hcju/isabelleps huggingface dataset:
unfold_unique: fixes f1 :: "'d \<Rightarrow> 'e JF1"   and f2 :: "'f \<Rightarrow> 'e JF2"   and s1 :: "'d \<Rightarrow> ('e, 'd, 'f) F1"   and s2 :: "'f \<Rightarrow> ('e, 'd, 'f) F2" assumes "F1map id f1 f2 \<circ> s1 = dtor1 \<circ> f1"   and "F2map id f1 f2 \<circ> s2 = dtor2 \<circ> f2" shows "f1 = unfold1 s1 s2 \<and> f2 = unfold2 s1 s2"
ideal_nsum_closedTr: fixes I :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R I" shows "(\<forall>j\<le>n. f j \<in> I) \<longrightarrow> \<Sigma>\<^sub>e R f n \<in> I"
Mul_Append_def: fixes n :: "nat" shows "Mul_Append n \<equiv> \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>Safe\<rbrace> \<acute>ind := 0;; \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>Safe \<and> \<acute>ind = 0\<rbrace>  WHILE \<acute>ind < length \<acute>M INV \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind \<le> length \<acute>M\<rbrace>  DO \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind < length \<acute>M\<rbrace>  IF \<acute>M ! \<acute>ind = Black THEN \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind < length \<acute>M \<and> \<acute>M ! \<acute>ind = Black\<rbrace> \<acute>M := \<acute>M[\<acute>ind := White] ELSE \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind < length \<acute>M \<and> \<acute>ind \<notin> Reach \<acute>E\<rbrace> \<acute>E := Append_to_free (\<acute>ind, \<acute>E) FI;; \<lbrace>\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv (\<acute>ind + 1) \<and> \<acute>ind < length \<acute>M\<rbrace> \<acute>ind := \<acute>ind + 1 OD"
shift_over_ao_dists_4: fixes a :: "'b word"   and b :: "'b word"   and c :: "nat" shows "a AND b << c = (a << c) AND (b << c)"
Derivation_take: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat"   and a :: "symbol list"   and b :: "symbol list" assumes "derivation_ge D i"   and "Derivation a D b" shows "take i a = take i b"
1: shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) \<^bold>& \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>) in v]"
Cons: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and v :: "val"   and s1 :: "state"   and es :: "expr list"   and vs :: "val list"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>es\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>e # es\<doteq>\<succ>v # vs\<midarrow>n\<rightarrow> s2"
map_poly_homo_polyneg: fixes f :: "'a \<Rightarrow> 'c"   and p1 :: "'a poly" assumes "\<And>x y. f (- x) = - f x" shows "map_poly f (~\<^sub>p p1) = ~\<^sub>p (map_poly f p1)"
Px: shows "simple_distributed M X (\<lambda>x. prob (X -` {x} \<inter> space M))"
l: shows "rat_of_int (d fs i) * fs.gs.gso i $ j \<in> \<int>"
inject_1: fixes x1 :: "name"   and y1 :: "name" shows "(Sconst x1 = Sconst y1) = (x1 = y1)"
terminates_sequence_appE: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and xs :: "('e, 'd, 'f) com list"   and ys :: "('e, 'd, 'f) com list"   and s :: "'e" assumes "\<Gamma>\<turnstile>sequence Seq (xs @ ys) \<down> Normal s" obtains "\<Gamma>\<turnstile>sequence Seq xs \<down> Normal s" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>sequence Seq ys \<down> s'"
coeff_powsum_mpoly_0_right: fixes X :: "nat set"   and mon :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (powsum_mpoly X 0) mon = (if mon = 0 then of_nat (card X) else (0::'b))"
Int_Diff: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and B :: "'d set" assumes "finite A" shows "sum g A = sum g (A \<inter> B) + sum g (A - B)"
refl: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and Z :: "('f \<Rightarrow> 'g) set" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l, Z\<rangle>"
assms: fixes T :: "'a set" assumes "x \<in> T"   and "open T" shows "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
comp: fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'i"   and i :: "'a \<Rightarrow> 'g"   and j :: "'b \<Rightarrow> 'h"   and k :: "'i \<Rightarrow> 'c" shows "map_gpv' f g h \<circ> map_gpv' i j k = map_gpv' (f \<circ> i) (g \<circ> j) (k \<circ> h)"
