hcju/isabelleps huggingface dataset:
interval_integral_Icc: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b" shows "interval_lebesgue_integral lborel (ereal a) (ereal b) f = set_lebesgue_integral lborel {a..b} f"
relImage_proj: fixes A :: "'d set"   and R :: "('d \<times> 'd) set" assumes "equiv A R" shows "BNF_Greatest_Fixpoint.relImage R (Equiv_Relations.proj R) \<subseteq> Id_on (A // R)"
assms_2: shows "a * \<xi>\<^sup>2 + b * \<xi> + c = 0"
shrK_keys_of_1: fixes A :: "agent"   and B :: "agent" shows "shrK A B \<in> keys_of A"
\<tau>rtrancl3p_trans: fixes s :: "'s1"   and tls :: "'tl1 list"   and s' :: "'s1"   and tls' :: "'tl1 list"   and s'' :: "'s1" assumes "s -\<tau>1-tls\<rightarrow>* s'"   and "s' -\<tau>1-tls'\<rightarrow>* s''" shows "s -\<tau>1-tls @ tls'\<rightarrow>* s''"
diff_right: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b"   and b' :: "'b" shows "a *\<^sub>V (b - b') = a *\<^sub>V b - a *\<^sub>V b'"
trg_cod: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.trg (Span.cod \<mu>) = Span.trg \<mu>"
htchLoc_def: fixes \<iota> :: "nat" shows "htchLoc \<iota> \<equiv> {(d, e). (d, e, \<iota>) \<in> lH}"
abelian_chain_group: fixes p :: "nat"   and X :: "'b topology" shows "comm_group (chain_group p X)"
preserves_reflects_arr: fixes f :: "'b" shows "C.arr (F.right_map f) = D.arr f"
subset_subseqs: fixes X :: "'b set"   and xs :: "'b list" assumes "X \<subseteq> set xs" shows "X \<in> set ` set (subseqs xs)"
rel_resource_conversep: fixes A :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_resource A\<inverse>\<inverse> B\<inverse>\<inverse> = (rel_resource A B)\<inverse>\<inverse>"
sorted_OAlist_tc_sorted_domain: fixes xs :: "('a, 'b) oalist_tc" shows "sorted_wrt (<) (OAlist_tc_sorted_domain xs)"
indRelSTEQ_impl_SRel_and_TRel_preserve_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelSTEQ SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
simplefact: fixes i :: "nat" assumes "i + 1 < length \<HH>" shows "simple_group (G\<lparr>carrier := H, carrier := \<HH> ! (i + 1)\<rparr> Mod \<HH> ! i)"
aform_ops_def: shows "aform_ops = \<lparr>appr_of_ivl = aforms_of_ivls, product_appr = product_aforms, msum_appr = msum_aforms', inf_of_appr = \<lambda>optns. inf_aforms (precision optns), sup_of_appr = \<lambda>optns. sup_aforms (precision optns), split_appr = split_aforms_largest_uncond_take, appr_inf_inner = \<lambda>optns. aform_inf_inner (precision optns), appr_sup_inner = \<lambda>optns. aform_sup_inner (precision optns), inter_appr_plane = \<lambda>optns xs. inter_aform_plane_lv (length xs) (precision optns) xs, reduce_appr = \<lambda>optns. reduce_aforms (precision optns), width_appr = \<lambda>optns. width_aforms (precision optns), approx_slp_dres = \<lambda>optns. aform_slp (precision optns), approx_euclarithform = \<lambda>optns. aform_form (precision optns), approx_isFDERIV = \<lambda>optns. aform_isFDERIV (precision optns)\<rparr>"
finite_in_span: fixes S :: "'c set"   and v :: "'c" assumes "finite S"   and "S \<subseteq> carrier V"   and "v \<in> V.module.span S" shows "\<exists>a. a \<in> S \<rightarrow> carrier K \<and> V.module.lincomb a S = v"
assms_2: fixes i :: "'a" assumes "i \<in> Basis" shows "(\<lambda>x. f x \<bullet> i) C1_differentiable_on s - t"
affine_hull_insert_span_gen: fixes a :: "'a"   and s :: "'a set" shows "affine hull insert a s = (+) a ` span ((+) (- a) ` s)"
min_height_size_log_if_incomplete: fixes kdt :: "'b kdt" assumes "\<not> KD_Tree.complete kdt" shows "real (min_height kdt) < log 2 (real (KD_Tree.size_kdt kdt))"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1"
Ch_singular': fixes h :: "'h"   and X :: "'x set" shows "inj_on Xd (Ch h X)"
network_with_ops_def: fixes history :: "nat \<Rightarrow> ('c \<times> 'd) event list" shows "network_with_ops history \<equiv> causal_network history fst"
