### Nuprl Lemma : list_extensionality

`∀[T:Type]. ∀[a,b:T List].`
`  (a = b ∈ (T List)) supposing ((∀i:ℕ. (i < ||a|| `` (a[i] = b[i] ∈ T))) and (||a|| = ||b|| ∈ ℤ))`

Proof

Definitions occuring in Statement :  select: `L[n]` length: `||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` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  so_apply: `x[s]` all: `∀x:A. B[x]` guard: `{T}` squash: `↓T` sq_stable: `SqStable(P)` uimplies: `b supposing a` nat: `ℕ` prop: `ℙ` implies: `P `` Q` so_lambda: `λ2x.t[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` top: `Top` so_apply: `x[s1;s2]` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` false: `False` subtract: `n - m` sq_type: `SQType(T)` ge: `i ≥ j ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` true: `True` not: `¬A` cons: `[a / b]` less_than: `a < b` nat_plus: `ℕ+` uiff: `uiff(P;Q)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` or: `P ∨ Q` decidable: `Dec(P)`
Rules used in proof :  axiomEquality isect_memberEquality isect_memberFormation universeEquality intEquality dependent_functionElimination imageElimination baseClosed imageMemberEquality independent_functionElimination natural_numberEquality independent_isectElimination hypothesisEquality cumulativity equalitySymmetry equalityTransitivity because_Cache rename setElimination functionEquality lambdaEquality sqequalRule hypothesis thin isectElimination sqequalHypSubstitution extract_by_obid introduction cut sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution lambdaFormation voidEquality voidElimination dependent_pairFormation sqequalIntensionalEquality applyEquality productElimination promote_hyp addEquality minusEquality multiplyEquality instantiate hyp_replacement dependent_set_memberEquality independent_pairFormation applyLambdaEquality unionElimination

Latex:
\mforall{}[T:Type].  \mforall{}[a,b:T  List].
(a  =  b)  supposing  ((\mforall{}i:\mBbbN{}.  (i  <  ||a||  {}\mRightarrow{}  (a[i]  =  b[i])))  and  (||a||  =  ||b||))

Date html generated: 2019_06_20-PM-00_41_09
Last ObjectModification: 2018_08_06-PM-02_09_12

Theory : list_0

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