Nuprl Lemma : funinv-permutes-permutations-list
∀n:ℕ. permutation({p:ℕn ⟶ ℕn| Inj(ℕn;ℕn;p)} ;permutations-list(n);map(λf.inv(f);permutations-list(n)))
Proof
Definitions occuring in Statement :
permutations-list: permutations-list(n),
permutation: permutation(T;L1;L2),
funinv: inv(f),
map: map(f;as),
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
true: True,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x]
Lemmas referenced :
nat_wf,
int_seg_wf,
inject_wf,
funinv_wf2,
permutation-of-permutations-list,
set_wf,
equal_wf,
iff_weakening_equal,
funinv-funinv,
sq_stable__inject,
true_wf,
squash_wf
Rules used in proof :
independent_functionElimination,
functionEquality,
setEquality,
applyEquality,
functionExtensionality,
because_Cache,
natural_numberEquality,
hypothesis,
dependent_set_memberEquality,
rename,
setElimination,
isectElimination,
lambdaEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalRule,
independent_pairFormation,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
universeEquality,
equalityTransitivity,
imageElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation
Latex:
\mforall{}n:\mBbbN{}
permutation(\{p:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;p)\} ;permutations-list(n);map(\mlambda{}f.inv(f);permutations-list(n)))
Date html generated:
2018_05_21-PM-08_22_41
Last ObjectModification:
2017_12_11-AM-11_13_30
Theory : general
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