Nuprl Lemma : do-apply-p-first

`∀[A,B:Type]. ∀[L:(A ⟶ (B + Top)) List]. ∀[x:A].`
`  do-apply(p-first(L);x) = do-apply(hd(filter(λf.can-apply(f;x);L));x) ∈ B supposing ↑can-apply(p-first(L);x)`

Proof

Definitions occuring in Statement :  p-first: `p-first(L)` do-apply: `do-apply(f;x)` can-apply: `can-apply(f;x)` hd: `hd(l)` filter: `filter(P;l)` list: `T List` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` top: `Top` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` union: `left + right` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` true: `True` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` bool: `𝔹` unit: `Unit` btrue: `tt` do-apply: `do-apply(f;x)` p-first: `p-first(L)` can-apply: `can-apply(f;x)` isl: `isl(x)` outl: `outl(x)` rev_implies: `P `` Q` rev_uimplies: `rev_uimplies(P;Q)` p-conditional: `[f?g]`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf assert_wf can-apply_wf p-first_wf top_wf less_than_transitivity1 less_than_irreflexivity equal-wf-T-base nat_wf colength_wf_list list-cases product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtype_rel_list subtype_rel_dep_function subtype_rel_union subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int list_wf p_first_nil_lemma filter_nil_lemma false_wf list_ind_cons_lemma list_ind_nil_lemma assert_functionality_wrt_uiff cons_wf squash_wf true_wf p-first-append nil_wf p-conditional_wf p-conditional-to-p-first p-conditional-domain decidable__assert append_wf filter_cons_lemma bool_wf bnot_wf not_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot reduce_hd_cons_lemma list_accum_cons_lemma list_induction all_wf list_accum_wf list_accum_nil_lemma p-first-singleton iff_weakening_equal do-apply_wf outl_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination axiomEquality because_Cache applyEquality equalityTransitivity equalitySymmetry functionEquality cumulativity unionEquality unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination universeEquality functionExtensionality imageMemberEquality equalityElimination inlEquality hyp_replacement inrFormation

Latex:
\mforall{}[A,B:Type].  \mforall{}[L:(A  {}\mrightarrow{}  (B  +  Top))  List].  \mforall{}[x:A].
do-apply(p-first(L);x)  =  do-apply(hd(filter(\mlambda{}f.can-apply(f;x);L));x)
supposing  \muparrow{}can-apply(p-first(L);x)

Date html generated: 2018_05_21-PM-06_44_41
Last ObjectModification: 2017_07_26-PM-04_55_12

Theory : general

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