univariate_mpoly_roots_finite: fixes p :: "'a mpoly"   and v :: "nat" assumes "vars p \<subseteq> {v}"   and "p \<noteq> 0" shows "finite {x. insertion (\<lambda>v. x) p = (0::'a)}"
scc_non_empty': fixes E :: "('b \<times> 'b) set"   and U :: "'b set" assumes "is_scc E U" shows "U \<noteq> {}"
not_in_net_ips_fst_init_missing: fixes i :: "nat"   and s :: "'s net_state" assumes "i \<notin> net_ips s" shows "fst (initmissing (netgmap sr s)) i = someinit i"
ptensor_mat_carrier: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "st3.ptensor_mat m1 m2 \<in> carrier_mat st3.d0 st3.d0"
entries_rbt_lookup: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t1 :: "('a, 'b) RBT_Impl.rbt"   and t2 :: "('a, 'b) RBT_Impl.rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t1"   and "ord.rbt_sorted less t2" shows "(RBT_Impl.entries t1 = RBT_Impl.entries t2) = (ord.rbt_lookup less t1 = ord.rbt_lookup less t2)"
norm_pre: shows "inner_prod ket_pre ket_pre = 1"
to_tree_order_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and child :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some parent"   and "parent \<in> set nodes" shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes"
intros_1: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'c"   and s :: "'d"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool" assumes "\<not> C r s" shows "(Some (r, s), Some (r, s)) \<in> whileLoop_results C B"
assms_4: shows "linear_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>1) (*s\<^sub>2) f"
no_change_if_ge_length_t: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and j :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "length t \<le> i"   and "i \<le> j" shows "distributed_system.s channel trans send recv init t i = distributed_system.s channel trans send recv init t j"
Joints_swap: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list"   and X :: "'b \<times> 'b pdevs"   and Y :: "'b \<times> 'b pdevs"   and XS :: "('b \<times> 'b pdevs) list" shows "(x # y # xs \<in> Joints (X # Y # XS)) = (y # x # xs \<in> Joints (Y # X # XS))"
fun_type_arg_cong': fixes f :: "'fun"   and g :: "'fun"   and X :: "('fun, 'var) Term.term list"   and x :: "('fun, 'var) Term.term"   and X' :: "('fun, 'var) Term.term list"   and Y :: "('fun, 'var) Term.term list"   and y :: "('fun, 'var) Term.term"   and Y' :: "('fun, 'var) Term.term list" assumes "f \<in> \<Union> (range \<Sigma>n) - \<C>"   and "g \<in> \<Union> (range \<Sigma>n) - \<C>"   and "\<Gamma> (Fun f (X @ x # X')) = \<Gamma> (Fun g (Y @ y # Y'))"   and "length X = length Y" shows "\<Gamma> x = \<Gamma> y"
filterlim_at_top_iff_smallomega: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" shows "filterlim f at_top F = (f \<in> \<omega>[F](\<lambda>_. 1) \<and> eventually_nonneg F f)"
one_closed: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.monoid G" shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
ev_nxt: fixes \<phi> :: "'b stream \<Rightarrow> bool" shows "ev \<phi> = (\<lambda>xs. \<phi> xs \<or> nxt (ev \<phi>) xs)"
distinct_entries: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "distinct (map fst (RBT_Impl.entries t))"
directly_attracted_def: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and S :: "'c set" assumes "ParityGame G" shows "ParityGame.directly_attracted G p S \<equiv> {v \<in> V\<^bsub>G\<^esub> - S. \<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w) \<and> (v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<longrightarrow> (\<exists>w. v \<rightarrow>\<^bsub>G\<^esub> w \<and> w \<in> S)) \<and> (v \<in> (if p** = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<longrightarrow> (\<forall>w. v \<rightarrow>\<^bsub>G\<^esub> w \<longrightarrow> w \<in> S))}"
wtC_list_glitOfC: fixes c :: "('fsym, 'psym) lit list" shows "GE.wtC (list (glitOfC c))"
skel_swap: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "good X" shows "skel (X #[x1 \<and> x2]_xs) = skel X"
