hcju/isabelleps huggingface dataset:
cases: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.siso aval tval cval indis a" obtains c :: "('d, 'c, 'a) cmd" where "a = c" and "\<And>s i. i < PL.brn c \<Longrightarrow> PL_Indis.siso aval tval cval indis (PL.cont aval tval cval c s i)" and "\<And>s t i. i < PL.brn c \<and> (s, t) \<in> indis \<Longrightarrow> (PL.eff aval tval cval c s i, PL.eff aval tval cval c t i) \<in> indis \<and> PL.wt aval tval cval c s i = PL.wt aval tval cval c t i \<and> PL.cont aval tval cval c s i = PL.cont aval tval cval c t i"
A1: shows "pr_conv_3_to_1 f \<in> PrimRec1'"
set_cases_1: fixes e :: "'a"   and a :: "('a, 'b) rbt" assumes "e \<in> set1_rbt a" obtains z1 :: "('a, 'b) rbt" and z2 :: "color" and z3 :: "'a" and z4 :: "'b" and z5 :: "('a, 'b) rbt" where "a = Node z1__ z2__ z3__ z4__ z5__" and "e \<in> set1_rbt z1__"   | z1 :: "('a, 'b) rbt" and z2 :: "color" and z4 :: "'b" and z5 :: "('a, 'b) rbt" where "a = Node z1a__ z2a__ e z4a__ z5a__"   | z1 :: "('a, 'b) rbt" and z2 :: "color" and z3 :: "'a" and z4 :: "'b" and z5 :: "('a, 'b) rbt" where "a = Node z1 z2 z3 z4 z5" and "e \<in> set1_rbt z5"
Rep_regular_inject: fixes x :: "'b regular"   and y :: "'b regular" shows "(Rep_regular x = Rep_regular y) = (x = y)"
mat_of_rows_carrier_2: fixes n :: "nat"   and vs :: "'b Matrix.vec list" shows "dim_row (mat_of_rows n vs) = length vs"
smult_add_distrib_vec: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec"   and a :: "'b" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "a \<cdot>\<^sub>v (v + w) = a \<cdot>\<^sub>v v + a \<cdot>\<^sub>v w"
assms_4: shows "NextRecvUpd' c0 c1 p q"
right_idem: fixes a :: "'e multiset"   and b :: "'e multiset" shows "a \<union># b \<union># b = a \<union># b"
rebase_poly_id: fixes p :: "'p mod_ring poly" shows "##p = #p"
labeled_list_insert_eq_ex_cases: fixes d :: "'a"   and D :: "('b \<times> 'a) list"   and i :: "'b" shows "List.insert d (unlabel D) = unlabel (List.insert (i, d) D) \<or> (\<exists>j. (j, d) \<in> set D \<and> List.insert d (unlabel D) = unlabel (List.insert (j, d) D))"
simps_3: fixes ops :: "'b arith_ops_record"   and y :: "'b"   and ys :: "'b list" shows "minus_poly_rev_list_i ops [] (y # ys) = []"
smult_distrib_left_diff: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a"   and m :: "'b"   and n :: "'b" assumes "RModule R smult M"   and "r \<in> R"   and "m \<in> M"   and "n \<in> M" shows "smult r (m - n) = smult r m - smult r n"
not_Arr_Null: shows "\<not> (C1.arr (fst (C1.null, C2.null)) \<and> C2.arr (snd (C1.null, C2.null)))"
dist_triangle_le: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and z :: "'c"   and y :: "'c"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist x z + dist y z \<le> e" shows "dist x y \<le> e"
homotopy_equivalent_space_imp_isomorphic_homology_groups: fixes X :: "'a topology"   and Y :: "'b topology"   and p :: "int" assumes "X homotopy_equivalent_space Y" shows "homology_group p X \<cong> homology_group p Y"
"11_6": shows "Ind (set b\<^sub>2) (sset u'')"
in_these_eq: fixes x :: "'d"   and A :: "\<langle>'d\<rangle>\<^sub>\<bottom> set" shows "(x \<in> Option.these A) = (\<lfloor>x\<rfloor> \<in> A)"
set_deg_shifts: fixes d :: "nat"   and fs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "set (deg_shifts d fs) = (\<Union>f\<in>set fs. (\<lambda>t. local.punit.monom_mult (1::'b) t f) ` deg_le_sect X (d - poly_deg f))"
count_greater_eq_one_iff: fixes M :: "'b multiset"   and x :: "'b" shows "(1 \<le> count M x) = (x \<in># M)"
is_lasso_prpl_conv: fixes prpl :: "'Q list \<times> 'Q list" shows "is_lasso_prpl prpl = (snd prpl \<noteq> [] \<and> is_lasso (lasso_of_prpl prpl))"
