hcju/isabelleps huggingface dataset:
finite_ring_finite_units: assumes "finite (carrier R)" shows "finite (Units R)"
HCP_adv_neg: shows "Negligible.negligible hcp_advantage"
remainder_extensionality: fixes \<phi> :: "'a"   and \<psi> :: "'a"   and A :: "'a set" assumes "Cn {\<phi>} = Cn {\<psi>}" shows "A .\<bottom>. \<phi> = A .\<bottom>. \<psi>"
bres_iso: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x y" shows "less_eq_prod (prod.bres z x) (prod.bres z y)"
assms_3: shows "c \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
lfinite_iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "\<not> lfinite (iterates f x)"
trans: fixes b :: "real \<times> real"   and a :: "real \<times> real"   and c :: "real \<times> real" assumes "le b a"   and "le c b" shows "le c a"
to_tree_order_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_to_tree_order_wf: shows "l_to_tree_order_wf heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.to_tree_order Shadow_DOM.get_parent Shadow_DOM.get_child_nodes"
mem_ball: fixes y :: "'d"   and x :: "'d"   and e :: "real" shows "(y \<in> ball x e) = (dist x y < e)"
sqmatrix_mult_1_left: fixes A :: "('c, 'd) sqmatrix" shows "1 \<cdot> A = A"
assms_5: shows "\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> P"
get_owner_document_owner_document_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_owner_document ptr \<rightarrow>\<^sub>r owner_document" shows "owner_document |\<in>| document_ptr_kinds h"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sim> y"   and "y \<sim> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sim> z"
comparator_option_simps_3: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c" shows "comparator_option comp\<^sub>'\<^sub>a (Some x) None = Gt"
assms: fixes i :: "'i" assumes "i \<in> I" shows "f i \<in> borel_measurable M"
Id_in_Hom: fixes A :: "'a" assumes "A \<in> Collect obj" shows "\<lbrakk>A\<rbrakk> \<in> Maps.Hom A A"
is_ideal: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R" shows "ideal I R"
ode_fa_def: fixes xs :: "floatarith list" shows "ode_fa xs = map (subst_floatarith ((!) xs)) ode_e"
as1: fixes n :: "nat" shows "bc n \<le> bc (Suc n)"
dvd_gcd_0_iff_2: fixes x :: "'a"   and a :: "'a" shows "(x dvd gcd a (0::'a)) = (x dvd a)"
B_theorem_3_b: shows "\<Theta> i * f i = ff (lead_coeff (H i))"
normalize_\<pi>: fixes \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl" assumes "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> the (\<pi> \<chi>) < max_rank_of \<Sigma> \<chi>"   and "accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (M_fin \<pi>, UNIV) w"   and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (Acc\<^sub>\<A> \<Sigma> \<pi> \<chi>) w" obtains \<pi>\<^sub>\<A> :: "'a ltl \<Rightarrow> nat option" where "dom \<pi> = dom \<pi>\<^sub>\<A>" and "\<And>\<chi>. \<chi> \<in> dom \<pi>\<^sub>\<A> \<Longrightarrow> \<pi>\<^sub>\<A> \<chi> = mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)) w {q. dom \<pi>\<^sub>\<A> \<up>\<Turnstile>\<^sub>P q}" and "accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (M_fin \<pi>\<^sub>\<A>, UNIV) w" and "\<And>\<chi>. \<chi> \<in> dom \<pi>\<^sub>\<A> \<Longrightarrow> accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (Acc\<^sub>\<A> \<Sigma> \<pi>\<^sub>\<A> \<chi>) w"