good_ex: fixes v :: "'a" assumes "v \<in> S" shows "\<exists>\<sigma>. \<sigma> \<in> good v"
must_sync_def2: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong>2 = (\<exists>taa x' m' s. r2.r_syntax t x m taa x' m' \<and> r2.actions_ok s t taa)"
tdghm_dghm_comp_NTMap_vdomain: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<NN> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> ((\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
set_insert: fixes x :: "'e"   and A :: "'e set" assumes "x \<in> A" obtains B :: "'e set" where "A = insert x B" and "x \<notin> B"
\<theta>: shows "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>a\<^sup>P,F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists>y. \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& F \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,y\<^sup>P\<rparr>) \<^bold>& \<^bold>\<not>\<lbrace>y\<^sup>P,F\<rbrace>)) in v]"
c4: fixes i :: "nat"   and x :: "real" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "c2 / min 1 ((bs ! i + (hs ! i) x / x) powr (p + 1)) \<le> c4"
self_in_upset: fixes x :: "'a" shows "x \<in> \<up>x"
no_start_end_points_iff: fixes x :: "'a" shows "no_start_end_points x = (no_start_points x \<and> no_end_points x)"
fps_compose_assoc: fixes c :: "'a fps"   and b :: "'a fps"   and a :: "'a fps" assumes "c $ 0 = (0::'a)"   and "b $ 0 = (0::'a)" shows "a oo (b oo c) = a oo b oo c"
even_flip_bit_iff: fixes m :: "nat"   and a :: "'a" shows "even (flip_bit m a) = (even a \<noteq> (m = 0))"
nFAND_Cons: fixes x :: "('a, 'b) aformula"   and xs :: "('a, 'b) aformula list" shows "nFAND (x # xs) = (if xs = [] then x else FAnd x (nFAND xs))"
inject_4: fixes x41 :: "sterm"   and x42 :: "Label"   and x43 :: "sterm"   and y41 :: "sterm"   and y42 :: "Label"   and y43 :: "sterm" shows "(Call x41 x42 x43 = Call y41 y42 y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
320: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - - (- x \<sqinter> y) = - x"
trg_vcomp: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.seq \<nu> \<mu>" shows "S.trg (\<nu> \<cdot>\<^sub>S \<mu>) = S.trg \<nu>"
l: shows "2 ^ 7 * 191 * 383 = 9363584"
STAR_sin_Infinitesimal: fixes x :: "'a star" assumes "x \<in> Infinitesimal" shows "(*f* sin) x \<approx> x"
inject_2: fixes x31 :: "('a, 'b) fm"   and x32 :: "('a, 'b) fm"   and y31 :: "('a, 'b) fm"   and y32 :: "('a, 'b) fm" shows "((x31 \<^bold>\<longrightarrow> x32) = (y31 \<^bold>\<longrightarrow> y32)) = (x31 = y31 \<and> x32 = y32)"
convergent_powser'_def: fixes cs :: "real msllist"   and f :: "real \<Rightarrow> real" shows "convergent_powser' cs f = (\<exists>R>0. \<forall>x. \<bar>x\<bar> < R \<longrightarrow> (\<lambda>n. lcoeff cs n * x ^ n) sums f x)"
busy_squareI: fixes A :: "state \<times> state \<Rightarrow> bool"   and s :: "state"   and t :: "state"   and v :: "state \<Rightarrow> 'b" assumes "A (s, t)" shows "(s, t) \<Turnstile> [A]_v"
graph_pred_it_def: fixes \<alpha> :: "'d \<Rightarrow> ('e, 'f) graph"   and invar :: "'d \<Rightarrow> bool"   and pred_list_it :: "'d \<Rightarrow> 'e \<Rightarrow> (('e \<times> 'f) list \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> ('e \<times> 'f) list \<Rightarrow> ('e \<times> 'f) list) \<Rightarrow> ('e \<times> 'f) list \<Rightarrow> ('e \<times> 'f) list" shows "graph_pred_it \<alpha> invar pred_list_it \<equiv> graph \<alpha> invar \<and> graph_pred_it_axioms \<alpha> invar pred_list_it"
infsetsum_mono_neutral_left: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set"   and g :: "'c \<Rightarrow> real"   and B :: "'c set" assumes "f abs_summable_on A"   and "g abs_summable_on B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"   and "A \<subseteq> B"   and "\<And>x. x \<in> B - A \<Longrightarrow> 0 \<le> g x" shows "infsetsum f A \<le> infsetsum g B"
g_orthogonal_to_i: shows "\<forall>x. g x \<bullet> i = 0"
forall_3: fixes P :: "3 \<Rightarrow> bool" shows "(\<forall>i. P i) = (P 1 \<and> P 2 \<and> P 3)"
