Nuprl Lemma : l_all_functionality_wrt_permutation
∀[A:Type]. ∀P:A ⟶ ℙ. ∀L1,L2:A List. (permutation(A;L1;L2) ⇒ {(∀x∈L1.P[x]) ⇐⇒ (∀x∈L2.P[x])})
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
l_all: (∀x∈L.P[x]),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
l_all_iff,
l_member_wf,
member-permutation,
l_all_wf,
permutation_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
hypothesis,
productElimination,
independent_functionElimination,
because_Cache,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}P:A {}\mrightarrow{} \mBbbP{}. \mforall{}L1,L2:A List. (permutation(A;L1;L2) {}\mRightarrow{} \{(\mforall{}x\mmember{}L1.P[x]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x\mmember{}L2.P[x])\})
Date html generated:
2016_05_14-PM-02_34_13
Last ObjectModification:
2015_12_26-PM-04_20_21
Theory : list_1
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