### Nuprl Lemma : rel-preserving-star-reachable

`∀[T1,T2:Type]. ∀[i2:T2]. ∀[R1:T1 ⟶ T1 ⟶ Type]. ∀[R2:T2 ⟶ T2 ⟶ Type].`
`  ∀f:T2 ⟶ T1`
`    ((∀x,y:{s:T2| i2 (R2^*) s} .  ((x R2 y) `` ((f x) (R1^*) (f y))))`
`    `` {∀x,y:{s:T2| i2 (R2^*) s} .  ((x (R2^*) y) `` ((f x) (R1^*) (f y)))})`

Proof

Definitions occuring in Statement :  rel_star: `R^*` uall: `∀[x:A]. B[x]` guard: `{T}` infix_ap: `x f y` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  guard: `{T}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` rel_star: `R^*` infix_ap: `x f y` exists: `∃x:A. B[x]` member: `t ∈ T` subtype_rel: `A ⊆r B` prop: `ℙ` 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)` not: `¬A` false: `False` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` uiff: `uiff(P;Q)` bfalse: `ff`
Lemmas referenced :  rel_star_wf subtype_rel_self istype-universe rel_exp_wf istype-void istype-le decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int 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 subtract_wf itermSubtract_wf int_term_value_subtract_lemma istype-less_than primrec-wf2 rel_star_weakening equal_wf squash_wf true_wf iff_weakening_equal eq_int_wf intformeq_wf int_formula_prop_eq_lemma assert_wf bnot_wf not_wf equal-wf-base int_subtype_base istype-assert 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 rel_rel_star
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation_alt lambdaFormation_alt sqequalHypSubstitution productElimination thin universeIsType cut applyEquality introduction extract_by_obid isectElimination hypothesisEquality functionExtensionality hypothesis instantiate functionEquality cumulativity universeEquality setElimination rename inhabitedIsType setIsType because_Cache functionIsType dependent_functionElimination independent_functionElimination dependent_set_memberEquality_alt natural_numberEquality independent_pairFormation voidElimination unionElimination independent_isectElimination approximateComputation dependent_pairFormation_alt lambdaEquality_alt int_eqEquality Error :memTop,  setEquality imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed intEquality equalityIstype sqequalBase

Latex:
\mforall{}[T1,T2:Type].  \mforall{}[i2:T2].  \mforall{}[R1:T1  {}\mrightarrow{}  T1  {}\mrightarrow{}  Type].  \mforall{}[R2:T2  {}\mrightarrow{}  T2  {}\mrightarrow{}  Type].
\mforall{}f:T2  {}\mrightarrow{}  T1
((\mforall{}x,y:\{s:T2|  i2  rel\_star(T2;  R2)  s\}  .    ((x  R2  y)  {}\mRightarrow{}  ((f  x)  rel\_star(T1;  R1)  (f  y))))
{}\mRightarrow{}  \{\mforall{}x,y:\{s:T2|  i2  (R2\^{}*)  s\}  .
((x  (R2\^{}*)  y)  {}\mRightarrow{}  ((f  x)  (R1\^{}*)  (f  y)))\})

Date html generated: 2020_05_20-AM-08_11_45
Last ObjectModification: 2020_01_26-PM-00_16_16

Theory : general

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