K: fixes k' :: "stack" assumes " [y]n\<ggreater>L \<mapsto> k' "   and "r = t \<star> k'" shows "P"
generalisation_of_preserves: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" assumes "is_generalisation_of t' t i r" shows "Label t = Label t' \<and> Arity t = Arity t' \<and> Outputs t = Outputs t'"
linear_comb0_1: fixes H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> {\<zero>\<^bsub>R\<^esub>}"   and "m \<in> {j. j \<le> n} \<rightarrow> H" shows "l_comb R M n s m = \<zero>"
diagonalize_strict_arr: fixes f :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f" shows "D f = f"
steps_z_complete: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and u :: "'e \<Rightarrow> 'f"   and l' :: "'d"   and u' :: "'e \<Rightarrow> 'f"   and Z :: "('e \<Rightarrow> 'f) set" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l', u'\<rangle>"   and "u \<in> Z" shows "\<exists>Z'. A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z'\<rangle> \<and> u' \<in> Z'"
le_fininf_empty: fixes w :: "'action stream" shows "[] \<preceq>\<^sub>F\<^sub>I w"
finite_product_sigma_finite_def: fixes M :: "'b \<Rightarrow> 'c measure"   and I :: "'b set" shows "finite_product_sigma_finite M I \<equiv> product_sigma_finite M \<and> finite_product_sigma_finite_axioms I"
lucas_lehmer_add_def: fixes m :: "nat" shows "lucas_lehmer_add m = (\<lambda>(a, b) (c, d). ((a + c) mod m, (b + d) mod m))"
compet_rand_def: fixes A :: "('state \<Rightarrow> 'b pmf) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> ('answer \<times> 'b) pmf)"   and c :: "real"   and S0 :: "'state set" shows "compet_rand A c S0 = (\<forall>s\<in>S0. \<exists>b\<ge>0. \<forall>rs. wf s rs \<longrightarrow> T_on_rand' A (fst A s \<bind> (\<lambda>is. return_pmf (s, is))) rs \<le> c * real (T_opt s rs) + b)"
step: fixes x :: "'b node \<times> 'b node set" assumes "(\<mu> and\<^sub>r \<eta> \<in> old (fst x) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst x) \<union> new (fst x)) \<and> (\<forall>nd\<in>snd x. \<mu> and\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd)" shows "f x \<le> SPEC (\<lambda>x'. \<forall>nd\<in>snd x'. \<mu> and\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd)"
isCont_powser_converges_everywhere: fixes c :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)" shows "isCont (\<lambda>x. \<Sum>n. c n * x ^ n) x"
RedAAcc: fixes h :: "'heap"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and v :: "'addr val"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>"   and "0 \<le>s i"   and "sint i < int n"   and "heap_read h a (ACell (nat (sint i))) v" shows "extTA,P,t \<turnstile> \<langle>addr a\<lfloor>Val (Intg i)\<rceil>,(h, l)\<rangle> -\<lbrace>ReadMem a (ACell (nat (sint i))) v\<rbrace>\<rightarrow> \<langle>Val v,(h, l)\<rangle>"
inf_bres: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "times_prod' x y = Inf_prod {z. less_eq_prod y (prod.bres x z)}"
P_3: fixes n :: "nat"   and x :: "'a"   and y :: "'a" assumes "P n x"   and "P n y" shows "dist x y < 1 / real (Suc n)"
eq_refl: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" assumes "x = y" shows "x \<preceq> y"
tmap_eq_TNil_conv: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and xs :: "('c, 'd) tllist"   and y :: "'b" shows "(tmap f g xs = TNil y) = (\<exists>y'. xs = TNil y' \<and> g y' = y)"
bij_\<alpha>: shows "bij (id(r := r', r' := r))"
cases: fixes x :: "jvm_state" obtains xp :: "nat option" and h :: "heap" and frs :: "frame list" and sh :: "sheap" where "x = (xp, h, frs, sh)"
prems: shows "s1 (t1 @ t @ [a])\<Longrightarrow>\<^bsub>SES\<^esub> sn"
zeta_2: shows "zeta 2 = complex_of_real (pi\<^sup>2 / 6)"
src_vcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.seq \<nu> \<mu>" shows "src\<^sub>C (\<nu> \<cdot>\<^sub>C \<mu>) = src\<^sub>C \<nu>"
