### Nuprl Lemma : first_index_equal

`∀[T:Type]. ∀[L1,L2:T List]. ∀[P,Q:T ⟶ 𝔹].`
`  index-of-first a in L1.P[a] ∧b Q[a] = index-of-first a in L2.P[a] ∧b Q[a] ∈ ℤ supposing L1 agree_on(T;a.↑P[a]) L2`

Proof

Definitions occuring in Statement :  agree_on: `agree_on(T;x.P[x])` first_index: `index-of-first x in L.P[x]` list: `T List` band: `p ∧b q` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` infix_ap: `x f y` so_apply: `x[s]` function: `x:A ⟶ B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` 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` guard: `{T}` or: `P ∨ Q` agree_on: `agree_on(T;x.P[x])` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` cand: `A c∧ B` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` cons: `[a / b]` colength: `colength(L)` nil: `[]` it: `⋅` sq_type: `SQType(T)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` infix_ap: `x f y` select: `L[n]` le: `A ≤ B` less_than': `less_than'(a;b)` less_than: `a < b` squash: `↓T` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` band: `p ∧b q` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` subtract: `n - m` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` true: `True` nat_plus: `ℕ+`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut thin lambdaFormation_alt extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType axiomEquality equalityTransitivity equalitySymmetry applyLambdaEquality inhabitedIsType functionIsTypeImplies unionElimination because_Cache voidEquality functionExtensionality instantiate applyEquality cumulativity universeEquality productEquality intEquality functionEquality productElimination functionIsType promote_hyp hypothesis_subsumption equalityIsType1 equalityIsType4 baseClosed dependent_set_memberEquality_alt imageElimination baseApply closedConclusion equalityElimination pointwiseFunctionality unionIsType addEquality productIsType inlFormation_alt inrFormation_alt imageMemberEquality hyp_replacement equalityIsType2

Latex:
\mforall{}[T:Type].  \mforall{}[L1,L2:T  List].  \mforall{}[P,Q:T  {}\mrightarrow{}  \mBbbB{}].
index-of-first  a  in  L1.P[a]  \mwedge{}\msubb{}  Q[a]  =  index-of-first  a  in  L2.P[a]  \mwedge{}\msubb{}  Q[a]
supposing  L1  agree\_on(T;a.\muparrow{}P[a])  L2

Date html generated: 2019_10_15-AM-10_58_20
Last ObjectModification: 2018_10_09-AM-09_57_25

Theory : list!

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