hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "Var list \<times> (int \<times> Instr) list \<times> int"   and x2 :: "int"   and x3 :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and x4 :: "nat"   and x5 :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and P :: "Var list \<times> (int \<times> Instr) list \<times> int \<Rightarrow> int \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> nat \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "(x1, x2, x3, x4, x5) \<in> Reachable"   and "\<And>k t s M l. \<lbrakk>k = 0; t = s\<rbrakk> \<Longrightarrow> P M l s k t"   and "\<And>M l s n ll r m t k. \<lbrakk>(M, l, s, n, ll, r) \<in> Step; (M, ll, r, m, t) \<in> Reachable; P M ll r m t; k = Suc m + n\<rbrakk> \<Longrightarrow> P M l s k t"   and "\<And>C m par code l0 M l s ops S h R ops1 n t k. \<lbrakk>mbody_is C m (par, code, l0); get_ins M l = Some (invokeS C m); s = (ops, S, h); (ops, par, R, ops1) \<in> Frame; ((par, code, l0), l0, ([], R, h), n, t) \<in> Reachable; P (par, code, l0) l0 ([], R, h) n t; k = Suc n\<rbrakk> \<Longrightarrow> P M l s k t" shows "P x1 x2 x3 x4 x5"
UNION_singleton_eq_range: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "(\<Union>x\<in>A. {f x}) = f ` A"
sinit_ext: fixes xs :: "'b seq"   and ys :: "'b seq" assumes "\<forall>i. sinit i xs = sinit i ys" shows "xs = ys"
length_nat_def: fixes x :: "nat" shows "length_nat x = (LEAST n. x < 2 ^ n)"
wf_vars_mono: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and W :: "'c set" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t (V \<union> W) S"
sset_siterate: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "sset (siterate f x) = {(f ^^ n) x |n. True}"
hb_iso: shows "hom_boundary (int n) (subtopology (nsphere n) {x. 0 \<le> x n}) {x. x n = 0} \<in> Group.iso (relative_homology_group (int n) (subtopology (nsphere n) {x. 0 \<le> x n}) {x. x n = 0}) (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))"
assms_2: shows "r \<subseteq>\<^sub>\<circ> s"
from_nat_to_nat_id: fixes x :: "'d" shows "mod_type_class.from_nat (mod_type_class.to_nat x) = x"
above_def: fixes L :: "(alt \<times> alt) set"   and a :: "alt"   and b :: "alt" shows "Arrow_Order.above L a b \<equiv> {(x, y). x <\<^bsub>L\<^esub> y \<and> x \<noteq> b \<and> y \<noteq> b} \<union> {(a, b)} \<union> {(x, b) |x. x <\<^bsub>L\<^esub> a \<and> x \<noteq> b} \<union> {(b, y) |y. a <\<^bsub>L\<^esub> y \<and> y \<noteq> b}"
next_state_consistent: shows "\<forall>s t u execs. vpeq (current s) s t \<and> vpeq u s t \<and> current s = current t \<longrightarrow> vpeq u (int.next_state s execs) (int.next_state t execs)"
last_index_size_conv: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "length xs = n" shows "(last_index xs x = n) = (x \<notin> set xs)"
simps: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr" shows "(\<sigma>, \<tau>, e)\<^sub>\<tau> = \<tau>"
stable_leads_to_eventually: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(\<box>(\<lambda>s. enabled s \<longrightarrow> (\<circle>(\<lambda>s. enabled s \<or> taken s)) s)) \<sigma>" shows "(\<box>(\<lambda>s. enabled s \<longrightarrow> (\<box>enabled) s \<or> (\<diamond>taken) s)) \<sigma>"
monomial_0I: fixes c :: "'b"   and t :: "'c" assumes "c = (0::'b)" shows "monomial c t = 0"
bottom_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a" shows "bottom r B = (\<forall>x. B \<sqsubseteq>\<^bsub>r\<^esub> x)"
order_iff: fixes a :: "nat"   and b :: "nat" shows "(a dvd b) = (a = gcd a b)"
cases: fixes a1 :: "'a list"   and a2 :: "sequent" assumes "(a1, a2) \<in> upRules" obtains c :: "sequent" and R :: "char list" and Fs :: "form_list" and ps :: "'a list" where "a1 = ps" and "a2 = c" and "NominalSequents.mset c \<equiv> \<LM> Cpd0 R Fs  \<RM>" and "ps \<noteq> []"
cone_iff: fixes S :: "'c set" assumes "S \<noteq> {}" shows "cone S = ((0::'c) \<in> S \<and> (\<forall>c>0. (*\<^sub>R) c ` S = S))"