prod_mset_image: fixes X :: "'a poly multiset" shows "\<Prod>\<^sub># (image_mset poly_x_mult_y X) = poly_x_mult_y (\<Prod>\<^sub># X)"
power_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" shows "power.power one times a zero_class.zero = one"
BSD_implies_SD_for_modified_view: fixes \<V>' :: "'a V_rec" assumes "BSD \<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 "SD \<V> Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"
3: shows "prv (imp \<phi> (cnj (PP \<langle>\<phi>\<rangle>) (PP (encF N \<langle>\<phi>\<rangle>))))"
last_index_image_set: fixes xs :: "'a list" assumes "distinct xs" shows "last_index xs ` set xs = {..<length xs}"
assms: shows "local.quadratic_hensel_loop j = (q, S, T, D, H)"
valid_reqs_def: fixes M :: "'a SecurityInvariant_configured list" shows "valid_reqs M \<equiv> \<forall>m\<in>set M. configured_SecurityInvariant m"
l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and create_character_data :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) character_data_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool" shows "l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs create_character_data known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_character_data known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr"
fps_add_nth: fixes f :: "'b fps"   and g :: "'b fps"   and n :: "nat" shows "(f + g) $ n = f $ n + g $ n"
l_get_shadow_root_axioms: shows "l_get_shadow_root ShadowRootClass.type_wf get_shadow_root get_shadow_root_locs"
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P P' A\<^sub>P \<Psi>\<^sub>P"   and "x \<sharp> \<Psi>"   and "x \<sharp> A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* C"   and "x \<sharp> C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) (\<lparr>\<nu>x\<rparr>P') (x # A\<^sub>P) \<Psi>\<^sub>P"
cases: fixes a1 :: "'a list"   and a2 :: "'b list"   and r :: "('a \<times> 'b) set" assumes "(a1, a2) \<in> listrel r" obtains "a1 = []" and "a2 = []"   | x :: "'a" and y :: "'b" and xs :: "'a list" and ys :: "'b list" where "a1 = x # xs" and "a2 = y # ys" and "(x, y) \<in> r" and "(xs, ys) \<in> listrel r"
enum_all_sdim_def: fixes P :: "sdim \<Rightarrow> bool" shows "enum_class.enum_all P = (P Length \<and> P Mass \<and> P Time \<and> P Current \<and> P Temperature \<and> P Amount \<and> P Intensity)"
lspace: fixes n :: "nat" shows "(\<lambda>x. max (- real n) (min (real n) (f x))) \<in> lspace (lebesgue_on S) (ennreal p)"
ConstFtor'_2: fixes A :: "('a, 'b, 'c) Category_scheme"   and B :: "('d, 'e, 'f) Category_scheme"   and b :: "'d" assumes "Category A"   and "Category B"   and "b \<in> obj\<^bsub>B\<^esub>" shows "FunctorM (ConstFunctor' A B b)"
finite_roots_with: fixes p :: "'a poly"   and P :: "'a \<Rightarrow> bool" assumes "p \<noteq> 0" shows "finite (roots_with P p)"
inverse_ennreal: fixes r :: "real" assumes "0 < r" shows "inverse (ennreal r) = ennreal (inverse r)"
cyclic_rotation': shows "map (cycle_of_list cs) cs = rotate1 cs"
filterlim_at_infinity_imp_eventually_ne: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and c :: "'c" assumes "filterlim f at_infinity F" shows "\<forall>\<^sub>F z in F. f z \<noteq> c"
wf_J_prog'_wf_J_prog: fixes P :: "'a J_prog" assumes "wf_J_prog' P" shows "wf_J_prog P"
filter_min_append_cases: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "transp rel"   and "x \<in> set xs \<union> set ys" obtains "x \<in> set (local.filter_min_append xs ys)"   | y :: "'a" where "y \<in> set (local.filter_min_append xs ys)" and "x \<notin> set (local.filter_min_append xs ys)" and "rel y x"
child_parent_dual: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some ptr"
star_rtc3: fixes x :: "'a" shows "((1::'a) + x \<cdot> x = x) = (x = x\<^sup>\<star>)"
