Nuprl Lemma : wellfounded_functionality_wrt_iff
∀[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,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
so_lambda: λ2x y.t[x; y],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x]
Lemmas referenced :
wellfounded_functionality_wrt_implies,
all_wf,
iff_wf,
equal_wf
Rules used in proof :
functionEquality,
universeEquality,
instantiate,
hyp_replacement,
applyEquality,
because_Cache,
lambdaEquality,
sqequalRule,
hypothesisEquality,
cumulativity,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
lambdaFormation,
rename,
thin,
hypothesis,
axiomEquality,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
independent_functionElimination,
independent_isectElimination,
productElimination,
dependent_functionElimination,
equalitySymmetry
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{}{}\mRightarrow{} r2[x;y])) {}\mRightarrow{} (WellFnd\{i\}(T1;x,y.r1[x;y]) \mLeftarrow{}{}\mRightarrow{} WellFnd\{i\}(T2;x,y.r2[x;y]))
supposing T1 = T2
Date html generated:
2019_06_20-AM-11_19_21
Last ObjectModification:
2018_10_15-PM-09_53_37
Theory : well_fnd
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