Nuprl Lemma : perm_induction_a
∀n:ℕ. ∀Q:Sym(n) ⟶ ℙ.
(Q[id_perm()] ⇒ (∀p:Sym(n). (Q[p] ⇒ (∀i:{1..n-}. Q[txpose_perm(i;0) O p]))) ⇒ {∀p:Sym(n). Q[p]})
Proof
Definitions occuring in Statement :
txpose_perm: txpose_perm,
sym_grp: Sym(n),
comp_perm: comp_perm,
id_perm: id_perm(),
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
sym_grp: Sym(n),
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T)
Lemmas referenced :
perm_mon_assoc,
triple_txpose_perm,
int_formula_prop_eq_lemma,
intformeq_wf,
int_subtype_base,
subtype_base_sq,
txpose_perm_sym,
perm_mon_ident,
iff_weakening_equal,
txpose_perm_id,
true_wf,
squash_wf,
decidable__equal_int,
nat_wf,
id_perm_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_seg_properties,
false_wf,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
txpose_perm_wf,
comp_perm_wf,
all_wf,
int_seg_wf,
perm_wf,
perm_induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesis,
independent_functionElimination,
functionEquality,
universeEquality,
because_Cache,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
cumulativity,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate
Latex:
\mforall{}n:\mBbbN{}. \mforall{}Q:Sym(n) {}\mrightarrow{} \mBbbP{}.
(Q[id\_perm()]
{}\mRightarrow{} (\mforall{}p:Sym(n). (Q[p] {}\mRightarrow{} (\mforall{}i:\{1..n\msupminus{}\}. Q[txpose\_perm(i;0) O p])))
{}\mRightarrow{} \{\mforall{}p:Sym(n). Q[p]\})
Date html generated:
2016_05_16-AM-07_35_35
Last ObjectModification:
2016_01_16-PM-11_13_35
Theory : list_2
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