rel_star_contr: fixes X :: "('b \<times> 'b) set"   and Y :: "('b \<times> 'c) set" shows "X\<^sup>* O Y = (\<Union>n. rel_dioid.power X n O Y)"
invfun_l: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and a :: "'e" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B"   and "a \<in> A" shows "(f\<inverse>\<^bsub>B,A\<^esub>) (f a) = a"
f_in_funcset: shows "f \<in> A \<rightarrow> A"
prems_1: shows "C \<in> f \<turnstile> folding_g.\<C>"
braun_if_braun_indices: fixes t :: "'b tree" assumes "braun_indices t = {1..size t}" shows "braun t"
ps_first_two: obtains q :: "real poly" and ps' :: "real poly list" where "ps = p # q # ps'"
rel_simps_24: fixes m :: "num"   and n :: "num" shows "(numeral m \<le> numeral n) = (m \<le> n)"
det_diagonal: fixes A :: "(('b, 'a) vec, 'a) vec" assumes "\<And>i j. i \<noteq> j \<Longrightarrow> A $ i $ j = (0::'b)" shows "det A = (\<Prod>i\<in>UNIV. A $ i $ i)"
not_in_id: fixes x :: "'d" assumes "x \<notin> iso_arcs hom ` A" shows "wrap_iso_arcs hom f x = x"
asymp_equiv_uminus: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "(\<lambda>x. - f x) \<sim>[F] (\<lambda>x. - g x)"
simps_4: fixes F :: "'b formula"   and G :: "'b formula" shows "nnf (\<^bold>\<not> (F \<^bold>\<or> G)) = nnf (\<^bold>\<not> F) \<^bold>\<and> nnf (\<^bold>\<not> G)"
ide_mkarr_Ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Ide C a" shows "partial_magma.ide (free_monoidal_category.comp C) (free_monoidal_category.mkarr C a)"
pivot_fun_stabilizes: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i1 :: "nat"   and i2 :: "nat" assumes "pivot_fun A f nc"   and "i1 \<le> i2"   and "i2 < dim_row A"   and "nc \<le> f i1" shows "f i2 = nc"
max_normal_subgroup_axioms_def: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" shows "max_normal_subgroup_axioms H G \<equiv> H \<noteq> carrier G \<and> (\<forall>J. J \<lhd> G \<longrightarrow> J \<noteq> H \<longrightarrow> J \<noteq> carrier G \<longrightarrow> \<not> H \<subseteq> J)"
dvd_Lcm_nat: fixes m :: "nat"   and M :: "nat set" assumes "m \<in> M" shows "m dvd Lcm M"
add_right: fixes a :: "'a"   and b :: "'b"   and b' :: "'b" shows "a ** (b + b') = a ** b + a ** b'"
beforexD1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and I :: "nat set"   and d :: "nat" assumes "P,C,M \<rhd> xt / I,d" shows "pcs xt \<subseteq> I"
cospoly_lb_imp_sinpoly_lb: fixes x :: "real"   and k :: "nat" assumes "0 \<le> x"   and "0 < k"   and "\<And>x. 0 \<le> x \<Longrightarrow> cospoly (k - 1) x \<le> cos x" shows "sinpoly k x \<le> sin x"
plus: fixes p1 :: "perm"   and p2 :: "perm" assumes "P p1"   and "P p2" shows "P (p1 + p2)"
mark_guards_simps_4: fixes f :: "'f"   and c\<^sub>1 :: "('d, 'e, 'g) com"   and c\<^sub>2 :: "('d, 'e, 'g) com" shows "mark_guards f (Seq c\<^sub>1 c\<^sub>2) = Seq (mark_guards f c\<^sub>1) (mark_guards f c\<^sub>2)"
lebesgue_integral_eq_integral: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "integrable (lebesgue_on S) f"   and "S \<in> sets lebesgue" shows "integral\<^sup>L (lebesgue_on S) f = integral S f"
rank_older_seniors_bounded: fixes y :: "nat"   and x :: "nat"   and n :: "nat"   and j :: "nat" assumes "y \<in> older_seniors x n"   and "rank x n = Some j" shows "\<exists>j'<j. rank y n = Some j'"
induced_in_sccs: shows "G \<restriction> S \<in> sccs"
LeastI2_ex: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>a. P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (ord.Least less_eq P)"
simps_1: fixes i :: "'a"   and x :: "int poly" shows "sub_trees (Factor_Leaf i x) = {Factor_Leaf i x}"
ex_bij_betw_finite_nat: fixes M :: "'d set" assumes "finite M" shows "\<exists>h. bij_betw h M {0..<card M}"
length_partition_by_median_5: fixes ps :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"   and "(l, m, r) = partition_by_median k ps" shows "0 < length l"
less_eq_Sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and u :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v \<in> A \<Longrightarrow> less_eq u v"   and "A \<noteq> {}" shows "less_eq u (Sup A)"