power_Suc: fixes a :: "'a"   and n :: "nat" shows "a +^ Suc n = a + a +^ n"
fds_zeta_commutes: fixes f :: "'b fds" shows "fds_zeta * f = f * fds_zeta"
sinks_cons_same: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and x :: "'f"   and xs :: "'f list" assumes "refl I" shows "sinks I D (D x) (x # xs) = insert (D x) (sinks I D (D x) xs)"
image_constant_conv: fixes c :: "'d"   and A :: "'e set" shows "(\<lambda>x. c) ` A = (if A = {} then {} else {c})"
relpow_0_I: fixes x :: "'b"   and R :: "('b \<times> 'b) set" shows "(x, x) \<in> R ^^ 0"
double_compl: fixes minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and uminus :: "'m \<Rightarrow> 'm"   and inf :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and less_eq :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and less :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and sup :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and bot :: "'m"   and top :: "'m"   and x :: "'m" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
a_closure': fixes x :: "'a" shows "antidomain_near_semiring.ans_d antirange_op ar x = ar x"
lead_coeff_pCons_1: fixes p :: "'c poly"   and a :: "'c" assumes "p \<noteq> 0" shows "lead_coeff (pCons a p) = lead_coeff p"
len_lexext_rev_compat_cons: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "len_lexext_rev gt ys xs" shows "len_lexext_rev gt (x # ys) (x # xs)"
mgp_no_loop: fixes g :: "graph"   and f :: "face"   and v :: "nat" assumes "minGraphProps g"   and "f \<in> \<F> g"   and "v \<in> \<V> f" shows "f \<bullet> v \<noteq> v"
static_mheadsD: fixes emh :: "ref_ty \<times> mhead"   and G :: "prog"   and S :: "qtname"   and t :: "ref_ty"   and sig :: "sig"   and E :: "env"   and e :: "expr" assumes "emh \<in> mheads G S t sig"   and "wf_prog G"   and "E\<turnstile>e\<Colon>-RefT t"   and "prg E = G"   and "invmode (mhd emh) e \<noteq> IntVir" shows "\<exists>m. ((\<exists>C. t = ClassT C \<and> accmethd G S C sig = Some m) \<or> (\<forall>C. t \<noteq> ClassT C \<and> accmethd G S Object sig = Some m)) \<and> declrefT emh = ClassT (declclass m) \<and> mhead (mthd m) = mhd emh"
trace_callee_None: fixes callee :: "'i \<Rightarrow> 'j \<Rightarrow> ('h \<times> 'i) spmf"   and xs :: "('j \<times> 'h) list"   and x :: "'j" shows "trace_callee callee (return_pmf None) xs x = return_pmf None"
fsubsetI: fixes A :: "'c fset"   and B :: "'c fset" assumes "\<And>x. x |\<in>| A \<Longrightarrow> x |\<in>| B" shows "A |\<subseteq>| B"
simps: fixes \<phi> :: "'b ltl"   and \<nu> :: "'b set" shows "af_G_letter_opt \<phi> \<nu> = Unf\<^sub>G (step \<phi> \<nu>)"
l1_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "l1_step2 Rb A B gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> dhlvl1.guessed_frame Rb xgnx = Some gnx \<and> dhlvl1.guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny))) \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xgny, xgnx, xsk}), signalsInit := if can_signal s A B then addSignal (signalsInit s) (Running A B (Exp gnx (NonceF (Rb $ ny)))) else signalsInit s\<rparr>}"
ShiftP_graph_aux_def: shows "ShiftP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>ShiftP_aux x x' y z f del k g. x1 = (f, k, del, g) \<and> x2 = SyntaxN.All z (Var z IN g IFF SyntaxN.Ex x (SyntaxN.Ex x' (SyntaxN.Ex y (Var z EQ HPair (Var x') (Var y) AND HaddP del (Var x) (Var x') AND HPair (Var x) (Var y) IN f AND Var x IN k)))) \<and> atom x \<sharp> (x', y, z, f, del, k) \<and> atom x' \<sharp> (y, z, f, del, k) \<and> atom y \<sharp> (z, f, del, k) \<and> atom z \<sharp> (f, del, g, k))"
i_not_b: shows "mod_type_class.from_nat i \<noteq> b"
hom_between_subgroups: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and A :: "'a set"   and B :: "'c set" assumes "group_hom G H h"   and "h ` A \<subseteq> B" shows "h \<in> hom (subgroup_generated G A) (subgroup_generated H B)"
