### Nuprl Lemma : eq-finite-seqs-implies-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]` 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` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` 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` decidable: `Dec(P)` or: `P ∨ Q` lelt: `i ≤ j < k` int_seg: `{i..j-}` guard: `{T}` true: `True` band: `p ∧b q` less_than': `less_than'(a;b)` le: `A ≤ B` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bnot: `¬bb` sq_type: `SQType(T)` bfalse: `ff` assert: `↑b` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` eq-finite-seqs: `eq-finite-seqs(a;b;x)` so_apply: `x[s]` so_lambda: `λ2x.t[x]`
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 istype-assert eq-finite-seqs_wf decidable__le intformnot_wf int_formula_prop_not_lemma istype-le subtract-1-ge-0 istype-nat int_seg_wf int_seg_properties bfalse_wf istype-false int_seg_subtype_nat eq_int_wf band_wf bool_cases btrue_wf int_term_value_subtract_lemma itermSubtract_wf subtract_wf primrec_wf less_than_wf assert_wf iff_weakening_uiff assert-bnot bool_subtype_base bool_wf subtype_base_sq bool_cases_sqequal eqff_to_assert assert_of_lt_int eqtt_to_assert lt_int_wf primrec-unroll decidable__equal_int equal-wf-base le_wf set_subtype_base int_subtype_base int_formula_prop_eq_lemma intformeq_wf iff_transitivity assert_of_band assert_of_eq_int decidable__lt
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination 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,  axiomEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  Error :dependent_set_memberEquality_alt,  unionElimination Error :functionIsType,  productElimination Error :functionExtensionality_alt,  Error :equalityIstype,  applyEquality because_Cache cumulativity instantiate promote_hyp Error :equalityIsType1,  equalitySymmetry equalityTransitivity equalityElimination productEquality sqequalBase closedConclusion Error :productIsType,  intEquality applyLambdaEquality

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

Date html generated: 2019_06_20-PM-03_07_24
Last ObjectModification: 2019_01_02-PM-00_36_10

Theory : continuity

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