hcju/isabelleps huggingface dataset:
assign\<^sub>2: fixes x :: "Eg1.var"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and e :: "aexp"   and t :: "bexp set"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P' :: "bexp set"   and P :: "bexp set" assumes "x \<in> dom \<Gamma>"   and "type_aexpr \<Gamma> e t"   and "type_stable \<S> t"   and "P' = restrict_preds_to_vars (assign_post P x e) {v. stable \<S> v}"   and "x \<notin> snd \<S> \<longrightarrow> subtype t P' (dma_type x)" shows "has_type \<Gamma> \<S> P (x \<leftarrow> e) (\<Gamma>(x \<mapsto> t)) \<S> P'"
sep_add_cancel: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x ## z"   and "y ## z" shows "(x + z = y + z) = (x = y)"
meet_glb: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<le> x \<and> z \<le> y" shows "z \<le> x \<sqinter> y"
assms_1: shows "isFDERIV (J + K) [0..<J + K] (es @ rs) xs"
Rep_formula_inject: fixes x :: "('a, 'b, 'c) formula"   and y :: "('a, 'b, 'c) formula" shows "(Rep_formula x = Rep_formula y) = (x = y)"
at_within_union: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and S :: "'c set"   and T :: "'c set" assumes "class.topological_space open" shows "topological_space.at_within open x (S \<union> T) = sup (topological_space.at_within open x S) (topological_space.at_within open x T)"
f_\<upsilon>_val: fixes a1 :: "'b"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "f_\<upsilon> (f_lam a1 \<tau>) \<tau> = f_\<upsilon> a1 \<tau>"
sum_with_transfer: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "right_total A"   and "bi_unique A"   and "bi_unique B" shows "rel_fun (rel_fun A (rel_fun A A)) (rel_fun A (rel_fun (rel_fun B A) (rel_fun (rel_set B) A))) sum_with sum_with"
smod_word_def: fixes v :: "'a word"   and w :: "'a word" shows "v smod w = word_of_int (sint v smod sint w)"
integrable: shows "set_integrable lborel (box (0, 0) (1, 1)) (\<lambda>(x, y). P x * P y / (1 - (1 - x * y) * z))"
sum_commutativity: fixes x :: "i"   and y :: "i" assumes "CEM" shows "x \<oplus> y = y \<oplus> x"
carrierE: fixes x :: "'c"   and charts :: "('c, 'd) chart set" assumes "x \<in> manifold.carrier charts" obtains c :: "('c, 'd) chart" where "c \<in> charts" and "x \<in> domain c"
g_set_xy_loc_def: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme" shows "g_set_xy_loc ops1 ops2 \<equiv> g_set_xy_defs_loc ops1 ops2 \<and> StdSet ops1 \<and> StdSet ops2"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "type_wf = ShadowRootClass.type_wf"   and "set_mode = set_mode"   and "set_mode_locs = set_mode_locs" shows "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs"
mask_over_length: fixes n :: "nat" assumes "LENGTH('b) \<le> n" shows "mask n = - 1"
of_complex_one_iff: fixes x :: "complex" shows "(of_complex x = 1\<^sub>h) = (x = 1)"
assms_2: shows "\<exists>x\<in>S. \<forall>y\<in>S. x \<le> y"
eq_add_iff: fixes x :: "'a"   and y :: "'a" shows "(x = x + y) = (y = (0::'a))"
SN_defs: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" shows "SN_on r A = (\<nexists>f. f 0 \<in> A \<and> (\<forall>i. (f i, f (Suc i)) \<in> r))"
similarity_scale_2: fixes I :: "complex \<times> complex \<times> complex \<times> complex"   and k :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" shows "similarity I (k *\<^sub>s\<^sub>m M) = k *\<^sub>s\<^sub>m similarity I M"
VInter_vempty: shows "\<Inter>\<^sub>\<circ> 0 = 0"
res: shows "(False, i - 1, dec_i (update_im1 (update_i f (fs ! (i - 1))) (fs ! i)), swap_mu m mu i (dmu_ij_state state i (i - 1)) (d_state state (i - 1)) (d_state state i) (d_state state (Suc i)), iarray_update ds i ((d_state state (Suc i) * d_state state (i - 1) + dmu_ij_state state i (i - 1) * dmu_ij_state state i (i - 1)) div d_state state i)) = (upw', i', state')"
p: shows "p \<in> carrier (K [X]) \<and> pirreducible K p \<and> lead_coeff p = \<one> \<and> a_kernel (K [X]) R (\<lambda>p. local.eval p x) = PIdl\<^bsub>K [X]\<^esub> p"
RModule_inner_dirsum: fixes Ns :: "'m set list" assumes "\<forall>N\<in>set Ns. RSubmodule N" shows "RModule R (\<cdot>) (\<Oplus>N\<leftarrow>Ns. N)"