INVS_SC_def: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and D :: "Class"   and m' :: "Method"   and T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and MI :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and Anno2 :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and M' :: "Var list \<times> (int \<times> Instr) list \<times> int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "INVS_SC C m l D m' T MI Anno Anno2 M' A I = (\<exists>M par code l0 T1 MI1. mbody_is C m M \<and> get_ins M l = Some (invokeS D m') \<and> MST\<down>(C, m) = Some (T1, MI1, Anno) \<and> MST\<down>(D, m') = Some (T, MI, Anno2) \<and> mbody_is D m' M' \<and> M' = (par, code, l0) \<and> (\<forall>Q. Anno\<down>l = Some Q \<longrightarrow> (\<forall>s0 s. A s0 s \<longrightarrow> Q s0 s)) \<and> (\<forall>s0 s. A s0 s \<longrightarrow> I s0 s (heap s)) \<and> (\<forall>s0 ops1 ops2 S R h t. (ops1, par, R, ops2) \<in> Frame \<longrightarrow> A s0 (ops1, S, h) \<longrightarrow> MI (R, h) t \<longrightarrow> I s0 (ops1, S, h) t))"
notStuck_CallDefinedD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "'d"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}" shows "\<Gamma> p \<noteq> None"
Subgroup_RSubmodule: fixes N :: "'m set" assumes "RSubmodule N" shows "VecGroup.Subgroup N"
steps_non_empty': fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "Graph_Defs.steps E xs" shows "xs \<noteq> []"
minim_isMinim: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "local.isMinim B (local.minim B)"
cbox_interval: fixes a :: "'b"   and b :: "'b" shows "cbox a b = {a..b}"
let1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e1 :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and n :: "char list option"   and e2 :: "exp0"   and bv :: "'b state \<times> (v, v) result" assumes "evaluate ck env s1 e1 (s2, Rval v1) \<and> evaluate ck (update_v (\<lambda>_. nsOptBind n v1 (sem_env.v env)) env) s2 e2 bv" shows "evaluate ck env s1 (exp0.Let n e1 e2) bv"
affine_hull_finite: fixes S :: "'a set" assumes "finite S" shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "star r x1 x2"   and "\<And>x. P x x"   and "\<And>x y z. \<lbrakk>r x y; star r y z; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x1 x2"
Step_Rrule: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and L :: "rrule"   and j :: "nat" shows "step_result (SG, C) (i, Rrule L j) = (close (SG @ Rrule_result L j (SG ! i)) (SG ! i), C)"
conjugate_zero_iff: fixes conjugate :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "(conjugate a = zero) = (a = zero)"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "P T"   and "P F"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
matrix_A_def: fixes sign_list :: "rat list list"   and subset_list :: "nat list list" shows "matrix_A sign_list subset_list = mat_of_rows_list (length sign_list) (map (mtx_row sign_list) subset_list)"
lattice_of_finsum: fixes fs :: "'a vec list" assumes "set fs \<subseteq> carrier_vec n" shows "lattice_of fs = range (\<lambda>c. \<Oplus>\<^bsub>V\<^esub>i\<in>{0..<length fs}. of_int (c i) \<cdot>\<^sub>v fs ! i)"
induct: fixes P :: "nat list \<Rightarrow> bool"   and a0 :: "nat list" assumes "\<And>n m l. P (ack m n # l) \<Longrightarrow> P (n # m # l)"   and "\<And>m. P [m]"   and "P []" shows "P a0"
AbGroup_inner_dirsum_el_decompI: fixes Gs :: "'a set list"   and x :: "'a" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "x \<in> (\<Oplus>G\<leftarrow>Gs. G)" shows "\<Oplus>Gs\<leftarrow> x \<in> listset Gs \<and> x = sum_list (\<Oplus>Gs\<leftarrow> x)"
std_simplex: shows "convex hull insert (0::'c) Basis = {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
foo'_ax1: fixes x :: "'a"   and y :: "bool"   and z :: "bool" assumes "foo' x y z" shows "z \<and> y"
decide_existential_R_def: fixes fml :: "real poly fml" shows "decide_existential_R fml = (let (fml_struct, polys) = convert fml; conds = find_consistent_signs_R polys in find (lookup_sem fml_struct) conds \<noteq> None)"
