Nuprl Lemma : permutation-generators4
∀n:ℕ
∀[P:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ⟶ ℙ]
(P[λx.x]
⇒ (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . ∀j:ℕ+n. (P[f] ⇒ P[f o (0, 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,
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 :
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
prop: ℙ,
int_seg: {i..j-},
uimplies: b supposing a,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
sq_type: SQType(T),
squash: ↓T,
sq_stable: SqStable(P),
compose: f o g,
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
flip: (i, j)
Lemmas referenced :
nat_wf,
identity-injection,
flip_wf,
decidable__le,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
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,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
int_seg_properties,
false_wf,
flip-injection,
compose-injections,
all_wf,
set_wf,
less_than_wf,
inject_wf,
int_seg_wf,
member-less_than,
permutation-generators3,
int_formula_prop_eq_lemma,
intformeq_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
sq_stable__inject,
inject-compose,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf
Rules used in proof :
cumulativity,
instantiate,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
productElimination,
independent_pairFormation,
universeEquality,
lambdaEquality,
sqequalRule,
dependent_set_memberEquality,
because_Cache,
natural_numberEquality,
functionEquality,
setEquality,
functionExtensionality,
applyEquality,
independent_isectElimination,
rename,
setElimination,
independent_functionElimination,
isectElimination,
isect_memberFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
hyp_replacement,
imageElimination,
baseClosed,
imageMemberEquality,
promote_hyp,
equalityElimination
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{}j:\mBbbN{}\msupplus{}n. (P[f] {}\mRightarrow{} P[f o (0, 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_59
Last ObjectModification:
2017_12_11-PM-02_20_47
Theory : list_1
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