axioms_1: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "module_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2" shows "module_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1"
weak_Nullstellensatz: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "finite X"   and "F \<subseteq> P[X]"   and "\<V> F = {}" shows "ideal F = UNIV"
fds_abs_converges_mult_const_left: fixes f :: "'b fds"   and s :: "'b"   and c :: "'b" assumes "fds_abs_converges f s" shows "fds_abs_converges (fds_const c * f) s"
bv_to_int_upper_limit_lem1: fixes w :: "bit list" assumes "bv_to_int w < - 1" shows "bv_to_int w < - (2 ^ (length (norm_signed w) - 2))"
outputFreshChainDerivative_2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and yvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "xvec \<sharp>* M"   and "distinct xvec"   and "yvec \<sharp>* P"   and "yvec \<sharp>* xvec" shows "yvec \<sharp>* P'"
inf_bot_right: fixes x :: "'a" shows "x \<cdot> 0 = 0"
simps_2: fixes x :: "'b"   and y :: "'b"   and ys :: "'b list" shows "PermutationLemmas.count x (y # ys) = (if x = y then 1 else 0) + PermutationLemmas.count x ys"
ode_d2_na_def: shows "ode_d2_na = (\<lambda>tx. flip_blinfun (flip_blinfun (ode_d2 (snd tx) o\<^sub>L snd_blinfun) o\<^sub>L snd_blinfun))"
rel_mset_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and X :: "'c multiset"   and Y :: "'d multiset" shows "rel_mset R X Y = (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"
strict_prefix_eq_strict_prefix_take_ex: fixes xs :: "'b list"   and ys :: "'b list" shows "strict_prefix xs ys = ((\<exists>n. ys \<down> n = xs) \<and> xs \<noteq> ys)"
A_authenticates_B: fixes servK :: "nat"   and T3 :: "nat"   and evs :: "event list"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and A :: "agent"   and Ta :: "nat"   and authTicket :: "msg" assumes "Crypt servK (Number T3) \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Says B A (Crypt servK (Number T3)) \<in> set evs"
Maclaurin_bi_le: fixes diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and n :: "nat"   and x :: "real" assumes "diff 0 = f"   and "\<forall>m t. m < n \<and> \<bar>t\<bar> \<le> \<bar>x\<bar> \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)" shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
infinite_imp_nonempty: fixes S :: "'c set" assumes "infinite S" shows "S \<noteq> {}"
final_read_final1: fixes c :: "'b"   and r :: "(bool \<times> 'b) rexp" shows "final (read c r) = final1 r c"
implc_get_offending_flows_complies: fixes M :: "('a SecurityInvariant \<times> 'a SecurityInvariant_configured) list"   and G :: "'a list_graph" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec"   and "wf_list_graph G" shows "set ` set (implc_get_offending_flows (get_impl M) G) = get_offending_flows (get_spec M) (list_graph_to_graph G)"
id_FSwImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'a, 'b, 'd) model_scheme" shows "FSwImorph id id MOD MOD"
1: shows "transp (\<lambda>a\<^sub>1 a\<^sub>2. g a\<^sub>1 = g a\<^sub>2)"
AE_borel_eq: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and f :: "'c \<Rightarrow> real"   and t :: "'d"   and g :: "'c \<Rightarrow> real" assumes "filtrated_prob_space M F"   and "f \<in> borel_measurable (F t)"   and "g \<in> borel_measurable (F t)"   and "AE w in M. f w = g w" shows "{w \<in> space M. f w \<noteq> g w} \<in> sets (F t) \<and> emeasure M {w \<in> space M. f w \<noteq> g w} = 0"
