### Nuprl Lemma : mu-dec_wf

`∀[A:Type]. ∀[P:A ⟶ ℕ ⟶ ℙ]. ∀[d:a:A ⟶ k:ℕ ⟶ Dec(P[a;k])]. ∀[a:A].  mu-dec(d;a) ∈ ℕ supposing ↓∃k:ℕ. P[a;k]`

Proof

Definitions occuring in Statement :  mu-dec: `mu-dec(d;a)` nat: `ℕ` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` exists: `∃x:A. B[x]` squash: `↓T` member: `t ∈ T` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  squash: `↓T` member: `t ∈ T` mu-dec: `mu-dec(d;a)` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` prop: `ℙ` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` or: `P ∨ Q` decidable: `Dec(P)` true: `True` btrue: `tt` ifthenelse: `if b then t else f fi ` assert: `↑b` isl: `isl(x)` false: `False` not: `¬A`
Lemmas referenced :  mu_wf isl_wf not_wf nat_wf squash_wf exists_wf decidable_wf assert_wf subtype_rel_self equal_wf
Rules used in proof :  sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity imageElimination cut introduction extract_by_obid isectElimination thin Error :lambdaEquality_alt,  applyEquality functionExtensionality hypothesisEquality cumulativity hypothesis because_Cache sqequalRule Error :universeIsType,  independent_isectElimination Error :functionIsType,  universeEquality Error :isect_memberFormation_alt,  axiomEquality equalityTransitivity equalitySymmetry Error :isect_memberEquality_alt,  productElimination Error :dependent_pairFormation_alt,  independent_functionElimination dependent_functionElimination lambdaFormation lemma_by_obid unionElimination natural_numberEquality voidElimination Error :inhabitedIsType,  Error :lambdaFormation_alt,  instantiate Error :equalityIsType1

Latex:
\mforall{}[A:Type].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[d:a:A  {}\mrightarrow{}  k:\mBbbN{}  {}\mrightarrow{}  Dec(P[a;k])].  \mforall{}[a:A].
mu-dec(d;a)  \mmember{}  \mBbbN{}  supposing  \mdownarrow{}\mexists{}k:\mBbbN{}.  P[a;k]

Date html generated: 2019_06_20-PM-01_17_27
Last ObjectModification: 2018_10_03-PM-10_14_54

Theory : int_2

Home Index