### 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)`
Lemmas referenced :  list_wf le_weakening length_wf less_than_transitivity1 sq_stable__le select_wf equal_wf less_than_wf nat_wf all_wf list_induction equal-wf-base-T nil_wf length_of_nil_lemma stuck-spread base_wf equal-wf-base length_of_cons_lemma non_neg_length length_wf_nat set_subtype_base le_wf int_subtype_base cons_wf less_than_irreflexivity equal-wf-T-base add-commutes subtract_wf minus-add add-associates minus-one-mul zero-add add-swap add-mul-special two-mul mul-distributes-right zero-mul add-zero one-mul subtype_base_sq minus-zero nat_properties and_wf true_wf squash_wf nat_plus_wf add_nat_plus false_wf le-add-cancel2 add_functionality_wrt_le minus-one-mul-top condition-implies-le le_antisymmetry_iff not-equal-2 decidable__int_equal less-iff-le not-lt-2 decidable__lt le-add-cancel not-le-2 decidable__le iff_weakening_equal select_cons_tl
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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