hcju/isabelleps huggingface dataset:
cc_restr_CCexp: fixes e :: "exp"   and a :: "Arity" shows "ccExp_syn a e G|` fv e = ccExp_syn a e"
subtopology_PiE: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and S :: "'c \<Rightarrow> 'd set" shows "subtopology (product_topology X I) (Pi\<^sub>E I S) = product_topology (\<lambda>i. subtopology (X i) (S i)) I"
pair_lessI1: fixes a :: "nat"   and b :: "nat"   and s :: "nat"   and t :: "nat" assumes "a < b" shows "((a, s), b, t) \<in> pair_less"
inf_right_idem: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) y = inf x y"
CutAux: fixes H :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "H \<rhd> c : A" shows "\<forall>G P D. H = insert P D \<longrightarrow> G \<rhd> Call : P \<longrightarrow> G \<subseteq> D \<longrightarrow> D \<rhd> c : A"
1: shows "t\<^bold>\<down> \<in> VHom t \<^bold>\<lfloor>t\<^bold>\<rfloor>"
check_ipresIGWls: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGWls check (errMOD MOD) MOD"
wqo_onI: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "transp_on P A"   and "almost_full_on P A" shows "wqo_on P A"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and ord :: "ord" shows "P (case ord of LESS \<Rightarrow> f1 | EQUAL \<Rightarrow> f2 | GREATER \<Rightarrow> f3) = ((ord = LESS \<longrightarrow> P f1) \<and> (ord = EQUAL \<longrightarrow> P f2) \<and> (ord = GREATER \<longrightarrow> P f3))"
upd_cs_empty: fixes cs :: "'edge list" assumes "upd_cs cs [] = []" shows "cs = []"
T'_mom_select_ge: fixes n :: "nat" shows "463 \<le> T'_mom_select n"
constrains_trans: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "F \<in> A co B"   and "F \<in> B co C" shows "F \<in> A co C"
extremum_unique: fixes a :: "'b" shows "(top \<le> a) = (a = top)"
Cons_in_subseqsD: fixes y :: "'b"   and ys :: "'b list"   and xs :: "'b list" assumes "y # ys \<in> set (subseqs xs)" shows "ys \<in> set (subseqs xs)"
connected_complement_homeomorphic_convex_compact: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "convex T"   and "compact T"   and "2 \<le> DIM('a)" shows "connected (- S)"
sb_inc_lem: fixes a :: "int"   and k :: "nat" assumes "a + 2 ^ k < 0" shows "a + 2 ^ k + 2 ^ Suc k \<le> (a + 2 ^ k) mod 2 ^ Suc k"
assms_2: shows "ui_edge re \<in> edges (black prb) \<and> (src re \<in> red_vertices prb \<and> tgt re \<notin> red_vertices prb \<and> red prb' = add_edge (red prb) re) \<and> src re \<notin> subsumees (subs prb) \<and> se (confs prb (src re)) (labelling (black prb) (ui_edge re)) c' \<and> prb' = \<lparr>red = red prb', black = black prb, subs = subs prb, init_conf = init_conf prb, confs = (confs prb)(tgt re := c'), marked = (marked prb)(tgt re := marked prb (src re)), strengthenings = strengthenings prb\<rparr>"
iapath_dist_ends: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "gen_iapath (verts3 (with_proj G)) u p v" shows "u \<noteq> v"
vsv_vimage_vsingleton: fixes a :: "V"   and aa :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (set {\<langle>aa, b\<rangle>})" shows "set {\<langle>aa, b\<rangle>} `\<^sub>\<circ> set {a} = set {set {\<langle>aa, b\<rangle>}\<lparr>a\<rparr>}"
pairwise_singleton: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b" shows "pairwise P {A}"
quadratic_normal_iff: fixes z :: "complex" shows "normal_poly [:(cmod z)\<^sup>2, - 2 * Re z, 1:] = (Re z \<le> 0 \<and> (cmod z)\<^sup>2 \<le> 4 * (Re z)\<^sup>2)"
