Nuprl Lemma : firstn_map

`∀[f:Top]. ∀[n:ℕ]. ∀[l:Top List].  (firstn(n;map(f;l)) ~ map(f;firstn(n;l)))`

Proof

Definitions occuring in Statement :  firstn: `firstn(n;as)` map: `map(f;as)` list: `T List` nat: `ℕ` uall: `∀[x:A]. B[x]` top: `Top` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` 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` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` guard: `{T}` firstn: `firstn(n;as)` 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]` 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)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff`
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 list_wf top_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf first0 map_wf map_nil_lemma equal-wf-T-base colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases list_ind_nil_lemma product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma le_wf equal_wf subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int map_cons_lemma lt_int_wf bool_wf equal-wf-base assert_wf le_int_wf bnot_wf list_ind_cons_lemma uiff_transitivity eqtt_to_assert assert_of_lt_int eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom unionElimination because_Cache functionExtensionality applyEquality promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate cumulativity imageElimination baseApply closedConclusion equalityElimination

Latex:
\mforall{}[f:Top].  \mforall{}[n:\mBbbN{}].  \mforall{}[l:Top  List].    (firstn(n;map(f;l))  \msim{}  map(f;firstn(n;l)))

Date html generated: 2017_04_17-AM-08_00_43
Last ObjectModification: 2017_02_27-PM-04_30_45

Theory : list_1

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