add_instr_sub1_def: fixes instr_name :: "sparc_operation"   and result :: "word32"   and rs1_val :: "word32"   and operand2 :: "word32" shows "add_instr_sub1 instr_name result rs1_val operand2 \<equiv> if instr_name \<in> {arith_type ADDcc, arith_type ADDXcc} then do psr_val \<leftarrow> gets (cpu_reg_val PSR); result_31 \<leftarrow> gets (\<lambda>s. ucast (result >> 31)); rs1_val_31 \<leftarrow> gets (\<lambda>s. ucast (rs1_val >> 31)); operand2_31 \<leftarrow> gets (\<lambda>s. ucast (operand2 >> 31)); new_n_val \<leftarrow> gets (\<lambda>s. result_31); new_z_val \<leftarrow> gets (\<lambda>s. if result = 0 then 1 else 0); new_v_val \<leftarrow> gets (\<lambda>s. rs1_val_31 AND operand2_31 AND NOT result_31 OR NOT rs1_val_31 AND NOT operand2_31 AND result_31); new_c_val \<leftarrow> gets (\<lambda>s. rs1_val_31 AND operand2_31 OR NOT result_31 AND (rs1_val_31 OR operand2_31)); new_psr_val \<leftarrow> gets (\<lambda>s. update_PSR_icc new_n_val new_z_val new_v_val new_c_val psr_val); write_cpu new_psr_val PSR; return () od else return ()"
sse_term: fixes R :: "('b \<times> 'b) set"   and \<Sigma>i :: "'b set" assumes "finite (R\<^sup>* `` \<Sigma>i)" shows "wf (sse_term_rel \<Sigma>i R)"
terminates_splice_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and h :: "'g \<Rightarrow> ('f, 'g) step" assumes "terminates g"   and "terminates h" shows "terminates (splice_raw g h)"
sortedP_Cons: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "linorder_list0.sortedP le (x # xs)" obtains "\<forall>x\<in>set xs. le x x" and "linorder_list0.sortedP le xs"
mono: fixes c :: "'d multiset"   and a :: "'d multiset"   and d :: "'d multiset"   and b :: "'d multiset" assumes "c \<subseteq># a"   and "d \<subseteq># b" shows "c \<union># d \<subseteq># a \<union># b"
Suc_leI: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "Suc m \<le> n"
Ord_Union: fixes X :: "V set" assumes "\<And>x. x \<in> X \<Longrightarrow> Ord x"   and "small X" shows "Ord (set (\<Union> (elts ` X)))"
lifelock_free_is_non_terminating: fixes P :: "'d process" shows "lifelock_free P = non_terminating P"
one_induced_iff_principal: shows "\<Phi>.induced_modulus 1 = (\<Phi> = principal_dchar d)"
separation_t0: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.t0_space open" shows "(x \<noteq> y) = (\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U))"
Rf_imp_ex_non_Rf: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set" assumes "C \<in> Rf N" shows "\<exists>CC. set_mset CC \<subseteq> N - Rf N \<and> (\<forall>I. I \<Turnstile>m CC \<longrightarrow> I \<Turnstile> C) \<and> (\<forall>C'. C' \<in># CC \<longrightarrow> C' < C)"
simps: fixes xs :: "'d"   and x :: "'e"   and ys :: "'d"   and y :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "qAFreshAbs xs x (qAbs ys y X) = ((xs \<noteq> ys \<or> x \<noteq> y) \<and> qAFresh xs x X)"
set_insert: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" obtains B :: "'b set" where "A = x \<triangleright> B" and "x \<notin> B"
l5_5_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Le C D" shows "\<exists>X. Bet A B X \<and> Cong A X C D"
psitau_init: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat"   and x :: "nat" shows "\<psi>\<^sup>\<tau> (f \<triangleright> n) x = (if x < Suc n then Some (the (f x)) else \<tau> (the (f 0)) x)"
simps_2: fixes h :: "QDelta \<times> QDelta"   and t :: "(QDelta \<times> QDelta) list" shows "\<delta>_min (h # t) = min (\<delta>_min t) (\<delta>0 (fst h) (snd h))"