lex_ext_unbounded_map_NS: fixes ss :: "'a list"   and ts :: "'a list"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>i. \<lbrakk>i < length ss; i < length ts; fst (r (ss ! i) (ts ! i))\<rbrakk> \<Longrightarrow> fst (r (map f ss ! i) (map f ts ! i))"   and "\<And>i. \<lbrakk>i < length ss; i < length ts; snd (r (ss ! i) (ts ! i))\<rbrakk> \<Longrightarrow> snd (r (map f ss ! i) (map f ts ! i))"   and "snd (lex_ext_unbounded r ss ts)" shows "snd (lex_ext_unbounded r (map f ss) (map f ts))"
between_exchange3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Bet A C D" shows "Bet B C D"
Ssub: shows "S \<subseteq> \<Union> (\<G> \<inter> Pow V)"
affine_hull_UNIV: fixes S :: "'a set" assumes "aff_dim S = int DIM('a)" shows "affine hull S = UNIV"
distinct_1: fixes x21 :: "'e"   and x22 :: "'e list" shows "[] \<noteq> Cons x21 x22"
Reachability_Problem_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set" shows "Reachability_Problem K S S1 S2 \<equiv> Finite_Markov_Decision_Process K S \<and> Reachability_Problem_axioms S S1 S2"
IH: assumes "A \<noteq> {}" shows "\<exists>y\<in>A. \<forall>z\<in>A. y \<le> z"
obj_class_CInst1: fixes obj :: "obj"   and C :: "qtname" assumes "tag obj = CInst C" shows "obj_class obj = C"
convex_closure_inter: fixes I :: "'a set set" assumes "\<forall>S\<in>I. convex S"   and "\<Inter> {rel_interior S |S. S \<in> I} \<noteq> {}" shows "closure (\<Inter> I) = \<Inter> {closure S |S. S \<in> I}"
intros_2: shows "OclVoid \<sqsubset>\<^sub>B UnlimitedNatural"
quasi_inverses_are_adjoint_pair: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.quasi_inverses V H \<a> \<i> src trg f g" shows "bicategory.adjoint_pair V H \<a> \<i> src trg f g"
hom_add: fixes x :: "'a poly"   and y :: "'a poly"   and a :: "'a" shows "poly (x + y) a = poly x a + poly y a"
mapMs': shows "map_of ms' M = \<lfloor>(Ts', T', m')\<rfloor>"
boffa_var: fixes x :: "'a" assumes "x \<cdot> x \<le> x" shows "x\<^sup>\<star> = (1::'a) + x"
cr_view\<^sub>m_Grp: shows "cr_view\<^sub>m = Grp UNIV to_view\<^sub>m"
is_extensional: fixes f :: "'a2 \<times> 'a1" assumes "\<not> A2xA1.arr f" shows "SF.map f = B.null"
partitions_imp_multiset: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p partitions n" shows "p \<in> multiset"
qfa_def: shows "qfa = (\<lambda>acts. foldr do_act acts Hood_Melville_Queue.empty)"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Pos x1 = Pos y1) = (x1 = y1)"
mat_of_rows_index: fixes i :: "nat"   and rs :: "'a Matrix.vec list"   and j :: "nat"   and n :: "nat" assumes "i < length rs"   and "j < n" shows "mat_of_rows n rs $$ (i, j) = rs ! i $v j"
mem_val_mod_cache: fixes asi :: "word8"   and a :: "word36"   and s :: "'b sparc_state"   and new_cache :: "cpu_cache" shows "mem_val_alt asi a s = mem_val_alt asi a (s\<lparr>cache := new_cache\<rparr>)"
apex_composite: shows "\<mu>\<nu>.apex = \<mu>.leg0 \<down>\<down> \<nu>.leg1"
convex_halfspace_ge: fixes b :: "real"   and a :: "'c" shows "convex {x. b \<le> a \<bullet> x}"
ptensor_mat_id: shows "st1.ptensor_mat (1\<^sub>m st2.d1) (1\<^sub>m st4.d2) = 1\<^sub>m st1.d0"
simps_1: fixes S :: "(char list \<Rightarrow> 'av) option"   and P :: "(char list \<Rightarrow> 'av) option" shows "step' S (SKIP {P}) = SKIP {S}"
