### Nuprl Lemma : subtype_rel_tuple-type

`∀[As,Bs:Type List].  tuple-type(As) ⊆r tuple-type(Bs) supposing (||As|| = ||Bs|| ∈ ℤ) ∧ (∀i:ℕ||As||. (As[i] ⊆r Bs[i]))`

Proof

Definitions occuring in Statement :  tuple-type: `tuple-type(L)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` and: `P ∧ Q` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` and: `P ∧ Q` prop: `ℙ` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` so_apply: `x[s]` subtype_rel: `A ⊆r B` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` ge: `i ≥ j ` le: `A ≤ B` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` less_than': `less_than'(a;b)` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` true: `True` select: `L[n]` cons: `[a / b]` nil: `[]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cand: `A c∧ B` subtract: `n - m` iff: `P `⇐⇒` Q`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination universeEquality sqequalRule lambdaEquality hypothesis isectEquality productEquality intEquality because_Cache hypothesisEquality natural_numberEquality setElimination rename independent_isectElimination equalityTransitivity equalitySymmetry productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll applyEquality cumulativity independent_functionElimination axiomEquality lambdaFormation equalityElimination promote_hyp baseClosed pointwiseFunctionality baseApply closedConclusion dependent_set_memberEquality imageMemberEquality applyLambdaEquality addEquality hypothesis_subsumption hyp_replacement imageElimination

Latex:
\mforall{}[As,Bs:Type  List].
tuple-type(As)  \msubseteq{}r  tuple-type(Bs)  supposing  (||As||  =  ||Bs||)  \mwedge{}  (\mforall{}i:\mBbbN{}||As||.  (As[i]  \msubseteq{}r  Bs[i]))

Date html generated: 2017_04_17-AM-09_29_07
Last ObjectModification: 2017_02_27-PM-05_31_29

Theory : tuples

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