older_seniors_subset: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and y :: "nat"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "y \<in> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n" shows "semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n \<subset> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n"
dcharacter_axioms_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" shows "dcharacter_axioms n \<chi> \<equiv> ((\<forall>a b. a \<in> totatives n \<longrightarrow> b \<in> totatives n \<longrightarrow> \<chi> (a * b) = \<chi> a * \<chi> b) \<and> (\<forall>a. \<not> coprime a n \<longrightarrow> \<chi> a = 0)) \<and> (\<forall>a. \<chi> (a + n) = \<chi> a) \<and> \<chi> 1 \<noteq> 0"
fps_nth_Abs_fps: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "Abs_fps f $ n = f n"
seq_comp_secure_aux_2_case_3: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and P :: "'d option process"   and Q :: "'d option process"   and ws :: "'d option list"   and ys :: "'d option list"   and Y :: "'d option set"   and xs :: "'d option list"   and zs :: "'d option list"   and y :: "'d option" assumes "secure_termination I D"   and "ref_union_closed P"   and "sequential P"   and "secure P I D"   and "secure Q I D"   and "ws \<in> sentences P"   and "(ys, Y) \<in> failures Q"   and "ys \<noteq> []"   and "ws @ ys = xs @ zs"   and "(xs @ [y], {}) \<in> seq_comp_failures P Q" shows "(xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Y) \<in> seq_comp_failures P Q"
invertible_lf_intro: fixes g :: "('a, 'c) vec \<Rightarrow> ('a, 'b) vec"   and A :: "(('a, 'b) vec, 'c) vec" assumes "g \<circ> (*v) A = id"   and "(*v) A \<circ> g = id" shows "invertible_lf (*s) (*s) ((*v) A)"
symmetric_poly_of_roots_in_subring: fixes C :: "'a set"   and q :: "'a mpoly"   and l :: "'b \<Rightarrow> 'a"   and A :: "nat set"   and cinv :: "'a"   and c :: "'b"   and p :: "'b poly"   and root :: "nat \<Rightarrow> 'b" assumes "ring_closed C"   and "\<forall>m. MPoly_Type.coeff q m \<in> C"   and "ring_homomorphism l"   and "finite A"   and "symmetric_mpoly A q"   and "vars q \<subseteq> A"   and "cinv * l c = (1::'a)"   and "cinv \<in> C"   and "p = Polynomial.smult c (\<Prod>i\<in>A. [:- root i, 1::'b:])"   and "\<forall>i. l (poly.coeff p i) \<in> C" shows "insertion (\<lambda>x. l (root x)) q \<in> C"
axioms: fixes A :: "V"   and f :: "V" assumes "binop_v11 A f" shows "nop_v11 A (2\<^sub>\<nat>) f"
assms: shows "wf_formula n (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2)"
ultra: fixes P :: "'a \<Rightarrow> bool" shows "eventually P F \<or> (\<forall>\<^sub>F x in F. \<not> P x)"
LCons_lprefix_LCons: fixes x :: "'c"   and xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "LCons x xs \<sqsubseteq> LCons y ys = (x = y \<and> xs \<sqsubseteq> ys)"
seqI: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "arr g"   and "local.dom g = cod f" shows "seq g f"
var_eq_transp: fixes x :: "('e, 'f, 'g) var0"   and y :: "('e, 'f, 'g) var0"   and z :: "('e, 'f, 'g) var0" assumes "x \<sim> y"   and "y \<sim> z" shows "x \<sim> z"
vcard_def: fixes a :: "V" shows "vcard a \<equiv> LEAST i. Ord i \<and> elts i \<approx> elts a"
antisym: fixes x :: "'a"   and y :: "'a" assumes "bo x y"   and "bo y x"   and "x \<in> A" shows "x = y"
RedCallStatic: fixes h :: "heap"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and P :: "(char list list \<times> char list exp) prog"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> char list exp"   and D :: "char list"   and vs :: "val list"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool" assumes "h a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>addr a\<bullet>M(map Val vs),(h, l, sh),b\<rangle> \<rightarrow> \<langle>THROW IncompatibleClassChangeError,(h, l, sh),b\<rangle>"