OrL_in_SNa: fixes M :: "trm"   and N :: "trm"   and x :: "name"   and y :: "name"   and z :: "name" assumes "SNa M"   and "SNa N" shows "SNa (OrL x.M y.N z)"
valid_path_no_deadends: fixes G :: "('c, 'd) Graph_scheme"   and P :: "'c llist"   and i :: "nat" assumes "Digraph G"   and "Digraph.valid_path G P"   and "enat (Suc i) < llength P" shows "\<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. P $ i \<rightarrow>\<^bsub>G\<^esub> w)"
sums_unique2: fixes a :: "'a"   and b :: "'a" assumes "f sums a"   and "f sums b" shows "a = b"
inner_sum_left: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and y :: "'c" shows "sum f A \<bullet> y = (\<Sum>x\<in>A. f x \<bullet> y)"
fmimage_drop: fixes a :: "'b"   and m :: "('b, 'a) fmap"   and A :: "'b fset" shows "fmimage (fmdrop a m) A = fmimage m (A |-| {|a|})"
P_of_Liminf_state_fin: shows "P_of_state (Liminf_state Sts') = P_of_state (Liminf_state Sts)"
union: fixes A :: "'d set"   and B :: "'d set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "Max (A \<union> B) = max (Max A) (Max B)"
assms_3: shows "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
nat_to_set_upper_bound: fixes x :: "nat"   and u :: "nat" assumes "x \<in> nat_to_set u" shows "2 ^ x \<le> u"
gbm_invar_split_3: fixes g :: "'m1"   and v :: "'V"   and m2 :: "'m2"   and v' :: "'V"   and s3 :: "'s3" assumes "gbm_invar g"   and "m1.\<alpha> g v = Some m2"   and "m2.\<alpha> m2 v' = Some s3" shows "s3.invar s3"
eWlsBinp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'b option" shows "eWlsBinp MOD delta (OKI binp) = igWlsBinp MOD delta binp"
algebraic_iff_represents: fixes x :: "'a" shows "algebraic x = (\<exists>p. p represents x)"
empty_bind: fixes f :: "'b \<Rightarrow> 'a set" shows "Set.bind {} f = {}"
restrictI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "restrict f A \<in> Pi A B"
restriction_card_mon_trans: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and v3 :: "view"   and v4 :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=v1--v2 \<and> v2=v3--v4 \<and> |restrict v3 f c| = 1 \<longrightarrow> 1 \<le> |restrict v f c|"
sorted_hd_le: fixes xs :: "'b list"   and x :: "'b" assumes "sorted xs"   and "x \<in> list.set xs" shows "hd xs \<le> x"
bisim1_fr_hext_mono: fixes P :: "'addr J1_prog"   and h :: "'heap"   and exs :: "'addr expr1 \<times> 'addr val list"   and fr :: "'addr frame"   and h' :: "'heap" assumes "bisim1_fr P h exs fr"   and "h \<unlhd> h'" shows "bisim1_fr P h' exs fr"
iso_runit: fixes f :: "'a" assumes "S.ide f" shows "S.iso \<r>[f]"
silent_moves_Nil: fixes ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" assumes "length ms = length s" shows "S,f \<turnstile> (ms,s) =[]\<Rightarrow>\<^sub>\<tau> (ms,s)"
simps_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and uv :: "'d"   and \<sigma> :: "'f" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.pol uv (Guard \<sigma>) = Fext"
cmod_square: fixes z :: "complex" shows "(cmod z)\<^sup>2 = Re (z * cnj z)"
of_nat_or_eq: fixes m :: "nat"   and n :: "nat" shows "of_nat (m OR n) = of_nat m OR of_nat n"
lambda_system_strong_additive: fixes z :: "'a set"   and x :: "'a set"   and y :: "'a set"   and f :: "'a set \<Rightarrow> ennreal" assumes "z \<in> lambda_system \<Omega> M f"   and "x \<inter> y = {}"   and "x \<in> lambda_system \<Omega> (lambda_system \<Omega> M f) f"   and "y \<in> lambda_system \<Omega> (lambda_system \<Omega> M f) f" shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