AE_integrable_fst': fixes f :: "'b \<times> 'a \<Rightarrow> 'c" assumes "integrable (M2 \<Otimes>\<^sub>M M1) f" shows "AE x in M2. integrable M1 (\<lambda>y. f (x, y))"
that_2: fixes d :: "nat"   and q :: "nat" assumes "\<And>m. \<lbrakk>0 < d; q < m\<rbrakk> \<Longrightarrow> g q \<le> d * f q" shows "thesis"
joinableE: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'c) set" assumes "(x, y) \<in> joinable R" obtains z :: "'c" where "(x, z) \<in> R" and "(y, z) \<in> R"
div_plus_div_distrib_dvd_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times c b" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
cd_ent: shows "CC \<union> DD \<Turnstile> EE"
lless_def: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" shows "(x \<sqsubset>\<^bsub>L\<^esub> y) = (x \<sqsubseteq>\<^bsub>L\<^esub> y \<and> x .\<noteq>\<^bsub>L\<^esub> y)"
step1: fixes B :: "'a set"   and cs :: "'a list"   and b :: "'a"   and C :: "'a set" assumes "B \<subseteq> A"   and "cs \<in> local.mk_linear_orders {} (B - {b})"   and "b \<in> f B"   and "f B \<subseteq> C"   and "P {} (B - {b}) cs" shows "P C B (b # cs)"
u: shows "unitary (tensor_P mat_O (1\<^sub>m K))"
set_fs: shows "set fs \<subseteq> carrier (Ring_Divisibility.mult_of R)"
vv: shows "v = [y, x] @ c @ [y, y]"
p_def: shows "p = butlast (prio_emb_pos prio (prio_emb_step prio t s) s)"
double_sum_equiv: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and n :: "nat" shows "(\<Sum>k = 1..n. \<Sum>j = 1..n. f k * g j) = (\<Sum>k = 1..n. \<Sum>j = 1..n. f j * g k)"
HFun_Sigma_Zero: fixes H :: "fm set" shows "H \<turnstile> HFun_Sigma Zero"
subset_UNIV: fixes A :: "'g set" shows "A \<subseteq> UNIV"
from_nat_to_nat_id: fixes x :: "'b" shows "mod_type_class.from_nat (mod_type_class.to_nat x) = x"
inf_cvec_z2_zero_iff: fixes z1 :: "complex"   and z2 :: "complex" assumes "(z1, z2) \<noteq> vec_zero" shows "((z1, z2) \<approx>\<^sub>v \<infinity>\<^sub>v) = (z2 = 0)"
chart_basis_inv_inverse: fixes x :: "'a" shows "chart_basis b (local.chart_basis_inv x) = x"
f5: shows "rk {A', C', C, \<beta>} = 2"
sym_mpoly_infinite: fixes A :: "nat set"   and k :: "nat" assumes "infinite A" shows "sym_mpoly A k = 0"
H: shows "\<Gamma> \<turnstile> (\<lambda>X<:S. s) : T"
All2_E': fixes H :: "fm set"   and i :: "name"   and t :: "tm"   and A :: "fm"   and x :: "tm"   and B :: "fm" assumes "H \<turnstile> All2 i t A"   and "H \<turnstile> x IN t"   and "insert (A(i::=x)) H \<turnstile> B"   and "atom i \<sharp> t" shows "H \<turnstile> B"
Abs: fixes T :: "Lambda_Example.type"   and env :: "Lambda_Example.type list"   and t :: "dB"   and U :: "Lambda_Example.type" assumes "T # env \<turnstile> t : U" shows "env \<turnstile> Abs T t : T \<Rightarrow> U"
PhiWhileOp_def: fixes b :: "BExpr"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" shows "PhiWhileOp b \<Phi> = (\<lambda>\<Psi> (s, t, \<beta>). (evalB b (fst t) \<longrightarrow> (\<exists>r. \<Phi> (r, t, \<beta>) \<and> (\<forall>w \<gamma>. r \<equiv>\<^sub>\<gamma> w \<longrightarrow> \<Psi> (s, w, \<gamma>)))) \<and> (\<not> evalB b (fst t) \<longrightarrow> s \<equiv>\<^sub>\<beta> t))"
admits_triangular: shows "\<forall>A. admits_triangular_reduction A"
prod_list_map_remove1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'c"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'b" assumes "class.comm_monoid_mult times one"   and "x \<in> set xs" shows "monoid_mult.prod_list one times (map f xs) = times (f x) (monoid_mult.prod_list one times (map f (remove1 x xs)))"
