### Nuprl Lemma : map_equal

`∀[T,T':Type]. ∀[a:T List]. ∀[f,g:T ⟶ T'].`
`  map(f;a) = map(g;a) ∈ (T' List) supposing ∀i:ℕ. (i < ||a|| `` ((f a[i]) = (g a[i]) ∈ T'))`

Proof

Definitions occuring in Statement :  select: `L[n]` length: `||as||` map: `map(f;as)` list: `T List` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  so_apply: `x[s]` and: `P ∧ Q` top: `Top` not: `¬A` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` or: `P ∨ Q` decidable: `Dec(P)` all: `∀x:A. B[x]` ge: `i ≥ j ` uimplies: `b supposing a` nat: `ℕ` prop: `ℙ` implies: `P `` Q` so_lambda: `λ2x.t[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` int_seg: `{i..j-}` lelt: `i ≤ j < k` true: `True` squash: `↓T` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  list_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties select_wf equal_wf length_wf less_than_wf nat_wf all_wf map_wf list_extensionality map_length istype-less_than istype-nat full-omega-unsat istype-int istype-void istype-le iff_weakening_equal squash_wf true_wf subtype_rel_self istype-universe map_select
Rules used in proof :  equalitySymmetry equalityTransitivity axiomEquality isect_memberFormation universeEquality computeAll independent_pairFormation voidEquality voidElimination isect_memberEquality intEquality int_eqEquality dependent_pairFormation unionElimination natural_numberEquality dependent_functionElimination independent_isectElimination functionExtensionality applyEquality hypothesisEquality cumulativity because_Cache rename setElimination functionEquality lambdaEquality sqequalRule hypothesis thin isectElimination sqequalHypSubstitution extract_by_obid introduction cut sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution Error :lambdaFormation_alt,  Error :dependent_set_memberEquality_alt,  approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  Error :isect_memberEquality_alt,  Error :universeIsType,  Error :productIsType,  imageElimination imageMemberEquality baseClosed productElimination Error :inhabitedIsType,  instantiate

Latex:
\mforall{}[T,T':Type].  \mforall{}[a:T  List].  \mforall{}[f,g:T  {}\mrightarrow{}  T'].
map(f;a)  =  map(g;a)  supposing  \mforall{}i:\mBbbN{}.  (i  <  ||a||  {}\mRightarrow{}  ((f  a[i])  =  (g  a[i])))

Date html generated: 2019_06_20-PM-01_45_16
Last ObjectModification: 2019_01_10-PM-08_46_14

Theory : list_1

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