Nuprl Lemma : permutation-generators5
∀n:ℕ
∀[P:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ⟶ ℙ]
(P[λx.x]
⇒ (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . ∀i:ℕn - 1. (P[f] ⇒ P[f o (i, i + 1)]))
⇒ (∀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],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
int_seg: {i..j-},
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
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,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
cand: A c∧ B,
uiff: uiff(P;Q),
sq_type: SQType(T),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
less_than': less_than'(a;b),
absval: |i|,
compose: f o g,
flip: (i, j),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
permutation-generators3,
member-less_than,
istype-less_than,
int_seg_wf,
inject_wf,
subtract_wf,
subtype_rel_self,
compose-injections,
flip-injection,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
int_seg_properties,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
flip_wf,
identity-injection,
istype-nat,
le_witness_for_triv,
set_subtype_base,
lelt_wf,
int_subtype_base,
istype-void,
absval_wf,
primrec-wf2,
all_wf,
isect_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
absval_ubound,
decidable__equal_int,
subtype_base_sq,
itermMinus_wf,
int_term_value_minus_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
compose_wf,
flip_symmetry,
iff_weakening_equal,
sq_stable__inject,
nat_wf,
le_wf,
istype-false,
subtract-add-cancel,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
absval-diff-symmetry
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation_alt,
isectElimination,
independent_functionElimination,
setElimination,
rename,
independent_isectElimination,
universeIsType,
applyEquality,
because_Cache,
sqequalRule,
inhabitedIsType,
natural_numberEquality,
setIsType,
functionIsType,
instantiate,
universeEquality,
productElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
voidElimination,
productIsType,
addEquality,
imageElimination,
independent_pairEquality,
functionIsTypeImplies,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
intEquality,
sqequalBase,
isectIsType,
functionEquality,
cumulativity,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
minusEquality,
functionExtensionality,
equalityElimination,
promote_hyp,
baseApply,
closedConclusion
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:\mBbbN{}n - 1. (P[f] {}\mRightarrow{} P[f o (i, i + 1)]))
{}\mRightarrow{} (\mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} . P[f]))
Date html generated:
2020_05_19-PM-09_44_47
Last ObjectModification:
2020_01_04-PM-08_24_27
Theory : list_1
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