### Nuprl Lemma : non-forking-rel_exp

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  (non-forking(T;x,y.R[x;y]) `` (∀n:ℕ. non-forking(T;x,y.x λx,y. R[x;y]^n y)))`

Proof

Definitions occuring in Statement :  non-forking: `non-forking(T;x,y.R[x; y])` rel_exp: `R^n` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]` nat: `ℕ` 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` non-forking: `non-forking(T;x,y.R[x; y])` so_apply: `x[s1;s2]` rel_exp: `R^n` eq_int: `(i =z j)` infix_ap: `x f y` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` decidable: `Dec(P)` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` nequal: `a ≠ b ∈ T ` so_apply: `x[s]` so_lambda: `λ2x y.t[x; y]`
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 infix_ap_wf rel_exp_wf equal_wf le_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int exists_wf nat_wf non-forking_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin 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 instantiate cumulativity because_Cache universeEquality applyEquality functionExtensionality equalityTransitivity equalitySymmetry dependent_set_memberEquality unionElimination equalityElimination productElimination promote_hyp hyp_replacement applyLambdaEquality productEquality functionEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(non-forking(T;x,y.R[x;y])  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  non-forking(T;x,y.x  rel\_exp(T;  \mlambda{}x,y.  R[x;y];  n)  y)))

Date html generated: 2018_05_21-PM-09_05_16
Last ObjectModification: 2017_07_26-PM-06_28_05

Theory : general

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