head_subst: fixes \<rho> :: "'d \<Rightarrow> ('c, 'd) tm"   and s :: "('c, 'd) tm" shows "head (subst \<rho> s) = head (subst \<rho> (Hd (head s)))"
fin: shows "finite (carrier (G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>))"
no_olen_add: fixes x :: "'b word"   and y :: "'b word" shows "(x \<le> x + y) = (uint x + uint y < 2 ^ LENGTH('b))"
fold_Cons_rev: fixes xs :: "'d list" shows "fold (#) xs = (@) (rev xs)"
substitute'_Cons: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree"   and x :: "'b"   and xs :: "'b list" shows "substitute' f T t (x # xs) = (possible t x \<and> substitute' (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x) xs)"
Q_relax_outgoing: fixes Q :: "'b"   and u :: "'v"   and du :: "nat"   and V :: "'a" assumes "Q_invar Q" shows "Q_invar (Q_relax_outgoing u du V Q) \<and> Q_lookup (Q_relax_outgoing u du V Q) = relax_outgoing' u du (M_lookup V) (Q_lookup Q)"
geom_rand_walk_diff_induct: fixes n :: "nat"   and w :: "bool stream" shows "geom_proc (Suc n) (spick w n True) - geom_proc (Suc n) (spick w n False) = geom_proc n w * (u - d)"
TS_config'_fin: fixes s :: "(nat list \<times> nat list) pmf"   and h :: "nat list"   and qs :: "nat list" assumes "finite (set_pmf s)" shows "finite (set_pmf (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h)) s qs))"
unit_factor: fixes a :: "'a"   and b :: "'a" assumes "a \<oplus> b \<in> Units (add_monoid R)"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<in> Units (add_monoid R)"
impl_mp1: fixes P :: "'d upred"   and Q :: "'d upred" shows "(P \<and> (P \<Rightarrow> Q)) = (P \<and> Q)"
sba5_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x \<squnion> y) = x"
rat_poly_unity: shows "rat_poly_times 1 1 = 1"
events_DF: fixes A :: "'a set" shows "events_of (DF A) = A"
hom_one: shows "poly.coeff 1 0 = (1::'a)"
pp_of_term_def: fixes v :: "'a" shows "local.punit.pp_of_term v = fst (to_pair_unit v)"
C1: fixes S :: "ccs"   and T :: "ccs"   and U :: "ccs" assumes "(S, T) \<in> Rel" shows "(S \<parallel> U, T \<parallel> U) \<in> Rel'"
ind_eq_fin': fixes a :: "'item"   and v :: "'item list" assumes "Ind {a} (set v)" shows "[a] @ v =\<^sub>F v @ [a]"
transition_eqvt: fixes P :: "'state"   and \<alpha>Q :: "('act, 'state) residual"   and p :: "perm" assumes "P \<rightarrow> \<alpha>Q" shows "p \<bullet> P \<rightarrow> p \<bullet> \<alpha>Q"
evnodd_insert: fixes i :: "nat"   and j :: "nat"   and C :: "(nat \<times> nat) set"   and b :: "nat" shows "evnodd (insert (i, j) C) b = (if (i + j) mod 2 = b then insert (i, j) (evnodd C b) else evnodd C b)"
listset_doubletonE: fixes ys :: "'c list"   and A :: "'c set"   and B :: "'c set" assumes "ys \<in> listset [A, B]" obtains a :: "'c" and b :: "'c" where "a \<in> A" and "b \<in> B" and "ys = [a, b]"
setdist_eq_0_closedin: fixes U :: "'b set"   and S :: "'b set"   and x :: "'b" assumes "closedin (top_of_set U) S"   and "x \<in> U" shows "(setdist {x} S = 0) = (S = {} \<or> x \<in> S)"
union_result: shows "\<forall>x\<in>{S. \<exists>m n. {m..n} = S}. \<forall>y\<in>{S. \<exists>m n. {m..n} = S}. x \<noteq> {} \<and> y \<noteq> {} \<and> Max x + 1 = Min y \<longrightarrow> x \<union> y \<in> {S. \<exists>m n. {m..n} = S}"
simps_2: fixes s :: "('c, 'd) tm"   and t :: "('c, 'd) tm" shows "args (App s t) = args s @ [t]"
lprefix_trans: fixes xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" assumes "lprefix xs ys"   and "lprefix ys zs" shows "lprefix xs zs"
not_less_iff_gr_or_eq: fixes x :: "'t"   and y :: "'t" shows "(\<not> x \<prec>\<^sub>t y) = (y \<prec>\<^sub>t x \<or> x = y)"
sq_le_cancel_1: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "0 \<le> b"   and "a\<^sup>2 \<le> b * a" shows "a \<le> b"
assms: shows "(a, b, c) \<noteq> (x, y, z)"
