Nuprl Lemma : permutation-generators3
∀n:ℕ
∀[P:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ⟶ ℙ]
(P[λx.x]
⇒ (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . ∀i,j:ℕn. P[f] ⇒ P[f o (i, j)] supposing i < j)
⇒ (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . P[f]))
Proof
Definitions occuring in Statement :
flip: (i, j),
inject: Inj(A;B;f),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
int_seg: {i..j-},
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
true: True,
squash: ↓T,
less_than: a < b,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
compose-flips: compose-flips(flips),
sq_stable: SqStable(P),
compose: f o g
Lemmas referenced :
nat_wf,
identity-injection,
less_than_wf,
isect_wf,
all_wf,
inject_wf,
int_seg_wf,
set_wf,
permutation-generators2,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
false_wf,
member-less_than,
iff_weakening_equal,
inject-compose,
rotate-injection,
rotate-as-flips,
equal_wf,
compose_wf,
list_wf,
compose-flips_wf,
compose-flips-injection,
reduce_cons_lemma,
map_cons_lemma,
reduce_nil_lemma,
map_nil_lemma,
list_induction,
sq_stable__inject,
flip-injection,
flip_wf,
comp_assoc,
true_wf,
squash_wf,
flip_symmetry,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_seg_properties,
flip_identity
Rules used in proof :
cumulativity,
instantiate,
universeEquality,
dependent_set_memberEquality,
setEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
sqequalRule,
because_Cache,
rename,
setElimination,
natural_numberEquality,
functionEquality,
independent_functionElimination,
isectElimination,
isect_memberFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
baseClosed,
imageMemberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
independent_pairFormation,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
productElimination,
productEquality,
levelHypothesis,
hyp_replacement,
addLevel,
imageElimination
Latex:
\mforall{}n:\mBbbN{}
\mforall{}[P:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} {}\mrightarrow{} \mBbbP{}]
(P[\mlambda{}x.x]
{}\mRightarrow{} (\mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} . \mforall{}i,j:\mBbbN{}n. P[f] {}\mRightarrow{} P[f o (i, j)] supposing i < j)
{}\mRightarrow{} (\mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} . P[f]))
Date html generated:
2018_05_21-PM-00_42_43
Last ObjectModification:
2017_12_10-PM-03_57_02
Theory : list_1
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