### Nuprl Lemma : is-list-if-has-value-fun-ax-mem

`∀[t:Base]. ∀[n:ℕ].  Ax ∈ is-list-if-has-value-fun(t;n) supposing is-list-if-has-value-fun(t;n)`

Proof

Definitions occuring in Statement :  is-list-if-has-value-fun: `is-list-if-has-value-fun(t;n)` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` member: `t ∈ T` base: `Base` axiom: `Ax`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` is-list-if-has-value-fun: `is-list-if-has-value-fun(t;n)` unit: `Unit` decidable: `Dec(P)` or: `P ∨ Q` exposed-it: `exposed-it` bool: `𝔹` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` subtype_rel: `A ⊆r B` has-value: `(a)↓` pi2: `snd(t)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` true: `True`
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 is-list-if-has-value-fun_wf base_wf primrec0_lemma unit_wf2 le_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma primrec-unroll eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int nat_wf assert_wf bnot_wf not_wf equal-wf-base int_subtype_base has-value-implies-dec-ispair-2 has-value_wf_base bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot isaxiom-implies
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality unionElimination because_Cache equalityElimination productElimination promote_hyp instantiate cumulativity applyEquality baseClosed baseApply closedConclusion impliesFunctionality

Latex:
\mforall{}[t:Base].  \mforall{}[n:\mBbbN{}].    Ax  \mmember{}  is-list-if-has-value-fun(t;n)  supposing  is-list-if-has-value-fun(t;n)

Date html generated: 2018_05_21-PM-10_19_24
Last ObjectModification: 2017_07_26-PM-06_36_59

Theory : eval!all

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