ik\<^sub>e\<^sub>s\<^sub>t_finite: fixes A :: "('fun, 'var) extstrand_step list" shows "finite (ik\<^sub>e\<^sub>s\<^sub>t A)"
adm_prod_split: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "adm (\<lambda>p. f (fst p) (snd p))" shows "adm (\<lambda>(x, y). f x y)"
is_meta_adjunction: shows "meta_adjunction (\<cdot>\<^sub>A) (\<cdot>\<^sub>C) (F \<circ> F') (G' \<circ> G) (\<lambda>z. \<phi>' z \<circ> \<phi> (F' z)) (\<lambda>x. \<psi> x \<circ> \<psi>' (G x))"
euler_increment_def: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b" shows "euler_increment f = rk_increment f 1 (\<lambda>i j. 0) (\<lambda>i. 1) (\<lambda>i. 0)"
induct: fixes P :: "'b rexp \<Rightarrow> 'b rexp \<Rightarrow> bool"   and a0 :: "'b rexp"   and a1 :: "'b rexp" assumes "\<And>uu_. P Zero uu_"   and "P One Zero"   and "\<And>v. P (Atom v) Zero"   and "\<And>v va. P (Plus v va) Zero"   and "\<And>v va. P (Times v va) Zero"   and "\<And>v. P (Star v) Zero"   and "P One One"   and "\<And>v. P One (Atom v)"   and "\<And>v va. P One (Plus v va)"   and "\<And>v va. P One (Times v va)"   and "\<And>v. P One (Star v)"   and "\<And>v. P (Atom v) One"   and "\<And>v va. P (Plus v va) One"   and "\<And>v va. P (Times v va) One"   and "\<And>v. P (Star v) One"   and "\<And>r s v. P s (Atom v) \<Longrightarrow> P (Times r s) (Atom v)"   and "\<And>r s v va. P s (Plus v va) \<Longrightarrow> P (Times r s) (Plus v va)"   and "\<And>r s v va. P s (Times v va) \<Longrightarrow> P (Times r s) (Times v va)"   and "\<And>r s v. P s (Star v) \<Longrightarrow> P (Times r s) (Star v)"   and "\<And>v va. P (Atom v) (Atom va)"   and "\<And>v va vb. P (Atom v) (Plus va vb)"   and "\<And>v va vb. P (Atom v) (Times va vb)"   and "\<And>v va. P (Atom v) (Star va)"   and "\<And>v va vb. P (Plus v va) (Atom vb)"   and "\<And>v va vb vc. P (Plus v va) (Plus vb vc)"   and "\<And>v va vb vc. P (Plus v va) (Times vb vc)"   and "\<And>v va vb. P (Plus v va) (Star vb)"   and "\<And>v va. P (Star v) (Atom va)"   and "\<And>v va vb. P (Star v) (Plus va vb)"   and "\<And>v va vb. P (Star v) (Times va vb)"   and "\<And>v va. P (Star v) (Star va)" shows "P a0 a1"
sup_assn_def: fixes P :: "assn"   and Q :: "assn" shows "P \<or>\<^sub>A Q \<equiv> Abs_assn (\<lambda>h. h \<Turnstile> P \<or> h \<Turnstile> Q)"
wn_triangle2: fixes z :: "complex"   and a :: "complex"   and b :: "complex"   and c :: "complex" assumes "z \<in> interior (convex hull {a, b, c})" shows "0 < Im ((b - a) * cnj (b - z)) \<and> 0 < Im ((c - b) * cnj (c - z)) \<and> 0 < Im ((a - c) * cnj (a - z)) \<or> Im ((b - a) * cnj (b - z)) < 0 \<and> 0 < Im ((b - c) * cnj (b - z)) \<and> 0 < Im ((a - b) * cnj (a - z)) \<and> 0 < Im ((c - a) * cnj (c - z))"
prj0_simps_3: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.cod \<p>\<^sub>0[f, g] = C.dom g"
same_prefix_prefix: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "prefix (xs @ ys) (xs @ zs) = prefix ys zs"
cases: fixes x :: "'b heap list" obtains "x = []"   | h :: "'b heap" and hs :: "'b heap list" where "x = h # hs"
reads_consistent_flush_independent: fixes sb :: "'b memref list"   and xs :: "'c memref list"   and pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs is_Write\<^sub>s\<^sub>b sb \<inter> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b xs = {}"   and "reads_consistent pending_write \<O> m xs" shows "reads_consistent pending_write \<O> (flush sb m) xs"
prems_2: shows "(l # ls) ! i = (p, RMW a t (D, f) cond ret A L R W # is, \<theta>, [], \<D>, \<O>)"
exhaust: fixes \<Delta> :: "('a, 'b) network_scheme"   and y :: "'c node" assumes "flow_attainability \<Delta>" obtains "y = SOURCE"   | x2 :: "'c" where "y = Inner x2"
imirror_cut_ge: fixes I :: "nat set"   and t :: "nat" assumes "finite I" shows "imirror I \<down>\<ge> mirror_elem t I = imirror_bounds (I \<down>\<le> t) (iMin I) (Max I)"