units_of_inv: fixes x :: "'a" assumes "x \<in> Units (add_monoid R)" shows "inv\<^bsub>units_of (add_monoid R)\<^esub> x = inv\<^bsub>add_monoid R\<^esub> x"
bind_assoc: fixes x :: "'m"   and f :: "'a multiset \<Rightarrow> 'm"   and g :: "'a multiset \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
hindered: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "hindrance \<Gamma> f"   and "current \<Gamma> f"   and "wave \<Gamma> f" shows "hindered \<Gamma>"
not_self_separated_by_wall: fixes H :: "'a set set set"   and C :: "'a set" assumes "H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}})" shows "\<not> separated_by H C C"
e_app_elim: fixes v3 :: "val"   and e1 :: "exp"   and e2 :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "v3 \<in> E (EApp e1 e2) \<rho>" obtains f :: "(val \<times> val) fset" and v2 :: "val" and v2' :: "val" and v3' :: "val" where "VFun f \<in> E e1 \<rho>" and "v2 \<in> E e2 \<rho>" and "(v2', v3') \<in> fset f" and "v2' \<sqsubseteq> v2" and "v3 \<sqsubseteq> v3'"
eq4: shows "exH_n * (2 \<cdot>\<^sub>m pre - 1\<^sub>m N) * exH_n = 2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N"
IdD: fixes a :: "'g"   and b :: "'g" assumes "(a, b) \<in> Id" shows "a = b"
to_bl_use_of_bl: fixes w :: "'a word"   and bl :: "bool list" shows "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
seqSubstSimps_1: fixes \<sigma> :: "(name list \<times> 'a list) list" shows "\<zero>[<\<sigma>>] = \<zero>"
DiaE: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<diamond>\<phi> in v]" shows "\<exists>v. [\<phi> in v]"
combine_evals_filter: fixes arch :: "nat list list list"   and p :: "nat" assumes "\<forall>g\<in>set arch. pre_iso_test g"   and "samet (tameEnumFilter p) arch" shows "fgraph ` TameEnum\<^bsub>p\<^esub> \<subseteq>\<^sub>\<simeq> set arch"
assms_3: shows "S_reduced_for w (alternating_list n s t @ cs)"
one_is_polynomial: fixes K :: "'a set" assumes "subring K R" shows "polynomial K [\<one>]"
D'': fixes qs :: "nat list"   and a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat"   and s :: "(nat list \<times> bool list \<times> nat list) pmf" assumes "qs \<in> Lxx a b"   and "a \<noteq> b"   and "{a, b} = {x, y}"   and "BIT_inv s a [x, y]"   and "set qs \<subseteq> {a, b}" shows "T\<^sub>p_on_rand' BIT s qs \<le> 175 / 10\<^sup>2 * real (T\<^sub>p [a, b] qs (OPT2 qs [a, b])) \<and> BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x, y]"
sort_by_mergesort: shows "Sorting_Algorithms.sort = mergesort"
c2: assumes "0 < c" shows "\<forall>y<min k (- 1). b * y + c < y\<^sup>2"
f1: shows "(s' \<otimes> r, s' \<otimes> s) \<in> carrier rel"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d stream" shows "smap g (smap f v) = smap (g \<circ> f) v"
cont_case_prodD2: fixes luba :: "'b set \<Rightarrow> 'b"   and lubb :: "'c set \<Rightarrow> 'c"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubc :: "'d set \<Rightarrow> 'd"   and ordc :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and y :: "'c" assumes "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc (\<lambda>(x, y). f x y)"   and "class.preorder ordb (mk_less ordb)"   and "lub_singleton lubb" shows "cont luba orda lubc ordc (\<lambda>x. f x y)"
triangular_imp_diagonal_1x2: fixes A :: "'a mat" assumes "A \<in> carrier_mat 1 2"   and "admits_triangular_reduction A" shows "admits_diagonal_reduction A"
