Nuprl Lemma : cond_rel_star_monotone
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[R1,R2:T ⟶ T ⟶ ℙ]. (when P, R1 => R2 ⇒ R1 preserves P ⇒ when P, R1^* => R2^*)
Proof
Definitions occuring in Statement :
rel_star: R^*,
cond_rel_implies: when P, R1 => R2,
preserved_by: R preserves P,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rel_star: R^*,
cond_rel_implies: when P, R1 => R2,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
rel_exp_wf,
exists_wf,
nat_wf,
preserved_by_wf,
all_wf,
cond_rel_exp_monotone
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
cut,
dependent_functionElimination,
hypothesis,
independent_functionElimination,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
lambdaEquality,
functionEquality,
Error :inhabitedIsType,
Error :functionIsType,
Error :universeIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[R1,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(when P, R1 => R2 {}\mRightarrow{} R1 preserves P {}\mRightarrow{} when P, rel\_star(T; R1) => rel\_star(T; R2))
Date html generated:
2019_06_20-PM-00_30_36
Last ObjectModification:
2018_09_26-PM-00_50_35
Theory : relations
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