d_restrict_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<le> y \<cdot> ar (ar x))"
power_one_right: fixes a :: "'a" shows "a +^ 1 = a"
z_def: shows "z = f (x \<sqinter> conjugate f y)"
focus_in_Polys: fixes X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "focus X p \<in> P[X]"
Disjoint_Eats_D2: fixes x :: "tm"   and y :: "tm"   and z :: "tm" shows "{Disjoint (Eats x y) z} \<turnstile> Neg (y IN z)"
fs: fixes f :: "int poly" assumes "f \<in># fs" shows "monic (Mp f)"
power_add: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m + n) = times (power.power one times a m) (power.power one times a n)"
intro: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and J :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ah \<Rightarrow> 'ag"   and a :: "'ag"   and \<chi> :: "'ah \<Rightarrow> 'ag" assumes "category C"   and "category J"   and "diagram J C D"   and "constant_functor J C a"   and "natural_transformation J C (constant_functor.map J C a) D \<chi>" shows "Limit.cone J C D a \<chi>"
rSym: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "(Q, P) \<in> X"
integrable_on_open_interval: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" shows "(f integrable_on box a b) = (f integrable_on cbox a b)"
sup_state_opt_None_any: fixes P :: "'b prog"   and s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> None \<le>' s"
eval_empty_graph: fixes N :: "'a set"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph \<lparr>nodes = N, edges = {}\<rparr>" shows "sinvar \<lparr>nodes = N, edges = {}\<rparr> nP"
prems_4: shows "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S'))"
C: shows "Max (offs_set_prev ns (x # xs) index key mi ma i) = j"
invrules_5: shows "(P\<^sub>1 \<union> wrap B\<^sub>1) \<inter> (P\<^sub>2 \<union> wrap B\<^sub>2) = {}"
hyps: fixes y :: "nat"   and e :: "nat" assumes "y < n" shows "(local.encode e y = 0) = (y = 0)"
conv_radius_geI_ex: fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'b" assumes "\<And>r. \<lbrakk>0 < r; ereal r < R\<rbrakk> \<Longrightarrow> \<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z ^ n)" shows "R \<le> conv_radius f"
surjective: fixes r :: "('b, 'c) simple_packet_scheme" shows "r = \<lparr>p_iiface = p_iiface r, p_oiface = p_oiface r, p_src = p_src r, p_dst = p_dst r, p_proto = p_proto r, p_sport = p_sport r, p_dport = p_dport r, p_tcp_flags = p_tcp_flags r, p_payload = p_payload r, \<dots> = simple_packet.more r\<rparr>"
allDecisionsThenUIP: fixes M :: "(Literal \<times> bool) list"   and c :: "Literal list"   and l :: "Literal" assumes "uniq (elements M)"   and "\<forall>l'. l' el c \<longrightarrow> opposite l' el decisions M"   and "isLastAssertedLiteral (opposite l) (oppositeLiteralList c) (elements M)" shows "isUIP l c M"
deg_belowI: fixes R :: "('c, 'd) ring_scheme"   and n :: "nat"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "n \<noteq> 0 \<Longrightarrow> coeff (UP R) p n \<noteq> \<zero>\<^bsub>R\<^esub>"   and "p \<in> carrier (UP R)" shows "n \<le> deg R p"
NSLIM_def: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and L :: "'d" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L = (\<forall>x. x \<noteq> star_of a \<and> x \<approx> star_of a \<longrightarrow> (*f* f) x \<approx> star_of L)"
maps_ide_in_hom: fixes a :: "'a" assumes "A.ide a" shows "\<guillemotleft>t a : F a \<rightarrow>\<^sub>B G a\<guillemotright>"
D1: fixes U :: "'a set" assumes "U \<in> \<D>" shows "open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
i_shrink_i_append_mod: fixes xs :: "'b message_af list"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "length xs mod k = 0" shows "(xs \<frown> f) \<div> k = xs \<div> k \<frown> f \<div> k"
that: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F"   and "red_single p q f (0::'a)" shows "thesis"
Rep_star_star_n: fixes X :: "nat \<Rightarrow> 'c" shows "X \<in> Rep_star (star_n X)"