### Nuprl Lemma : tree2fun_wf

`∀[A,B,C:Type]. ∀[eq:EqDecider(A)].  ∀[w:type2tree(A;B;C)]. ∀[g:A ⟶ B].  (tree2fun(eq;w;g) ∈ C) supposing value-type(B)`

Proof

Definitions occuring in Statement :  tree2fun: `tree2fun(eq;w;g)` type2tree: `type2tree(A;B;C)` deq: `EqDecider(T)` value-type: `value-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` member: `t ∈ T` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` type2tree: `type2tree(A;B;C)` all: `∀x:A. B[x]` tree2fun: `tree2fun(eq;w;g)` Wsup: `Wsup(a;b)` so_lambda: `λ2x.t[x]` implies: `P `` Q` prop: `ℙ` so_apply: `x[s]` and: `P ∧ Q` subtype_rel: `A ⊆r B` pcw-pp-barred: `Barred(pp)` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` cw-step: `cw-step(A;a.B[a])` pcw-step: `pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b])` spreadn: spread3 less_than: `a < b` less_than': `less_than'(a;b)` true: `True` squash: `↓T` isr: `isr(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` btrue: `tt` ext-eq: `A ≡ B` unit: `Unit` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` ext-family: `F ≡ G` pi1: `fst(t)` nat_plus: `ℕ+` W-rel: `W-rel(A;a.B[a];w)` param-W-rel: `param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w)` pcw-steprel: `StepRel(s1;s2)` pi2: `snd(t)` isl: `isl(x)` pcw-step-agree: `StepAgree(s;p1;w)` cand: `A c∧ B` guard: `{T}` sq_type: `SQType(T)` le: `A ≤ B` sq_stable: `SqStable(P)` has-value: `(a)↓` deq: `EqDecider(T)`
Lemmas referenced :  type2tree_wf value-type_wf deq_wf W-elimination-facts equal_wf subtype_rel_self int_seg_wf subtract_wf nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_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_formula_prop_wf decidable__lt lelt_wf top_wf less_than_wf false_wf true_wf add-subtract-cancel itermAdd_wf int_term_value_add_lemma W-ext param-co-W-ext unit_wf2 it_wf param-co-W_wf pcw-steprel_wf subtype_rel_dep_function subtype_base_sq nat_wf set_subtype_base le_wf int_subtype_base decidable__equal_int intformeq_wf int_formula_prop_eq_lemma subtype_rel_function int_seg_subtype sq_stable__le value-type-has-value ifthenelse_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin sqequalHypSubstitution dependent_functionElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry sqequalRule axiomEquality functionEquality isect_memberEquality isectElimination because_Cache extract_by_obid universeEquality lambdaFormation unionElimination unionEquality lambdaEquality voidEquality independent_functionElimination productElimination strong_bar_Induction applyEquality instantiate functionExtensionality natural_numberEquality setElimination rename dependent_set_memberEquality independent_pairFormation independent_isectElimination approximateComputation dependent_pairFormation int_eqEquality intEquality voidElimination lessCases axiomSqEquality imageMemberEquality baseClosed imageElimination addEquality int_eqReduceTrueSq promote_hyp hypothesis_subsumption equalityElimination dependent_pairEquality inlEquality productEquality hyp_replacement applyLambdaEquality cumulativity callbyvalueReduce

Latex:
\mforall{}[A,B,C:Type].  \mforall{}[eq:EqDecider(A)].
\mforall{}[w:type2tree(A;B;C)].  \mforall{}[g:A  {}\mrightarrow{}  B].    (tree2fun(eq;w;g)  \mmember{}  C)  supposing  value-type(B)

Date html generated: 2019_06_20-PM-03_08_29
Last ObjectModification: 2018_08_21-PM-01_57_47

Theory : continuity

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