maximal_independent_subset: fixes V :: "('d, 'e) vec set" obtains B :: "('d, 'e) vec set" where "B \<subseteq> V" and "vec.independent B" and "V \<subseteq> vec.span B"
Labeling_axioms: shows "Labeling c s t f (relabel_effect f l u)"
cd_path_shift: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and j :: "nat"   and k :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob"   and "j \<le> k"   and "IFC_def.is_path prob \<pi>" shows "IFC_def.is_cdi prob i \<pi> k = IFC_def.is_cdi prob (i - j) (\<pi> \<guillemotleft> j) (k - j)"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "Min A \<le> a"
unit_simps_1: fixes a :: "'a" assumes "C.obj a" shows "D.arr (G.unit a)"
ev_in_ball: shows "\<forall>\<^sub>F n in sequentially. flow0 x (t n) \<in> ball p \<delta>"
tendsto_add_const_iff: fixes c :: "'d"   and f :: "'c \<Rightarrow> 'd"   and d :: "'d"   and F :: "'c filter" shows "((\<lambda>x. c + f x) \<longlongrightarrow> c + d) F = (f \<longlongrightarrow> d) F"
hyps: shows "(0::'a) \<le> eval_monom_list \<alpha> m"
theFTOne_sym: fixes cl :: "('a, 'b, 'c) cmd list"   and dl :: "('d, 'e, 'f) cmd list" shows "theFTOne cl dl = theFTOne dl cl"
S: shows "secure (c_process step out s\<^sub>0) I (c_dom D)"
map_fst: fixes G :: "nat graph_rec"   and x :: "'b"   and xs :: "'a list" assumes "dom_sl G" shows "map fst (map (\<lambda>pc. (pc, x)) xs) = xs"
emeasure_PiM: fixes I :: "'i set"   and A :: "'i \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> events i" shows "emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
prems_4: shows "A $ i $ j \<noteq> (0::'a)"
degree_map_poly_hom: fixes hom :: "'b \<Rightarrow> 'c"   and p :: "'b poly" assumes "zero_hom_0 hom" shows "degree (map_poly hom p) = degree p"
invariant_Int: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> invariant A"   and "F \<in> invariant B" shows "F \<in> invariant (A \<inter> B)"
le_lt_per2__lt: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Per Bet Cong A' B' C'"   and "Tarski_neutral_dimensionless.Le Bet Cong A B A' B'"   and "Tarski_neutral_dimensionless.Lt Bet Cong B C B' C'" shows "Tarski_neutral_dimensionless.Lt Bet Cong A C A' C'"
simps_2: fixes x :: "'b"   and l :: "'b list" shows "free_tv (x # l) = free_tv x \<union> free_tv l"
path_hits_winning_region_is_winning: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.strategy G p \<sigma>'"   and "\<And>v. v \<in> ParityGame.winning_region G p \<Longrightarrow> ParityGame.winning_strategy G p \<sigma>' v"   and "\<And>v. v \<in> ParityGame.winning_region G p \<Longrightarrow> \<sigma>' v = \<sigma> v"   and "lset P \<inter> ParityGame.winning_region G p \<noteq> {}" shows "ParityGame.winning_path G p P"
assms_2: shows "L\<^sub>1 \<subseteq> L\<^sub>2"
simps_3: fixes i :: "String.literal"   and a :: "fol" shows "free_in i (Atom a) = (case a of Rl x l \<Rightarrow> occurs_in_list i l)"
eq_iff: fixes a :: "'k"   and b :: "'k" shows "(a = b) = (b \<le> a \<and> a \<le> b)"
filter_IFS_no_violations_accu_no_IFS: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and accu :: "('b \<times> 'b) list"   and edgesList :: "('b \<times> 'b) list" assumes "valid_reqs (get_IFS M)"   and "wf_graph G"   and "get_IFS M = []"   and "set accu \<union> set edgesList \<subseteq> edges G" shows "filter_IFS_no_violations_accu G M accu edgesList = rev edgesList @ accu"