iso_in_list_imp_iso_in_remisodups: fixes Us :: "('b, 'a) aezfun set list" assumes "\<exists>U\<in>set Us. isomorphic (*) U" shows "\<exists>U\<in>set (ActingGroup.remisodups Us). isomorphic (*) U"
prime_card_elements_with_ord: fixes d :: "nat" assumes "\<psi> d \<noteq> 0"   and "prime n" shows "\<psi> d = totient d"
smc_Comp_vempty_if_Arr_vempty: assumes "\<AA>\<lparr>Arr\<rparr> = []\<^sub>\<circ>" shows "\<AA>\<lparr>Comp\<rparr> = []\<^sub>\<circ>"
cring_simprules_27: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
icsSp_Append_call: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "n ics-ns\<rightarrow>\<^sub>d* n''"   and "n'' s-p\<rightarrow>\<^bsub>call\<^esub> n'" shows "n ics-ns @ [n'']\<rightarrow>\<^sub>d* n'"
P6: shows "Bet PO b'' B \<and> Cong Po b PO b''"
less_refl: fixes v :: "val" shows "v \<sqsubseteq> v"
finite_Collect_disjI: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "finite {x. P x \<or> Q x} = (finite {x. P x} \<and> finite {x. Q x})"
CondFalse: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>2 :: "('d, 'e, 'f) com"   and c\<^sub>1 :: "('d, 'e, 'f) com" assumes "s \<notin> b"   and "\<Gamma>\<turnstile>c\<^sub>2 \<down> Normal s" shows "\<Gamma>\<turnstile>Cond b c\<^sub>1 c\<^sub>2 \<down> Normal s"
vert_dim: shows "rat_poly.row_length (brickmat vert) = 2 \<and> length (brickmat vert) = 2"
C\<^sub>0: fixes x :: "i"   and y :: "i" shows "E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)"
multiplicity_eq_Max: fixes x :: "'a"   and p :: "'a" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p" shows "multiplicity p x = Max {n. p ^ n dvd x}"
s0g_init_def: shows "s0g_init \<equiv> s0g_secrecy \<inter> s0g_dom"
in_not_in_iff1: fixes n :: "nat"   and i :: "nat_int" shows "(n \<^bold>\<in> i) = (\<not> n \<^bold>\<notin> i)"
order_max: fixes a :: "'b"   and k :: "nat"   and p :: "'b poly" assumes "[:- a, 1::'b:] ^ k dvd p"   and "p \<noteq> 0" shows "k \<le> order a p"
assms_1: shows "x * a $ 0 = (1::'a)"
ovalid_def: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "ovalid P f Q \<equiv> \<forall>s r. P s \<and> f s = Some r \<longrightarrow> Q r s"
ih2: shows "opnet onp p\<^sub>2 \<Turnstile> (otherwith S (net_tree_ips p\<^sub>2) (oarrivemsg I), other U (net_tree_ips p\<^sub>2) \<rightarrow>) global (\<lambda>\<sigma>. \<forall>i\<in>net_tree_ips p\<^sub>2. P i \<sigma>)"
Diff_insert_absorb: fixes x :: "'q"   and A :: "'q set" assumes "x \<notin> A" shows "insert x A - {x} = A"
OclAsBoolean\<^sub>I\<^sub>n\<^sub>t_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "X->oclAsType\<^sub>I\<^sub>n\<^sub>t(Boolean) = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> then \<lfloor>\<lfloor>\<lceil>\<lceil>X \<tau>\<rceil>\<rceil> \<noteq> 0\<rfloor>\<rfloor> else invalid \<tau>)"
wfzf_subset: fixes S :: "(ZF \<times> ZF) set"   and R :: "(ZF \<times> ZF) set" assumes "wfzf S"   and "R \<subseteq> S" shows "wfzf R"
cont_id: fixes Sup :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "cont Sup ord Sup ord id"
less_than_char_linear: fixes a :: "char"   and b :: "char" shows "(a, b) \<in> less_than_char \<or> a = b \<or> (b, a) \<in> less_than_char"
max_density_gr_zero: fixes G :: "nat set \<times> nat set set" assumes "finite_graph G"   and "nonempty_graph G"   and "uwellformed G" shows "0 < max_density G"
continuous_on_sequentially: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>x. \<forall>a\<in>s. (\<forall>n. x n \<in> s) \<and> x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
reachable_adjI: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^bsub>del_vert u\<^esub> v" shows "u \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> v"
hoare_partial_sound: fixes dims :: "nat list"   and P :: "complex mat"   and S :: "com"   and Q :: "complex mat" assumes "state_sig.hoare_partial dims P S Q"   and "state_sig.well_com dims S" shows "state_sig.hoare_partial_correct dims P S Q"