lattice_partial_function_definition: 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" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "partial_function_definitions less_eq Sup"
embed_bool_bounded_by_1: fixes P :: "'a \<Rightarrow> bool" shows "bounded_by 1 \<guillemotleft> P \<guillemotright>"
hyps: fixes x :: "'a" shows "\<exists>ys y. x # xs = ys @ [y]"
terminal_char2: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and t :: "'b" assumes "set_category_given_img S img \<AA>" shows "category.terminal S t = (partial_magma.ide S t \<and> set_category_data.set S img t = {t})"
F'_def: shows "F' \<equiv> (\<lambda>y. y - discrete_evolution (rk2_increment p (\<lambda>t x. f (t, x))) (t + h) t (x t)) ` R"
intros_1: fixes A :: "fmla" shows "A IMP A \<in> boolean_axioms"
ODE_vars_lr: fixes x :: "'d"   and I :: "('e, 'f, 'd) interp"   and ODE :: "('e, 'd) ODE" shows "(Inl x \<in> semBV I ODE) = (Inr x \<in> semBV I ODE)"
le_\<I>_def: fixes \<I> :: "('e, 'f) \<I>"   and \<I>' :: "('e, 'f) \<I>" shows "(\<I> \<le> \<I>') = (outs_\<I> \<I> \<subseteq> outs_\<I> \<I>' \<and> (\<forall>x\<in>outs_\<I> \<I>. responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x))"
simps_7: fixes uw :: "'d list"   and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and F' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>uw\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> = trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F'"
fresh_abs_funE: fixes x :: "'a"   and b :: "'b"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp x)"   and "b \<sharp> [a].x"   and "b \<noteq> a" shows "b \<sharp> x"
Gk_eq_P'A: shows "Gauss_Jordan_upt_k A k = P' ** A"
belongs_standard_open_iff: fixes x :: "'a"   and \<pp> :: "'a set" assumes "x \<in> R"   and "\<pp> \<in> Spec" shows "(x \<notin> \<pp>) = (\<pp> \<in> \<D>(x))"
Close2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "Q \<longmapsto> a<x> \<prec> Q'"   and "x \<sharp> Q" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> <\<nu>x>(P' \<parallel> Q')"
abs_eq: fixes x :: "((real, 'a) vec, 'a) vec" shows "($$) (to_mtx x) = ($) x"
snoc: shows "norm_fold (q, \<nu>, p) xs = ((q, \<nu>, p) \<in> set xs, q, \<nu>, p)"
b_e_type_br_2: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Br i" shows "\<exists>ts_c ts''. ts = ts_c @ ts'' \<and> label \<C> ! i = ts''"
fold_maybe_or_invalid_base: fixes l :: "trilean list" shows "fold (\<or>?) l invalid = invalid"
decomp_ik: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list" assumes "Ana t = (K, M)" shows "ik\<^sub>s\<^sub>t (decomp t) = set M"
choose_two: fixes n :: "nat" shows "n choose 2 = n * (n - 1) div 2"
conts''': fixes x2 :: "'b"   and i :: "'a" assumes "x2 \<noteq> x1"   and "dist x2 x1 < (SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a)))" shows "dist (f i x2) (f i x1) < \<epsilon> / real CARD('a)"
strict_dom_def: fixes G :: "nat graph_rec"   and i :: "nat"   and j :: "nat" assumes "cfg_doms G" shows "cfg_doms.strict_dom G i j \<equiv> let res = cfg_doms.dom_kildall G ([] # replicate (length (g_V G) - 1) (rev [0..<length (g_V G)])) ! j in i \<in> set res"
mono: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "a \<le> c"   and "b \<le> d" shows "min a b \<le> min c d"
seE_fringe2: fixes re :: "('d \<times> nat) edge"   and prb :: "('d, 'e, 'f) pre_RedBlack"   and prb' :: "('d, 'e, 'f) pre_RedBlack"   and c :: "('e, 'f) conf" assumes "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>"   and "marked prb (src re)" shows "fringe prb' = fringe prb"
f_eq_g: fixes x :: "'a" assumes "x \<in> S \<union> UF \<and> x \<in> S \<union> \<Union> G" shows "f x = g x"
HEndPhase0_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p" shows "blocksOf s' q \<subseteq> blocksOf s q \<union> {dblock s' q}"
simps_3: fixes a :: "'a"   and \<nu> :: "'a set" shows "af_letter p(a) \<nu> = (if a \<in> \<nu> then true else false)"
