Nuprl Lemma : apply_uncurry

`∀[B:Type]. ∀[n:ℕ]. ∀[m:ℕn + 1]. ∀[q:ℕm + 1]. ∀[A:ℕn ⟶ Type]. ∀[lst:k:{q..n-} ⟶ (A k)].`
`∀[f:(k:{q..n-} ⟶ (A k)) ⟶ funtype(n - m;λx.(A (x + m));B)].`
`  ((uncurry-gen(n) m f lst) = (apply_gen(n;lst) m (f lst)) ∈ B)`

Proof

Definitions occuring in Statement :  apply_gen: `apply_gen(n;lst)` uncurry-gen: `uncurry-gen(n)` funtype: `funtype(n;A;T)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` nat: `ℕ` int_seg: `{i..j-}` guard: `{T}` ge: `i ≥ j ` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` sq_type: `SQType(T)` squash: `↓T` le: `A ≤ B` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` apply_gen: `apply_gen(n;lst)` uncurry-gen: `uncurry-gen(n)` funtype: `funtype(n;A;T)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` rev_implies: `P `` Q` less_than: `a < b` subtract: `n - m` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  subtract_wf int_seg_properties nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_term_value_add_lemma int_formula_prop_wf le_wf decidable__equal_int intformeq_wf int_formula_prop_eq_lemma equal_wf subtype_base_sq int_subtype_base lelt_wf squash_wf true_wf iff_weakening_equal ge_wf less_than_wf int_seg_wf primrec_wf primrec-unroll eq_int_wf bool_wf uiff_transitivity equal-wf-T-base assert_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int iff_transitivity bnot_wf not_wf iff_weakening_uiff assert_of_bnot decidable__lt subtype_rel-equal minus-add minus-minus add-associates minus-one-mul add-mul-special add-swap add-commutes mul-distributes-right two-mul zero-add one-mul itermMultiply_wf int_term_value_mul_lemma zero-mul add-zero apply_gen_wf subtype_rel_dep_function int_seg_subtype subtype_rel_self nat_wf funtype_wf add-member-int_seg1
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut dependent_pairFormation dependent_set_memberEquality extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis hypothesisEquality natural_numberEquality addEquality productElimination dependent_functionElimination unionElimination independent_isectElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll instantiate cumulativity equalityTransitivity equalitySymmetry independent_functionElimination applyEquality imageElimination imageMemberEquality baseClosed universeEquality intWeakElimination lambdaFormation axiomEquality functionEquality functionExtensionality equalityElimination promote_hyp impliesFunctionality multiplyEquality minusEquality

Latex:
\mforall{}[B:Type].  \mforall{}[n:\mBbbN{}].  \mforall{}[m:\mBbbN{}n  +  1].  \mforall{}[q:\mBbbN{}m  +  1].  \mforall{}[A:\mBbbN{}n  {}\mrightarrow{}  Type].  \mforall{}[lst:k:\{q..n\msupminus{}\}  {}\mrightarrow{}  (A  k)].
\mforall{}[f:(k:\{q..n\msupminus{}\}  {}\mrightarrow{}  (A  k))  {}\mrightarrow{}  funtype(n  -  m;\mlambda{}x.(A  (x  +  m));B)].
((uncurry-gen(n)  m  f  lst)  =  (apply\_gen(n;lst)  m  (f  lst)))

Date html generated: 2017_10_01-AM-09_03_39
Last ObjectModification: 2017_07_26-PM-04_44_29

Theory : bags

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