Nuprl Lemma : apply-Id-alist-function

`∀[x:Id]. ∀[F:Top]. ∀[L:Id List].  apply-alist(IdDeq;map(λx.<x, F[x]>;L);x) ~ inl F[x] supposing (x ∈ L)`

Proof

Definitions occuring in Statement :  id-deq: `IdDeq` Id: `Id` apply-alist: `apply-alist(eq;L;x)` l_member: `(x ∈ l)` map: `map(f;as)` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` top: `Top` so_apply: `x[s]` lambda: `λx.A[x]` pair: `<a, b>` inl: `inl x` sqequal: `s ~ 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` iff: `P `⇐⇒` Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` apply-alist: `apply-alist(eq;L;x)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` pi1: `fst(t)` pi2: `snd(t)` eq_id: `a = b` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` Id: `Id` bfalse: `ff` bnot: `¬bb` assert: `↑b`
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 l_member_wf Id_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases nil_member nil_wf 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 subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int cons_member map_cons_lemma list_ind_cons_lemma eq_id_wf bool_wf eqtt_to_assert assert-eq-id atom2_subtype_base eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot cons_wf list_wf top_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 sqequalAxiom equalityTransitivity equalitySymmetry applyEquality because_Cache unionElimination productElimination promote_hyp hypothesis_subsumption applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate cumulativity imageElimination equalityElimination

Latex:
\mforall{}[x:Id].  \mforall{}[F:Top].  \mforall{}[L:Id  List].
apply-alist(IdDeq;map(\mlambda{}x.<x,  F[x]>L);x)  \msim{}  inl  F[x]  supposing  (x  \mmember{}  L)

Date html generated: 2017_04_17-AM-09_18_39
Last ObjectModification: 2017_02_27-PM-05_22_13

Theory : decidable!equality

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