### Nuprl Lemma : fps-compose-single-disjoint

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[v:bag(X)].`
`    ((¬x ↓∈ v) `` (∀[f:PowerSeries(X;r)]. (<v>(x:=f) = <v> ∈ PowerSeries(X;r)))) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-compose: `g(x:=f)` fps-single: `<c>` power-series: `PowerSeries(X;r)` bag-member: `x ↓∈ bs` bag: `bag(T)` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A` implies: `P `` Q` universe: `Type` equal: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` implies: `P `` Q` squash: `↓T` exists: `∃x:A. B[x]` prop: `ℙ` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` all: `∀x:A. B[x]` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` empty-bag: `{}` uiff: `uiff(P;Q)` fps-one: `1` fps-coeff: `f[b]` fps-single: `<c>` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` crng: `CRng` rng: `Rng` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` cons-bag: `x.b` top: `Top` fps-atom: `atom(x)` rev_uimplies: `rev_uimplies(P;Q)` sq_or: `a ↓∨ b`
Lemmas referenced :  bag_to_squash_list not_wf bag-member_wf list_induction list-subtype-bag equal_wf power-series_wf fps-compose_wf fps-single_wf list_wf nil_wf cons_wf bag_wf crng_wf deq_wf valueall-type_wf squash_wf true_wf fps-compose-one fps-one_wf iff_weakening_equal fps-ext empty-bag_wf bag-eq_wf bool_wf eqtt_to_assert assert-bag-eq bag-null_wf assert-bag-null rng_one_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot equal-wf-T-base rng_zero_wf single-bag_wf fps-mul_wf cons-bag-as-append fps-mul-single fps-compose-mul fps-compose-atom-neq bag-member-cons
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality imageElimination productElimination promote_hyp hypothesis equalitySymmetry hyp_replacement applyLambdaEquality cumulativity rename sqequalRule lambdaEquality functionEquality applyEquality independent_isectElimination independent_functionElimination voidEquality voidElimination dependent_functionElimination isect_memberEquality axiomEquality equalityTransitivity universeEquality natural_numberEquality imageMemberEquality baseClosed unionElimination equalityElimination setElimination dependent_pairFormation instantiate inlFormation inrFormation

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[x:X].  \mforall{}[v:bag(X)].
((\mneg{}x  \mdownarrow{}\mmember{}  v)  {}\mRightarrow{}  (\mforall{}[f:PowerSeries(X;r)].  (<v>(x:=f)  =  <v>)))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-10_10_15
Last ObjectModification: 2017_07_26-PM-06_34_20

Theory : power!series

Home Index