invert_side_of_triangle_1: fixes h :: "'b"   and f :: "'b"   and g :: "'b" assumes "D.arr h"   and "f \<cdot>\<^sub>D g = h"   and "D.iso f" shows "D.seq (D.inv f) h \<and> g = D.inv f \<cdot>\<^sub>D h"
s_mul_extE: fixes A :: "'a multiset"   and B :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(A, B) \<in> s_mul_ext ns s" obtains A1 :: "'a multiset" and A2 :: "'a multiset" and B1 :: "'a multiset" and B2 :: "'a multiset" where "A = A1 + A2" and "B = B1 + B2" and "(A1, B1) \<in> multpw ns" and "A2 \<noteq> {#}" and "\<And>b. b \<in># B2 \<Longrightarrow> \<exists>a. a \<in># A2 \<and> (a, b) \<in> s"
listE_nth_in: fixes xs :: "'a list"   and n :: "nat"   and A :: "'a set"   and i :: "nat" assumes "xs \<in> list n A"   and "i < n" shows "xs ! i \<in> A"
graph_nodes_it_axioms: shows "graph_nodes_it gbm_\<alpha> gbm_invar gbm_nodes_list_it"
rga_ops_def: fixes list :: "('b \<times> 'b option) list" shows "rga_ops list \<equiv> crdt_ops list set_option"
norm_M: fixes v :: "(real, 2) vec" shows "vector2_append1 v \<bullet> (M *v vector2_append1 v) = (norm v)\<^sup>2 - 1"
S_alt_def: shows "S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u)) = {r} \<union> fst ` A (Q' Q \<pi> u) (\<pi>' Q \<pi> u)"
simps_8: fixes t :: "'c local.term" shows "\<^bold>\<lfloor>\<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
select_convs_1: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getSATFlag \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getSATFlag"
lookup_alt_def: fixes x :: "'v" shows "lookup x = get (\<lambda>s. case apply s x of None \<Rightarrow> fail | Some y \<Rightarrow> return y)"
filter_leD: fixes F :: "'e filter"   and F' :: "'e filter"   and P :: "'e \<Rightarrow> bool" assumes "F \<le> F'"   and "eventually P F'" shows "eventually P F"
Conjunctive_def: fixes Inf_b :: "'d set \<Rightarrow> 'd"   and Inf_c :: "'e set \<Rightarrow> 'e"   and times_abc :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "Conjunctive.Conjunctive Inf_b Inf_c times_abc = {x. \<forall>X. times_abc x (Inf_b X) = Inf_c (times_abc x ` X)}"
accI: fixes x :: "'s"   and y :: "'s" assumes "x \<in> X"   and "y \<in> X" shows "(x, y) \<in> acc"
fun_upd_other: fixes z :: "'b"   and x :: "'b"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "z \<noteq> x" shows "(f(x := y)) z = f z"
assms: fixes n :: "nat" assumes "n < length cl" shows "{} \<notin> P n"
conjugation_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and g :: "'c" assumes "Group.group G"   and "subgroup H G"   and "g \<in> carrier G" shows "subgroup (g <#\<^bsub>G\<^esub> (H #>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> g)) G"
dvd_div_eq_iff: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c dvd a"   and "c dvd b" shows "(a / c = b / c) = (a = b)"
rep_eq: fixes x :: "CoCalls"   and xa :: "CoCalls" shows "(x \<sqsubseteq> xa) = (Rep_CoCall x \<subseteq> Rep_CoCall xa)"
op_set_npick_remove_def': fixes X :: "'c set" assumes "X \<noteq> {}" shows "op_set_npick_remove X = ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_pick X \<bind> (\<lambda>x. op_set_ndelete x X \<bind> (\<lambda>X'. RETURN (x, X'))))"
axioms_2: fixes x :: "'a \<Longrightarrow> 'b" assumes "bij_lens x" shows "bij_lens_axioms x"
