### Nuprl Lemma : rel-preserving-star

`∀[T1,T2:Type]. ∀[R1:T1 ⟶ T1 ⟶ Type]. ∀[R2:T2 ⟶ T2 ⟶ Type].`
`  ∀f:T2 ⟶ T1. (λx.f[x]:T2->T1 takes R2 into R1*) `` λx.f[x]:T2->T1 takes R2^* into R1*))`

Proof

Definitions occuring in Statement :  rel-preserving: `λx.f[x]:T2->T1 takes R2 into R1*)` rel_star: `R^*` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` rel-preserving: `λx.f[x]:T2->T1 takes R2 into R1*)` rel_star: `R^*` infix_ap: `x f y` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rel_exp: `R^n` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bfalse: `ff`
Lemmas referenced :  rel_star_wf rel-preserving_wf infix_ap_wf rel_exp_wf false_wf le_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_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_var_lemma int_formula_prop_less_lemma int_formula_prop_wf all_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma set_wf less_than_wf primrec-wf2 nat_wf rel_star_weakening and_wf equal_wf eq_int_wf intformeq_wf int_formula_prop_eq_lemma assert_wf bnot_wf not_wf equal-wf-base int_subtype_base bool_cases subtype_base_sq bool_wf bool_subtype_base eqtt_to_assert assert_of_eq_int eqff_to_assert iff_transitivity iff_weakening_uiff assert_of_bnot rel_star_transitivity
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution sqequalRule productElimination thin applyEquality cut introduction extract_by_obid isectElimination cumulativity hypothesisEquality functionExtensionality hypothesis lambdaEquality functionEquality universeEquality dependent_functionElimination independent_functionElimination instantiate because_Cache dependent_set_memberEquality natural_numberEquality independent_pairFormation rename setElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll equalitySymmetry applyLambdaEquality equalityTransitivity baseClosed impliesFunctionality

Latex:
\mforall{}[T1,T2:Type].  \mforall{}[R1:T1  {}\mrightarrow{}  T1  {}\mrightarrow{}  Type].  \mforall{}[R2:T2  {}\mrightarrow{}  T2  {}\mrightarrow{}  Type].
\mforall{}f:T2  {}\mrightarrow{}  T1
(\mlambda{}x.f[x]:T2->T1  takes  R2  into  R1*)  {}\mRightarrow{}  \mlambda{}x.f[x]:T2->T1  takes  R2\^{}*  into  R1*))

Date html generated: 2018_05_21-PM-08_00_45
Last ObjectModification: 2017_07_26-PM-05_37_36

Theory : general

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