Nuprl Lemma : inductively-defined-unique
∀R:Set{i:l} ⟶ Set{i:l} ⟶ ℙ'. ∀s1,s2:Set{i:l}.
(inductively-defined{i:l}(x,a.R[x;a];s1) ⇒ inductively-defined{i:l}(x,a.R[x;a];s2) ⇒ seteq(s1;s2))
Proof
Definitions occuring in Statement :
inductively-defined: inductively-defined{i:l}(x,a.R[x; a];s),
seteq: seteq(s1;s2),
Set: Set{i:l},
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
inductively-defined: inductively-defined{i:l}(x,a.R[x; a];s),
guard: {T},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
uall: ∀[x:A]. B[x],
prop: ℙ,
cand: A c∧ B,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
Set_wf,
inductively-defined_wf,
seteq-iff-setsubset
Rules used in proof :
universeEquality,
cumulativity,
functionEquality,
applyEquality,
lambdaEquality,
sqequalRule,
isectElimination,
independent_pairFormation,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}R:Set\{i:l\} {}\mrightarrow{} Set\{i:l\} {}\mrightarrow{} \mBbbP{}'. \mforall{}s1,s2:Set\{i:l\}.
(inductively-defined\{i:l\}(x,a.R[x;a];s1)
{}\mRightarrow{} inductively-defined\{i:l\}(x,a.R[x;a];s2)
{}\mRightarrow{} seteq(s1;s2))
Date html generated:
2018_05_29-PM-01_54_25
Last ObjectModification:
2018_05_25-PM-05_22_13
Theory : constructive!set!theory
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