simps: fixes a :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set" shows "(a \<in> grid b ds) = (a = b \<or> (\<exists>p d dir. a = child p dir d \<and> p \<in> grid b ds \<and> d \<in> ds))"
l2_obs_iagreement_Init: shows "oreach l2 \<subseteq> l2_iagreement_Init"
axioms_1: fixes x :: "'d \<Longrightarrow> 'e" assumes "bij_lens x" shows "weak_lens x"
isomorphic_integer_zeroth_homology_group: fixes X :: "'b topology" assumes "path_connected_space X"   and "topspace X \<noteq> {}" shows "homology_group 0 X \<cong> integer_group"
count_add_mset: fixes b :: "'d"   and A :: "'d multiset"   and a :: "'d" shows "count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"
digitToTree_inv_1: fixes n :: "nat"   and d :: "('a, 'b) Digit" assumes "FingerTreeStruc.is_leveln_digit n d" shows "FingerTreeStruc.is_leveln_ftree n (FingerTreeStruc.digitToTree d)"
conjugate_mat_def: fixes m :: "'b mat" shows "conjugate m = mat (dim_row m) (dim_col m) (\<lambda>(i, j). conjugate (m $$ (i, j)))"
naturality: fixes f :: "'a" assumes "arr f" shows "src (cod f) \<cdot> src f = src f \<cdot> src (local.dom f)"
or_gt_0: fixes \<G> :: "'a cyclic_group" assumes "chaum_ped_\<Sigma>_base \<G>" shows "0 < order \<G>"
rep_eq: fixes x :: "integer" shows "word_of_integer x = word_of_int (int_of_integer x)"
rep_eq: fixes x :: "'d" shows "set_nebset (nebsingleton x) = {x}"
TAbs_type: fixes \<Gamma> :: "binding list"   and S :: "POPLmark.type"   and s :: "trm"   and T :: "POPLmark.type"   and U :: "POPLmark.type"   and U' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> (\<lambda><:S. s) : T"   and "\<Gamma> \<turnstile> T <: (\<forall><:U. U')" obtains S' :: "POPLmark.type" where "\<Gamma> \<turnstile> U <: S" and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'" and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'"
intros: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "('a, 'c) com"   and s :: "'b"   and c' :: "('a, 'c) com"   and s' :: "'b" assumes "PL.transC tval aval (c, s) (c', s') \<and> PL.mayDiverge tval aval c' s'" shows "PL.mayDiverge tval aval c s"
df_system_axioms: shows "df_system s\<^sub>0 lstep"
suma_comm: 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"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I" shows "Tarski_neutral_dimensionless.SumA Bet Cong C B A F E D I H G"
Restrp_transitive: shows "transitive UNIV ((\<sqsubseteq>) \<restriction> A)"
extend_map_affine_to_sphere2: fixes S :: "'a set"   and U :: "'b set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and L :: "'a set" assumes "compact S"   and "convex U"   and "bounded U"   and "affine T"   and "S \<subseteq> T"   and "aff_dim T \<le> aff_dim U"   and "continuous_on S f"   and "f ` S \<subseteq> rel_frontier U"   and "\<And>C. C \<in> components (T - S) \<Longrightarrow> C \<inter> L \<noteq> {}" obtains K :: "'a set" and g :: "'a \<Rightarrow> 'b" where "finite K" and "K \<subseteq> L" and "K \<subseteq> T" and "disjnt K S" and "continuous_on (T - K) g" and "g ` (T - K) \<subseteq> rel_frontier U" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
intersect_list'_nodes_card: fixes AA :: "('c, 'd) dba list" assumes "list_all (finite \<circ> DBA.nodes) AA" shows "card (DGBA.nodes (intersect_list' AA)) \<le> prod_list (map (card \<circ> DBA.nodes) AA)"
FadmitG: shows "Fadmit \<sigma> (foldr (&&) \<Gamma> TT)"
cring_simprules_15: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "cring R" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
fsubset_antisym: fixes A :: "'a fset"   and B :: "'a fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| A" shows "A = B"
exists_bijective_linear_map_W_vec: fixes P :: "'a mod_ring poly set" assumes "finite P"   and "u = \<Prod>P"   and "P \<subseteq> {q. irreducible q \<and> monic q}" shows "\<exists>f. linear_map class_ring (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}) (module_vec TYPE('a mod_ring) (card P)) f \<and> bij_betw f (carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})) (carrier_vec (card P))"
alw_nxt: fixes \<phi> :: "'c stream \<Rightarrow> bool" shows "alw \<phi> = (\<lambda>xs. \<phi> xs \<and> nxt (alw \<phi>) xs)"
