### Nuprl Lemma : nat-pred-as-sub

`∀n:ℕ. (0 < n `` (n-1 = (n - 1) ∈ ℕ))`

Proof

Definitions occuring in Statement :  nat-pred: `n-1` nat: `ℕ` less_than: `a < b` all: `∀x:A. B[x]` implies: `P `` Q` subtract: `n - m` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  int_upper: `{i...}` nequal: `a ≠ b ∈ T ` assert: `↑b` ifthenelse: `if b then t else f fi ` bnot: `¬bb` guard: `{T}` sq_type: `SQType(T)` bfalse: `ff` less_than': `less_than'(a;b)` le: `A ≤ B` prop: `ℙ` top: `Top` not: `¬A` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` or: `P ∨ Q` decidable: `Dec(P)` ge: `i ≥ j ` uimplies: `b supposing a` and: `P ∧ Q` uiff: `uiff(P;Q)` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat-pred: `n-1` implies: `P `` Q` all: `∀x:A. B[x]`
Lemmas referenced :  nat_wf less_than_wf int_formula_prop_le_lemma intformle_wf decidable__le int_term_value_subtract_lemma int_formula_prop_not_lemma itermSubtract_wf intformnot_wf int_upper_properties zero-add nequal-le-implies int_upper_subtype_nat neg_assert_of_eq_int assert-bnot bool_subtype_base subtype_base_sq bool_cases_sqequal equal_wf eqff_to_assert le_wf false_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_and_lemma intformless_wf itermConstant_wf itermVar_wf intformeq_wf intformand_wf satisfiable-full-omega-tt subtract_wf decidable__equal_int nat_properties assert_of_eq_int eqtt_to_assert bool_wf eq_int_wf
Rules used in proof :  hypothesis_subsumption int_eqReduceFalseSq independent_functionElimination cumulativity instantiate promote_hyp dependent_set_memberEquality computeAll independent_pairFormation voidEquality voidElimination isect_memberEquality intEquality int_eqEquality lambdaEquality dependent_pairFormation dependent_functionElimination hypothesisEquality int_eqReduceTrueSq sqequalRule independent_isectElimination productElimination equalitySymmetry equalityTransitivity equalityElimination unionElimination natural_numberEquality hypothesis because_Cache rename setElimination thin isectElimination sqequalHypSubstitution extract_by_obid introduction cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}n:\mBbbN{}.  (0  <  n  {}\mRightarrow{}  (n-1  =  (n  -  1)))

Date html generated: 2017_04_21-AM-11_21_48
Last ObjectModification: 2017_04_20-PM-03_50_09

Theory : continuity

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