### Nuprl Lemma : dist_hom_over_mon_for

`∀T:Type. ∀m,n:IMonoid. ∀f:MonHom(m,n). ∀a:T List. ∀g:T ⟶ |m|.`
`  ((f (For{m} x ∈ a. g[x])) = (For{n} x ∈ a. (f g[x])) ∈ |n|)`

Proof

Definitions occuring in Statement :  mon_for: `For{g} x ∈ as. f[x]` list: `T List` so_apply: `x[s]` all: `∀x:A. B[x]` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T` monoid_hom: `MonHom(M1,M2)` imon: `IMonoid` grp_car: `|g|`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` imon: `IMonoid` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` monoid_hom: `MonHom(M1,M2)` infix_ap: `x f y`
Lemmas referenced :  istype-universe grp_car_wf list_wf monoid_hom_wf imon_wf nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 list-cases mon_for_nil_lemma product_subtype_list colength-cons-not-zero colength_wf_list istype-false le_wf subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le mon_for_cons_lemma nat_wf grp_id_wf equal_wf squash_wf true_wf monoid_hom_id subtype_rel_self iff_weakening_equal mon_for_wf infix_ap_wf grp_op_wf monoid_hom_op
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut hypothesis functionIsType introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality universeIsType setElimination rename inhabitedIsType universeEquality intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation axiomEquality functionIsTypeImplies unionElimination promote_hyp hypothesis_subsumption productElimination equalityIsType1 because_Cache dependent_set_memberEquality_alt instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination equalityIsType4 baseApply closedConclusion baseClosed applyEquality intEquality imageMemberEquality

Latex:
\mforall{}T:Type.  \mforall{}m,n:IMonoid.  \mforall{}f:MonHom(m,n).  \mforall{}a:T  List.  \mforall{}g:T  {}\mrightarrow{}  |m|.
((f  (For\{m\}  x  \mmember{}  a.  g[x]))  =  (For\{n\}  x  \mmember{}  a.  (f  g[x])))

Date html generated: 2019_10_16-PM-01_03_14
Last ObjectModification: 2018_10_08-PM-00_13_25

Theory : list_2

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