Nuprl Lemma : wellfounded_functionality_wrt_implies
∀[T1,T2:Type]. ∀[r1:T1 ⟶ T1 ⟶ ℙ]. ∀[r2:T2 ⟶ T2 ⟶ ℙ].
(∀x,y:T1. {r1[x;y] ⇐ r2[x;y]}) ⇒ {WellFnd{i}(T1;x,y.r1[x;y]) ⇒ WellFnd{i}(T2;x,y.r2[x;y])}
supposing T1 = T2 ∈ Type
Proof
Definitions occuring in Statement :
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
guard: {T},
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
squash: ↓T,
true: True,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
uall_wf,
rev_implies_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
universeEquality,
instantiate,
because_Cache,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[r1:T1 {}\mrightarrow{} T1 {}\mrightarrow{} \mBbbP{}]. \mforall{}[r2:T2 {}\mrightarrow{} T2 {}\mrightarrow{} \mBbbP{}].
(\mforall{}x,y:T1. \{r1[x;y] \mLeftarrow{}{} r2[x;y]\}) {}\mRightarrow{} \{WellFnd\{i\}(T1;x,y.r1[x;y]) {}\mRightarrow{} WellFnd\{i\}(T2;x,y.r2[x;y])\}
supposing T1 = T2
Date html generated:
2016_10_21-AM-09_35_54
Last ObjectModification:
2016_07_12-AM-05_00_01
Theory : well_fnd
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