set_shadow_root_def: shows "set_shadow_root = (\<lambda>element_ptr. put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt_update)"
snoc_5: shows "\<forall>a\<in>set (vs @ [v]). case a of (idn, a) \<Rightarrow> typ_ok \<Theta> a"
reachable_induct: fixes m :: "'e"   and y :: "'f"   and s :: "'e process_name \<Rightarrow> ('g, 'e, 'h, 'f) Model.local_state"   and P :: "'f \<Rightarrow> bool" assumes "mut_m.reachable m y s"   and "\<And>x. x \<in> roots (s (mutator m)) \<Longrightarrow> P x"   and "\<And>x. x \<in> ghost_honorary_root (s (mutator m)) \<Longrightarrow> P x"   and "\<And>x. x \<in> mut_m.tso_store_refs m s \<Longrightarrow> P x"   and "\<And>x y. \<lbrakk>mut_m.reachable m x s; (x points_to y) s; P x\<rbrakk> \<Longrightarrow> P y" shows "P y"
Lyndon_mono_def: fixes ws :: "'a list list" shows "Lyndon_mono ws = ((\<forall>u\<in>set ws. Lyndon u) \<and> linorder.sorted (\<le>lex) (rev ws))"
alpha_fresh': fixes a :: "name"   and x :: "'b"   and b :: "name"   and y :: "'b"   and c :: "name" assumes "[a].x = [b].y"   and "c \<noteq> a"   and "c \<noteq> b"   and "c \<sharp> x"   and "c \<sharp> y" shows "x = [(a, c)] \<bullet> [(b, c)] \<bullet> y"
mkClockAuto_implements: shows "Robot.Clock.implements (mkClockAuto Agents (\<lambda>_. kbp) envInit envAction envTrans envVal (\<lambda>_. envObs))"
lconsNlist_inv: fixes n :: "nat"   and t :: "('a, 'b) FingerTreeStruc"   and xs :: "('a, 'b) Node list" assumes "FingerTreeStruc.is_leveln_ftree n t"   and "FingerTreeStruc.is_measured_ftree t"   and "\<forall>x\<in>set xs. FingerTreeStruc.is_leveln_node n x \<and> FingerTreeStruc.is_measured_node x" shows "FingerTreeStruc.is_leveln_ftree n (FingerTreeStruc.lconsNlist xs t) \<and> FingerTreeStruc.is_measured_ftree (FingerTreeStruc.lconsNlist xs t)"
simps: fixes p :: "int poly"   and uu :: "rat"   and uv :: "rat" shows "poly_real_alg_1 (p, uu, uv) = p"
1: shows "D.cones_map (induced_arrow' b F) \<pi> = D.mkCone F"
append_rows_le: fixes A :: "'a mat"   and nr1 :: "nat"   and nc :: "nat"   and B :: "'a mat"   and nr2 :: "nat"   and a :: "'a vec"   and v :: "'a vec"   and b :: "'a vec" assumes "A \<in> carrier_mat nr1 nc"   and "B \<in> carrier_mat nr2 nc"   and "a \<in> carrier_vec nr1"   and "v \<in> carrier_vec nc" shows "((A @\<^sub>r B) *\<^sub>v v \<le> a @\<^sub>v b) = (A *\<^sub>v v \<le> a \<and> B *\<^sub>v v \<le> b)"
ctx_red_in_ctx: fixes t :: "lam"   and t' :: "lam"   and E :: "ctx" assumes "t \<longrightarrow>x t'" shows "E\<lbrakk>t\<rbrakk> \<longrightarrow>x E\<lbrakk>t'\<rbrakk>"
f7: fixes n :: "nat" shows "min j n = length (take n (take j a))"
ipresCons_imp_ipresWlsAll: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" assumes "ipresCons h hA MOD"   and "igConsIPresIGWls MOD" shows "ipresWlsAll h hA MOD"
takeWhile_raw_def: fixes P :: "'e \<Rightarrow> bool"   and g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f" shows "takeWhile_raw P g s = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip s' | Yield a s' \<Rightarrow> if P a then Yield a s' else Done)"
induct: fixes x1 :: "nat"   and x2 :: "Lholed"   and x3 :: "e list"   and x4 :: "e list"   and P :: "nat \<Rightarrow> Lholed \<Rightarrow> e list \<Rightarrow> e list \<Rightarrow> bool" assumes "Lfilled x1 x2 x3 x4"   and "\<And>vs lholed es' es. \<lbrakk>const_list vs; lholed = LBase vs es'\<rbrakk> \<Longrightarrow> P 0 lholed es (vs @ es @ es')"   and "\<And>vs lholed n es' l es'' k es lfilledk. \<lbrakk>const_list vs; lholed = LRec vs n es' l es''; Lfilled k l es lfilledk; P k l es lfilledk\<rbrakk> \<Longrightarrow> P (k + 1) lholed es (vs @ [Label n es' lfilledk] @ es'')" shows "P x1 x2 x3 x4"
