### Nuprl Lemma : mu-property2

`∀[P:ℕ ⟶ ℙ]. ∀d:∀n:ℕ. Dec(P[n]). {P[mu(d)] ∧ (∀i:ℕ. ¬P[i] supposing i < mu(d))} supposing ∃n:ℕ. P[n]`

Proof

Definitions occuring in Statement :  mu: `mu(f)` nat: `ℕ` less_than: `a < b` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` prop: `ℙ` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` mu: `mu(f)` nat: `ℕ` int_upper: `{i...}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` top: `Top` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` bfalse: `ff` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)`
Lemmas referenced :  mu-ge-property2 subtype_rel_function nat_wf int_upper_wf upper_subtype_nat istype-false subtype_rel_self all_wf decidable_wf mu-ge_wf2 subtype_rel_dep_function top_wf subtype_rel_union not_wf istype-void assert_wf btrue_wf bfalse_wf less_than_wf nat_properties int_upper_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf le_wf
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity dependent_functionElimination thin natural_numberEquality Error :isect_memberFormation_alt,  hypothesis isectElimination hypothesisEquality applyEquality instantiate cumulativity universeEquality because_Cache independent_isectElimination sqequalRule independent_pairFormation Error :lambdaFormation_alt,  Error :lambdaEquality_alt,  Error :universeIsType,  productElimination Error :dependent_pairFormation_alt,  promote_hyp Error :productIsType,  Error :functionIsType,  unionEquality Error :isect_memberEquality_alt,  voidElimination Error :unionIsType,  functionExtensionality Error :inhabitedIsType,  unionElimination Error :equalityIsType1,  equalityTransitivity equalitySymmetry independent_functionElimination Error :functionIsTypeImplies,  setElimination rename Error :dependent_set_memberEquality_alt,  applyLambdaEquality approximateComputation int_eqEquality

Latex:
\mforall{}[P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  \mforall{}d:\mforall{}n:\mBbbN{}.  Dec(P[n]).  \{P[mu(d)]  \mwedge{}  (\mforall{}i:\mBbbN{}.  \mneg{}P[i]  supposing  i  <  mu(d))\}  supposing  \mexists{}n:\mBbbN{}.  P[n]

Date html generated: 2019_06_20-PM-01_17_23
Last ObjectModification: 2018_10_06-AM-11_21_44

Theory : int_2

Home Index