ennreal_mult_strict_left_mono: fixes a :: "ennreal"   and c :: "ennreal"   and b :: "ennreal" assumes "a < c"   and "0 < b"   and "b < \<top>" shows "b * a < b * c"
app_vrangeD: fixes b :: "V"   and r :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "\<exists>a. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
nn_cond_exp_sum: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. nn_cond_exp M F f x + nn_cond_exp M F g x = nn_cond_exp M F (\<lambda>x. f x + g x) x"
polyhedron_linear_image_eq: fixes h :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear h"   and "bij h" shows "polyhedron (h ` S) = polyhedron S"
adjust_assign_before: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and y :: "char list" assumes "HT \<pi> P c Q" shows "HT \<pi> (\<lambda>s. P (s(x := s y))) (x[] ::= y;; c) (\<lambda>s\<^sub>0. Q (s\<^sub>0(x := s\<^sub>0 y)))"
distr_component: fixes i :: "'a" shows "distr (interval_measure (\<lambda>x. x)) (Pi\<^sub>M {i} (\<lambda>_. interval_measure (\<lambda>x. x))) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} (\<lambda>_. interval_measure (\<lambda>x. x))"
take_0: fixes xs :: "'g list" shows "take 0 xs = []"
mult_smult_left: fixes a :: "'c"   and p :: "'c poly"   and q :: "'c poly" shows "smult a p * q = smult a (p * q)"
emp_gen_iff: fixes G :: "'b list set" shows "(G\<^sub>+ = {}) = (\<langle>G\<rangle> = {\<epsilon>})"
Abs_3: shows "(x', T') # \<Gamma> \<turnstile> t' : U"
assms_2: assumes "\<And>x y. x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> y\<cdot>C\<^sup>\<infinity> \<Longrightarrow> f\<cdot>x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> g\<cdot>y\<cdot>C\<^sup>\<infinity>" shows "P"
rmlab_other_label: fixes l :: "'a"   and l' :: "'a"   and A :: "'a \<Rightarrow> 'b set" assumes "l \<noteq> l'" shows "rmlab l A l' = A l'"
subspace_def: fixes S :: "('b, 'c) vec set" shows "vec.subspace S = (0 \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *s x \<in> S))"
foundation14: fixes \<tau> :: "'a state \<times> 'a state"   and A :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> A \<triangleq> false) = (\<tau> \<Turnstile> not A)"
rev_lists_conv: fixes A :: "'b set" shows "rev ` lists A = lists A"
b2: shows "steps0 (1, l', r') B n3 = (0, l'', r'')"
gcd_mult_left_left_cancel: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "coprime b c" shows "gcd (c * a) b = gcd a b"
globally_sound_modes_compatible: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.globally_sound_mode_use eval (cms, mem)" shows "sifum_security_init.compatible_modes (map snd cms)"
partial_castTE: fixes s :: "'c net_state"   and R :: "nat set"   and m :: "'d"   and s' :: "'c net_state"   and S :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and T :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set" assumes "(s, R:*cast(m), s') \<in> pnet_sos S T" obtains sa :: "'c net_state" and s'a :: "'c net_state" and t :: "'c net_state" and H :: "nat set" and K :: "nat set" and t' :: "'c net_state" where "s = SubnetS sa__ t__" and "s' = SubnetS s'a__ t'__" and "(sa__, R:*cast(m), s'a__) \<in> S" and "(t__, H__\<not>K__:arrive(m), t'__) \<in> T" and "H__ \<subseteq> R" and "K__ \<inter> R = {}"   | sa :: "'c net_state" and H :: "nat set" and K :: "nat set" and s'a :: "'c net_state" and t :: "'c net_state" and t' :: "'c net_state" where "s = SubnetS sa t" and "s' = SubnetS s'a t'" and "(sa, H\<not>K:arrive(m), s'a) \<in> S" and "(t, R:*cast(m), t') \<in> T" and "H \<subseteq> R" and "K \<inter> R = {}"
hyps_4: fixes k' :: "'a" assumes "k' \<in> fst ` set xs" shows "lt k'__ k'"
additive_def: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "Transformers.additive t \<equiv> \<forall>P Q. sound P \<and> sound Q \<longrightarrow> t (\<lambda>s. P s + Q s) = (\<lambda>s. t P s + t Q s)"
simps: fixes s1 :: "'e"   and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" shows "P16 (s1, s2) = (dstate s2 \<noteq> \<bottom> \<longrightarrow> initialized s2)"