deriv_boundE: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and n :: "nat" assumes "deriv_bound r a n" obtains "\<And>b m. \<lbrakk>n < m; (a, b) \<in> r ^^ m\<rbrakk> \<Longrightarrow> False"
assms_1: shows "\<forall>m a. P m \<longrightarrow> match_list \<gamma> (f m) a p = matches \<gamma> m a p"
induct: fixes P :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> 'd"   and a1 :: "'c list" assumes "\<And>f. P f []"   and "\<And>f x xs. (\<And>xa xb y. \<lbrakk>xa = partition_tailrec (\<lambda>y. f x = f y) xs; (xb, y) = xa\<rbrakk> \<Longrightarrow> P f y) \<Longrightarrow> P f (x # xs)" shows "P a0 a1"
dom_dom: fixes f :: "'a" assumes "B.arr f" shows "B.dom (B.dom f) = B.dom f"
emb_injective: fixes J :: "'i set"   and L :: "'i set"   and X :: "('i \<Rightarrow> 'a) set"   and Y :: "('i \<Rightarrow> 'a) set" assumes "J \<subseteq> L"   and "L \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)"   and "Y \<in> sets (Pi\<^sub>M J M)"   and "emb L J X = emb L J Y" shows "X = Y"
vbrelation_vsubset: fixes s :: "V"   and r :: "V" assumes "vbrelation s"   and "r \<subseteq>\<^sub>\<circ> s" shows "vbrelation r"
leq_refl: fixes x :: "'a" shows "x \<le> x"
assms_3: shows "\<forall>n''\<ge>n. n'' < n' \<longrightarrow> \<not> \<parallel>c\<parallel>\<^bsub>t n''\<^esub>"
path_unconc: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and la :: "'b list"   and lb :: "'b list"   and w :: "'b" assumes "path E u (la @ lb) w" obtains v :: "'b" where "path E u la v" and "path E v lb w"
bsok_simps_16: fixes V :: "nat"   and T :: "ty"   and vo :: "'a val option"   and e :: "'a expr1"   and n :: "nat" shows "bsok {V:T=vo; e} n = (bsok e (Suc n) \<and> V = n)"
set_map_filter: fixes g :: "'e \<Rightarrow> 'd option"   and xs :: "'e list" shows "set (List.map_filter g xs) = {y. \<exists>x. x \<in> set xs \<and> g x = Some y}"
strict_iff_order: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subset># b) = (a \<subseteq># b \<and> a \<noteq> b)"
stutstep_notempty_sucnextnat: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "notemptyseq (s |\<^sub>s n)"   and "stutstep s n" shows "nextnat (s |\<^sub>s n) = Suc (nextnat (s |\<^sub>s Suc n))"
euclidean_all_zero_iff: fixes x :: "'a" shows "(\<forall>u\<in>Basis. inner x u = 0) = (x = (0::'a))"
BSI_implies_SI_for_modified_view: fixes \<V>' :: "'a V_rec" assumes "BSI \<V>' Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>" shows "SI \<V> Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"
of_nat_0_le_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (semiring_1.of_nat one plus zero n)"
coerce_Lifted: fixes x :: "'b" shows "COERCE('b\<cdot>lifted, 'a\<cdot>lifted)\<cdot>(Lifted\<cdot>x) = Lifted\<cdot>(COERCE('b, 'a)\<cdot>x)"
bij: shows "\<forall>b\<in>f ` A. bij_betw (bij b) {a \<in> A. f a = b} {a \<in> A'. f' a = b}"
A: shows "cring (R\<lparr>carrier := H\<rparr>)"
flat_interpretation: fixes b :: "'a" shows "partial_function_definitions (flat_ord b) (flat_lub b)"
diverge_denot_empty: fixes e :: "exp" assumes "diverge e"   and "FV e = {}" shows "E e [] = {}"
scale_cancel_left: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "(a * x = a * y) = (x = y \<or> a = (0::'a))"