### Nuprl Lemma : fun_exp-mul

`∀[T:Type]. ∀[f:T ⟶ T]. ∀[n,m:ℕ]. ∀[x:T].  ((f^n * m x) = (λx.(f^m x)^n x) ∈ T)`

Proof

Definitions occuring in Statement :  fun_exp: `f^n` nat: `ℕ` uall: `∀[x:A]. B[x]` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` multiply: `n * m` universe: `Type` equal: `s = t ∈ 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` sq_type: `SQType(T)` guard: `{T}` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` fun_exp: `f^n` primrec: `primrec(n;b;c)` compose: `f o g` bool: `𝔹` unit: `Unit` it: `⋅` 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 fun_exp0_lemma decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf zero-mul subtype_base_sq int_subtype_base decidable__equal_int intformeq_wf itermMultiply_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_mul_lemma int_term_value_add_lemma fun_exp_add-sq mul_bounds_1a le_wf equal_wf squash_wf true_wf fun_exp_wf iff_weakening_equal fun_exp_unroll eq_int_wf bool_wf equal-wf-base assert_wf bnot_wf not_wf compose_wf uiff_transitivity eqtt_to_assert assert_of_eq_int iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot
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 axiomEquality because_Cache unionElimination functionEquality cumulativity universeEquality instantiate equalityTransitivity equalitySymmetry dependent_set_memberEquality multiplyEquality applyEquality imageElimination functionExtensionality imageMemberEquality baseClosed productElimination baseApply closedConclusion equalityElimination impliesFunctionality

Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  T].  \mforall{}[n,m:\mBbbN{}].  \mforall{}[x:T].    ((f\^{}n  *  m  x)  =  (\mlambda{}x.(f\^{}m  x)\^{}n  x))

Date html generated: 2017_04_14-AM-09_13_12
Last ObjectModification: 2017_02_27-PM-03_50_17

Theory : int_2

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