distinct_19: fixes x3 :: "'b formula"   and x41 :: "'b formula"   and x42 :: "'b formula" shows "\<^bold>\<not> x3 \<noteq> x41 \<^bold>\<and> x42"
SubstComp: fixes A :: "'t"   and e :: "('t, 'f) Expression"   and B :: "'t"   and g :: "'m"   and d :: "('t, 'f) Expression"   and C :: "'t"   and h :: "'m" assumes "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> IMor g"   and "L\<lbrakk>Vx : B \<turnstile> d : C\<rbrakk> \<rightarrow> IMor h" shows "L\<lbrakk>Vx : A \<turnstile> sub e in d : C\<rbrakk> \<rightarrow> IMor (g ;;\<^bsub>iC\<^esub> h)"
enat_mult_minus_distrib: fixes x :: "nat"   and y :: "enat"   and z :: "enat" shows "enat x * (y - z) = enat x * y - enat x * z"
poly_sgnx_values_2: fixes p :: "real poly"   and a :: "real" assumes "p \<noteq> 0" shows "sgnx (poly p) (at_right a) = 1 \<or> sgnx (poly p) (at_right a) = - 1"
assms_2: shows "the (flag rt ip) = Aodv_Basic.inv"
exact_decomp_card_X: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and m :: "nat" assumes "valid_decomp X ps"   and "card X \<le> m" shows "exact_decomp m ps"
lens_quotient_comp: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e" assumes "weak_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "X /\<^sub>L Y ;\<^sub>L Y = X"
SubstClosure_def: fixes x :: "'c"   and e :: "('c, 'd) Expr" shows "SubstClosure x e \<equiv> \<forall>(d', e', \<iota>')\<in>lH. (d', e'<Var x\e>, \<iota>') \<in> lH"
elims_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'c"   and xa :: "('a, 'b) lit" assumes "U.Struct wtFsym wtPsym arOf parOf D intF intP"   and "U.Struct.satL intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'c" and a :: "('a, 'b) atm" where "x = \<xi>__" and "xa = Pos a__" and "U.Struct.satA intF intP \<xi>__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'c" and a :: "('a, 'b) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> U.Struct.satA intF intP \<xi> a"
case_eq_if: fixes f :: "exesignature \<Rightarrow> Core.term list \<Rightarrow> 'a"   and exetheory :: "exetheory" shows "(case exetheory of ExeTheory x xa \<Rightarrow> f x xa) = f (exesig exetheory) (exeaxioms_of exetheory)"
JF1map: fixes u :: "'d \<Rightarrow> 'e" shows "dtor1 \<circ> JF1map u = F1map u (JF1map u) (JF2map u) \<circ> dtor1"
fmfilter_alt_defs_1: fixes a :: "'d" shows "fmdrop a = fmfilter (\<lambda>a'. a' \<noteq> a)"
inf_commute: fixes x :: "'a"   and y :: "'a" shows "inf x y = inf y x"
cur_le: shows "cur \<le> \<tau> \<sigma> ne"
quotient_of_int: fixes a :: "int" shows "quotient_of (Rat.of_int a) = (a, 1)"
minimal_unsat_core_tabl_atomsD_3: fixes I :: "'a set"   and t :: "(nat \<times> linear_poly) list"   and as :: "('a \<times> 'b atom) set"   and J :: "'a set" assumes "minimal_unsat_core_tabl_atoms I t as"   and "distinct_indices_atoms as"   and "J \<subset> I" shows "\<exists>v. v \<Turnstile>\<^sub>t t \<and> (J, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>e\<^sub>s as"
restriction_stable2: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u\<parallel>w \<longrightarrow> restrict v f c = restrict w f c"
simps_11: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x21 :: "'b"   and x22 :: "'b list"   and y21 :: "'c"   and y22 :: "'c list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
gen_dfs_pre_imp_wf: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a" assumes "gen_dfs_pre E U S V u0" shows "wf (gen_dfs_var U)"
weak_inf_pres_def: fixes X :: "('a, 'b) gorder_scheme"   and Y :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "weak_inf_pres X Y f \<equiv> complete_lattice X \<and> complete_lattice Y \<and> (\<forall>A\<subseteq>carrier X. A \<noteq> {} \<longrightarrow> f (\<Sqinter>\<^bsub>X\<^esub>A) = \<Sqinter>\<^bsub>Y\<^esub>f ` A)"