Nuprl Lemma : flip-permutes-permutations-list
∀n:ℕ. ∀i,j:ℕn. permutation({p:ℕn ⟶ ℕn| Inj(ℕn;ℕn;p)} ;permutations-list(n);map(λf.(f o (i, j));permutations-list(n)))
Proof
Definitions occuring in Statement :
permutations-list: permutations-list(n),
permutation: permutation(T;L1;L2),
flip: (i, j),
map: map(f;as),
inject: Inj(A;B;f),
compose: f o g,
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: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
injection: A →⟶ B,
guard: {T},
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
squash: ↓T,
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
uimplies: b supposing a,
compose: f o g,
label: ...$L... t,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P)
Lemmas referenced :
permutation-of-permutations-list,
inject_wf,
int_seg_wf,
compose_wf-injection,
flip-injection,
flip_wf,
subtype_rel_self,
istype-nat,
compose_wf,
inject-compose,
equal_wf,
squash_wf,
true_wf,
istype-universe,
flip_twice,
int_seg_properties,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
iff_weakening_equal,
sq_stable__inject
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesis,
because_Cache,
functionIsType,
inhabitedIsType,
closedConclusion,
dependent_set_memberEquality_alt,
universeIsType,
applyEquality,
sqequalRule,
setEquality,
functionEquality,
setIsType,
independent_functionElimination,
equalityIstype,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
instantiate,
universeEquality,
functionExtensionality_alt,
productElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i,j:\mBbbN{}n.
permutation(\{p:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;p)\} ;permutations-list(n);
map(\mlambda{}f.(f o (i, j));permutations-list(n)))
Date html generated:
2019_10_15-AM-11_21_55
Last ObjectModification:
2018_11_27-AM-00_31_09
Theory : general
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