wf decidable__le intformnot_wf int_formula_prop_not_lemma istype-le int_seg_wf subtype_rel_function nat_wf int_seg_subtype_nat istype-false subtype_rel_self subtract-1-ge-0 istype-nat int_seg_properties primrec0_lemma primrec-unroll lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf less_than_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma set_subtype_base le_wf int_subtype_base bool_cases band_wf btrue_wf eq_int_wf bfalse_wf equal-wf-base iff_transitivity assert_of_band assert_of_eq_int decidable__equal_nat decidable__equal_int intformeq_wf int_formula_prop_eq_lemma decidable__lt int_seg_subtype not-le-2 condition-implies-le add-associates minus-add minus-one-mul add-swap minus-one-mul-top add-mul-special zero-mul add-zero add-commutes le-add-cancel2 equal_functionality_wrt_subtype_rel2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  productElimination independent_pairEquality axiomEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  Error :dependent_set_memberEquality_alt,  unionElimination Error :equalityIstype,  Error :functionIsType,  because_Cache applyEquality Error :functionExtensionality_alt,  equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity Error :productIsType,  intEquality sqequalBase productEquality applyLambdaEquality baseApply closedConclusion baseClosed addEquality minusEquality multiplyEquality functionEquality

Latex:
\mforall{}a,b:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \mforall{}x:\mBbbN{}.    (\muparrow{}eq-finite-seqs(a;b;x)  \mLeftarrow{}{}\mRightarrow{}  a  =  b)

Date html generated: 2019_06_20-PM-03_07_31
Last ObjectModification: 2018_12_19-PM-05_36_41

Theory : continuity

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