inj_on_rev: fixes A :: "'b list set" shows "inj_on rev A"
induct: fixes x1 :: "boolean_unop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'a OCL_Types.type"   and P :: "boolean_unop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "boolean_unop_type x1 x2 x3"   and "\<And>\<tau>. \<tau> \<le> Boolean[?] \<Longrightarrow> P NotOp \<tau> \<tau>" shows "P x1 x2 x3"
cyclic_group_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier \<G>"   and "b \<in> carrier \<G>"   and "c \<in> carrier \<G>" shows "a \<otimes> b \<otimes> c = a \<otimes> (b \<otimes> c)"
valid_sops_expr_valid_sop: fixes t :: "nat"   and e :: "expr" assumes "valid_sops_expr t e" shows "valid_sop (eval_expr t e)"
MakeCatMor: fixes C :: "('h, 'g, 'i) Category_scheme" shows "mor\<^bsub>MakeCat C\<^esub> = mor\<^bsub>C\<^esub>"
piecewise_continuous_on_def: fixes a :: "'c"   and b :: "'c"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "piecewise_continuous_on a b I f = (continuous_on ({a..b} - I) f \<and> finite I \<and> (\<forall>i\<in>I. (i \<in> {a<..b} \<longrightarrow> (\<exists>l. (f \<longlongrightarrow> l) (at_left i))) \<and> (i \<in> {a..<b} \<longrightarrow> (\<exists>u. (f \<longlongrightarrow> u) (at_right i)))))"
Pair_Sign: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "\<langle>X, Y\<rangle> \<noteq> Sign X' Y'"
some_src_dom: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_src (local.dom \<mu>) = some_src \<mu>"
member_rec_2: fixes y :: "'a" shows "List.member [] y = False"
theNFT_length: fixes n :: "nat"   and cl :: "('a, 'b, 'c) cmd list" assumes "n \<in> theNFT cl" shows "n < length cl"
Refl: fixes H :: "fm set"   and t :: "tm" shows "H \<turnstile> t EQ t"
rep_eq: fixes x :: "'c monom"   and xa :: "'c monom" shows "Rep_monom (x * xa) = monom_mult_list (Rep_monom x) (Rep_monom xa)"
assms_2: fixes r :: "nat"   and c :: "nat" assumes "r < dim_row A"   and "c < dim_col A" shows "A $$ (r, c) = f r * g c"
fresh_star_zero: fixes as :: "atom set" shows "as \<sharp>* 0"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>"   and "a \<in>\<^sub>\<circ> A"   and "r\<lparr>a\<rparr> = b" shows "thesis"
finitely_many_partition_on: fixes A :: "'b set" assumes "finite A" shows "finite {P. partition_on A P}"
hyps_5: assumes "finite_Psegments P a b"   and "b \<in> {a..s}"   and "closed ({a. P a} \<inter> {a..s})" shows "finite_Psegments P a s"
low_mds_eq_def: fixes mds :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1 :: "'Var\<^sub>C \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var\<^sub>C \<Rightarrow> 'Val" shows "conc.low_mds_eq mds mem\<^sub>1 mem\<^sub>2 \<equiv> \<forall>x. dma\<^sub>C mem\<^sub>1 x = Low \<and> (x \<in> \<C>\<^sub>C \<or> x \<notin> mds AsmNoReadOrWrite) \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
antisym: fixes b :: "'t"   and a :: "'t" assumes "b \<preceq>\<^sub>t a"   and "a \<preceq>\<^sub>t b" shows "a = b"
eucl_less_def: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and a :: "'e"   and b :: "'e" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "euclidean_space.eucl_less inner Basis a b = (\<forall>i\<in>Basis. inner a i < inner b i)"
lookup_PP_oalist: fixes xs :: "('b, 'c) oalist_tc" shows "lookup_pp (PP_oalist xs) = OAlist_tc_lookup xs"
is_strict_monoidal_functor: shows "strict_monoidal_functor (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C \<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D local.map"
p': shows "\<forall>y\<in>S. norm (v - p) \<le> norm (v - y)"
sup_least: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> x" shows "y \<squnion> z \<le> x"
flowsfix: shows "flows_fix (generate_valid_stateful_policy_IFSACS G M edgesList) = edges G"
red: shows "\<langle>ls, (ts, m), ws, is\<rangle> -t'\<triangleright>\<lbrace>\<rbrace>\<rightarrow> \<langle>ls', (ts', m'), ws', is'\<rangle>"
less_finite_2_def: fixes x :: "finite_2"   and y :: "finite_2" shows "(x < y) = (x = a\<^sub>1 \<and> y = a\<^sub>2)"