skl1_inv1_def: shows "skl1_inv1 \<equiv> {s. \<forall>Ra A B gny Nr. 0 < signalsInit s (Commit A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) \<longrightarrow> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Ra xnr = Some Nr \<and> sklvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))}"
semiring_simprules_5: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
G\<^sub>1_props: assumes "C.obj a"   and "C.obj a'" shows "\<exists>\<eta> \<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>"
valueChosen_equal: fixes s :: "state"   and v :: "InputsOrNi"   and w :: "InputsOrNi" assumes "valueChosen s v"   and "valueChosen s w" shows "v = w"
snd_conv: fixes x1 :: "'b"   and x2 :: "'a" shows "snd (x1, x2) = x2"
simps_3: fixes F :: "'b formula"   and G :: "'b formula" shows "is_mini_formula (F \<^bold>\<rightarrow> G) = (is_mini_formula F \<and> is_mini_formula G)"
explicationFWB2: shows "\<forall>c x. \<lfloor>\<^bold>\<diamond>\<^sub>pFWB x\<rfloor>\<^sub>c"
cs1l: shows "cs\<^bsup>\<pi>\<^sub>1\<^esup> l = cs\<^bsup>\<pi>\<^esup> l"
nth_tl: fixes xs :: "'a list"   and n :: "nat" assumes "xs \<noteq> []" shows "tl xs ! n = xs ! Suc n"
Ana_keys_wf': fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list"   and k :: "('fun, 'var) Term.term" assumes "Ana t = (K, T)"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "k \<in> set K" shows "wf\<^sub>t\<^sub>r\<^sub>m k"
Un_Int_assoc_eq: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "(A \<inter> B \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
implies_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<leadsto> x \<sqinter> y = x \<leadsto> y"
simps: fixes a1 :: "'a list"   and a2 :: "'b list"   and R :: "('a \<times> 'b) set" shows "((a1, a2) \<in> seq_lift R) = (a1 = [] \<and> a2 = [] \<or> (\<exists>xs ys x y. a1 = x # xs \<and> a2 = y # ys \<and> (xs, ys) \<in> seq_lift R \<and> (x, y) \<in> R))"
inj_on_imp_bij_betw: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "bij_betw f A (f ` A)"
gt_sym_irrefl: fixes f :: "'s" shows "\<not> f >\<^sub>s f"
one_Gp_one: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "\<one>\<^bsub>\<natural>H\<^esub> = \<one>"
cf_def: shows "cf \<equiv> \<lambda>(u, v). if (u, v) \<in> E then c (u, v) - f (u, v) else if (v, u) \<in> E then f (v, u) else (0::'capacity)"
app\<^sub>i: shows "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')"
filter_append: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list" shows "filter P (xs @ ys) = filter P xs @ filter P ys"
real_sqrt_sum_squares_ge1: fixes x :: "real"   and y :: "real" shows "x \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
assms_1: shows "receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
simps: fixes s :: "'a" shows "local.while s = (if guard s then body s \<bind> local.while else return_spmf s)"
subscene_eliminate: fixes Y :: "'a scene"   and X :: "'a scene"   and s\<^sub>1 :: "'a"   and s\<^sub>2 :: "'a"   and s\<^sub>3 :: "'a" assumes "idem_scene Y"   and "X \<subseteq>\<^sub>S Y" shows "s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on X \<oplus>\<^sub>S s\<^sub>3 on Y = s\<^sub>1 \<oplus>\<^sub>S s\<^sub>3 on Y"
rprD: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "rpr A r" shows "complete A r \<and> refl_on A r \<and> trans r"