### Nuprl Lemma : assert-eq-seg-nat-seq

`∀[n,m:finite-nat-seq()].  (↑eq-seg-nat-seq(n;m) `⇐⇒` n = m ∈ finite-nat-seq())`

Proof

Definitions occuring in Statement :  eq-seg-nat-seq: `eq-seg-nat-seq(n;m)` finite-nat-seq: `finite-nat-seq()` assert: `↑b` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` eq-seg-nat-seq: `eq-seg-nat-seq(n;m)` finite-nat-seq: `finite-nat-seq()` pi1: `fst(t)` pi2: `snd(t)` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` nat: `ℕ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` rev_implies: `P `` Q` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` sq_type: `SQType(T)` uiff: `uiff(P;Q)` bfalse: `ff` band: `p ∧b q` ifthenelse: `if b then t else f fi `
Lemmas referenced :  istype-le subtype_rel_function int_seg_wf nat_wf int_seg_subtype istype-false subtype_rel_self nat_properties decidable__le full-omega-unsat intformnot_wf intformle_wf itermVar_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf set_subtype_base le_wf int_subtype_base int_seg_properties istype-less_than intformand_wf itermConstant_wf int_formula_prop_and_lemma int_term_value_constant_lemma subtype_rel_dep_function le_weakening zero-le-nat intformeq_wf int_formula_prop_eq_lemma subtype_base_sq subtype_rel_wf assert-init-seg-nat-seq2 istype-assert init-seg-nat-seq_wf iff_weakening_uiff assert_wf bool_cases bool_wf bool_subtype_base eqtt_to_assert band_wf btrue_wf bfalse_wf assert_of_band assert_witness eq-seg-nat-seq_wf finite-nat-seq_wf decidable__equal_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalHypSubstitution productElimination thin sqequalRule independent_pairFormation Error :lambdaFormation_alt,  hypothesis Error :productIsType,  extract_by_obid isectElimination setElimination rename hypothesisEquality Error :equalityIstype,  Error :inhabitedIsType,  applyEquality natural_numberEquality because_Cache independent_isectElimination dependent_functionElimination unionElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination Error :universeIsType,  hyp_replacement equalitySymmetry Error :dependent_set_memberEquality_alt,  equalityTransitivity applyLambdaEquality baseApply closedConclusion baseClosed intEquality sqequalBase Error :functionExtensionality_alt,  instantiate cumulativity independent_pairEquality Error :functionIsType,  Error :dependent_pairEquality_alt,  promote_hyp productEquality axiomEquality Error :functionIsTypeImplies,  Error :isectIsTypeImplies

Latex:
\mforall{}[n,m:finite-nat-seq()].    (\muparrow{}eq-seg-nat-seq(n;m)  \mLeftarrow{}{}\mRightarrow{}  n  =  m)

Date html generated: 2019_06_20-PM-03_04_17
Last ObjectModification: 2018_11_25-PM-05_58_00

Theory : continuity

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