### Nuprl Lemma : combine-list-cons

`∀[A:Type]. ∀[f:A ⟶ A ⟶ A].`
`  ∀[L:A List]. ∀[a:A]. (combine-list(x,y.f[x;y];[a / L]) = f[a;combine-list(x,y.f[x;y];L)] ∈ A) supposing 0 < ||L|| `
`  supposing Assoc(A;λx,y. f[x;y])`

Proof

Definitions occuring in Statement :  combine-list: `combine-list(x,y.f[x; y];L)` length: `||as||` cons: `[a / b]` list: `T List` assoc: `Assoc(T;op)` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` or: `P ∨ Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` false: `False` and: `P ∧ Q` cons: `[a / b]` top: `Top` combine-list: `combine-list(x,y.f[x; y];L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` assoc: `Assoc(T;op)` infix_ap: `x f y`
Lemmas referenced :  list-cases length_of_nil_lemma product_subtype_list length_of_cons_lemma reduce_hd_cons_lemma reduce_tl_cons_lemma list_accum_cons_lemma less_than_wf length_wf list_wf assoc_wf list_induction all_wf equal_wf list_accum_wf list_accum_nil_lemma squash_wf true_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesisEquality extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis dependent_functionElimination unionElimination sqequalRule imageElimination productElimination voidElimination promote_hyp hypothesis_subsumption isect_memberEquality voidEquality axiomEquality because_Cache natural_numberEquality cumulativity equalityTransitivity equalitySymmetry lambdaEquality applyEquality functionExtensionality functionEquality universeEquality independent_functionElimination lambdaFormation rename imageMemberEquality baseClosed independent_isectElimination hyp_replacement applyLambdaEquality

Latex:
\mforall{}[A:Type].  \mforall{}[f:A  {}\mrightarrow{}  A  {}\mrightarrow{}  A].
\mforall{}[L:A  List]
\mforall{}[a:A].  (combine-list(x,y.f[x;y];[a  /  L])  =  f[a;combine-list(x,y.f[x;y];L)])
supposing  0  <  ||L||
supposing  Assoc(A;\mlambda{}x,y.  f[x;y])

Date html generated: 2017_04_14-AM-08_41_36
Last ObjectModification: 2017_02_27-PM-03_31_29

Theory : list_0

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