### Nuprl Lemma : weak-update-alist_wf

`∀[A,T:Type]. ∀[eq:T ⟶ T ⟶ 𝔹]. ∀[x:T]. ∀[L:(T × A) List]. ∀[z:A]. ∀[f:A ⟶ A].`
`  (weak-update-alist(eq;L;x;z;v.f[v]) ∈ (T × A) List)`

Proof

Definitions occuring in Statement :  weak-update-alist: `weak-update-alist(eq;L;x;z;v.f[v])` list: `T List` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]` product: `x:A × B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` weak-update-alist: `weak-update-alist(eq;L;x;z;v.f[v])` update-alist: `update-alist(eq;L;x;z;v.f[v])` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases list_ind_nil_lemma cons_wf nil_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int list_ind_cons_lemma ifthenelse_wf list_wf bool_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut thin lambdaFormation introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry productEquality cumulativity applyEquality because_Cache unionElimination independent_pairEquality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination functionExtensionality functionEquality universeEquality

Latex:
\mforall{}[A,T:Type].  \mforall{}[eq:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[x:T].  \mforall{}[L:(T  \mtimes{}  A)  List].  \mforall{}[z:A].  \mforall{}[f:A  {}\mrightarrow{}  A].
(weak-update-alist(eq;L;x;z;v.f[v])  \mmember{}  (T  \mtimes{}  A)  List)

Date html generated: 2017_04_17-AM-09_08_52
Last ObjectModification: 2017_02_27-PM-05_17_11

Theory : decidable!equality

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