Nuprl Lemma : strongwf-monotone
∀[T:Type]. ∀[R1,R2:T ⟶ T ⟶ Type]. (R2 => R1 ⇒ SWellFounded(R1[x;y]) ⇒ SWellFounded(R2[x;y]))
Proof
Definitions occuring in Statement :
strongwellfounded: SWellFounded(R[x; y]),
rel_implies: R1 => R2,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
rel_implies: R1 => R2,
infix_ap: x f y,
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
all_wf,
less_than_wf,
nat_wf,
strongwellfounded_wf,
rel_implies_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
cut,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
because_Cache,
lemma_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
setElimination,
rename,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R1,R2:T {}\mrightarrow{} T {}\mrightarrow{} Type]. (R2 => R1 {}\mRightarrow{} SWellFounded(R1[x;y]) {}\mRightarrow{} SWellFounded(R2[x;y]))
Date html generated:
2016_05_14-PM-03_52_16
Last ObjectModification:
2015_12_26-PM-06_57_11
Theory : relations2
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