full_embedding_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" shows "full_embedding_functor A B F \<equiv> embedding_functor A B F \<and> full_functor A B F"
unit_vecs_def: fixes n :: "nat" shows "unit_vecs n = map (unit_vec n) [0..<n]"
exists_base: fixes n :: "nat"   and E :: "'a set" assumes "dimension n K E" shows "\<exists>Vs. set Vs \<subseteq> carrier R \<and> local.independent K Vs \<and> length Vs = n \<and> Span K Vs = E"
prod_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and c :: "'b"   and g :: "'b"   and b :: "'b"   and d :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.in_hom C f a c"   and "partial_magma.in_hom C g b d" shows "partial_magma.in_hom C (elementary_category_with_binary_products.prod C pr0 pr1 f g) (elementary_category_with_binary_products.prod C pr0 pr1 a b) (elementary_category_with_binary_products.prod C pr0 pr1 c d)"
linear_hensel_binary_def: fixes p :: "int"   and n :: "nat"   and C :: "int poly"   and D :: "int poly"   and H :: "int poly" shows "linear_hensel_binary p n C D H = (let (S, T) = euclid_ext_poly_dynamic p D H in linear_hensel_main C p S T D H n)"
H_R1: fixes p :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "rel_aka.H p (rel_R p q) q"
more_arith_simps_11: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * (b * c)"
rbt_greater_prop: fixes k :: "'a"   and t :: "('a, 'd) rbt" shows "(k \<guillemotleft>| t) = (\<forall>x\<in>set (keys t). k < x)"
INT_iff: fixes b :: "'b"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "(b \<in> \<Inter> (B ` A)) = (\<forall>x\<in>A. b \<in> B x)"
alphaAbs_ex_distinct_qFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_ex_distinct_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
permutations_of_set_doubleton: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "permutations_of_set {x, y} = {[x, y], [y, x]}"
HDomain_Incl_Eats_I: fixes H :: "fm set"   and r :: "tm"   and d :: "tm"   and d' :: "tm" assumes "H \<turnstile> HDomain_Incl r d" shows "H \<turnstile> HDomain_Incl (Eats r (HPair d d')) (SUCC d)"
is_binqueue_append: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list"   and ys :: "('c, 'd) bintree option list" assumes "is_binqueue l xs"   and "is_binqueue (length xs + l) ys" shows "is_binqueue l (xs @ ys)"
OPT2_is_lb: fixes \<sigma> :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set \<sigma> \<subseteq> {x, y}"   and "x \<noteq> y" shows "T\<^sub>p [x, y] \<sigma> (OPT2 \<sigma> [x, y]) \<le> T\<^sub>p_opt [x, y] \<sigma>"
add2: fixes B :: "'a"   and k :: "'a"   and b :: "'a"   and a :: "'a" assumes "B \<equiv> k + b" shows "a + B \<equiv> k + (a + b)"
hypnat_le_add_diff_inverse: fixes n :: "nat star"   and m :: "nat star" assumes "n \<le> m" shows "n + (m - n) = m"
\<R>\<^sub>G_inv: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and \<beta> :: "'e \<Rightarrow> 'e"   and s' :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" assumes "bij \<alpha>"   and "bij \<beta>" shows "(\<R>\<^sub>G (inv \<alpha>) (inv \<beta>) s' = s) = (\<R>\<^sub>G \<alpha> \<beta> s = s')"
bourbaki_witt_fixpoint_axioms: shows "bourbaki_witt_fixpoint Inf leq link"
sp_fundef_same_arities: fixes F1 :: "'a \<Rightarrow> 'b fundef option"   and F2 :: "'a \<Rightarrow> 'c fundef option" assumes "all_same_arities F1 F2" shows "sp_fundef F1 = sp_fundef F2"
iT_Div_finite_iff: fixes k :: "nat"   and I :: "nat set" assumes "0 < k" shows "finite (I \<oslash> k) = finite I"
ps_frontier_in_imps_wl: fixes c :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t" assumes "inv_imps_work_sum c"   and "0 < zcount (zmset_of (mset_set (set_antichain (frontier (c_pts c loc))))) t" shows "0 < zcount (c_imp c loc + c_work c loc) t"
Union_rel_internal: fixes L :: "('d \<times> 'f set) set"   and S :: "('f \<times> 'e set) set" shows "Union_rel L S = L O \<langle>S\<rangle>set_rel O br \<Union> top"
valid_decompD_1: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "valid_decomp X ps"   and "(h, U) \<in> set ps" shows "h \<in> P[X]"