SD_inefficientE: fixes p :: "'alt pmf" assumes "\<not> SD_efficient R p" obtains q :: "'alt pmf" and i :: "'agent" where "q \<in> lotteries" and "\<And>i. i \<in> agents \<Longrightarrow> SD (R i) p q" and "i \<in> agents" and "p \<prec>[SD (R i)] q"
interval_integral_eq_integral: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b" assumes "a \<le> b"   and "set_integrable lborel {a..b} f" shows "interval_lebesgue_integral lborel (ereal a) (ereal b) f = integral {a..b} f"
binop: shows "binop \<real>\<^sub>\<circ> vreal_mult"
fps_exp_nth: fixes a :: "'a"   and n :: "nat" shows "fps_nth (fps_exp a) n = a ^ n / of_nat (fact n)"
tauActionUnfold: fixes P :: "('a, 'b, 'c) psi"   and C :: "'d" obtains x :: "name" where "x \<sharp> P" and "x \<sharp> C" and "\<tau>.P = \<lparr>\<nu>x\<rparr>((nameTerm x\<lparr>\<lambda>*[] nameTerm x\<rparr>.\<zero>) \<parallel> nameTerm x\<langle>nameTerm x\<rangle>.P)"
freeliftid_Abs_freeword_conv_sum_list: fixes xs :: "('w \<times> bool) list" assumes "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "freeword_funlift id (Abs_freeword xs) = sum_list (map fst xs)"
simps_6: fixes fv :: "'a \<Rightarrow> 'b set"   and safe :: "safety \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "modality"   and g :: "safety"   and r :: "'a Regex.regex" shows "Regex.safe_regex fv safe m g (regex.Star r) = ((g = Lax \<or> Regex.fv_regex fv r = {}) \<and> Regex.safe_regex fv safe m g r)"
simps_9: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Dom \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Dom t \<^bold>\<otimes> Dom u) \<^bold>\<otimes> Dom v"
orthogonal_def: fixes x :: "'a"   and y :: "'a" shows "real_inner_class.orthogonal x y = (x \<bullet> y = 0)"
f_non_negative: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and e :: "nat \<times> nat" assumes "Preflow c s t f" shows "(0::'a) \<le> f e"
hom1_1: shows "homotopic_with (\<lambda>x. True) X X (g1 \<circ> f1) id"
Afix_lookup_fresh: fixes v :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "atom v \<sharp> \<Gamma>" shows "(Afix \<Gamma>\<cdot>ae) v = ae v"
mcont_outputs: shows "mcont resumption_lub resumption_ord \<Union> (\<subseteq>) outputs"
convex_hull_subset: fixes s :: "'b set"   and t :: "'b set" assumes "s \<subseteq> convex hull t" shows "convex hull s \<subseteq> convex hull t"
rep_eq: fixes x :: "'a dense_filter_type"   and xa :: "'a dense_filter_type" shows "(x < xa) = (Rep_dense_filter_type x < Rep_dense_filter_type xa)"
C_eq_rd: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "p \<noteq> []" shows "C (list2FWpolicy (remdups p)) = C (list2FWpolicy p)"
Nil_ap: fixes xs :: "'d list" shows "[] \<diamondop> xs = []"
axioms_1: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" assumes "effective_redundancy_criterion \<Gamma> Rf Ri" shows "redundancy_criterion \<Gamma> Rf Ri"
not_in_Gromov_boundary: fixes x :: "'a Gromov_completion" assumes "x \<notin> Gromov_boundary" shows "\<exists>a. x = to_Gromov_completion a"
CAS_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and D :: "String.literal"   and F :: "String.literal"   and e' :: "'addr expr1"   and e'' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e\<bullet>compareAndSwap(D\<bullet>F, e', e'')) h s s'"
USUP_image_eq: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Squnion> i \<in> f ` A \<bullet> g i) = (\<Squnion> i \<in> A \<bullet> g (f i))"
seqllD: fixes i :: "'a"   and P :: "('b \<times> 'c) \<times> 'd \<times> 'b \<times> 'c \<Rightarrow> bool"   and \<sigma> :: "'a \<Rightarrow> 'b"   and p :: "'c"   and a :: "'d"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and p' :: "'c" assumes "seqll i P ((\<sigma>, p), a, \<sigma>', p')" shows "P ((\<sigma> i, p), a, \<sigma>' i, p')"
select_convs_3: fixes V :: "'a set"   and N :: "'a set"   and C :: "'a set"   and more :: "'b" shows "C \<lparr>V = V, N = N, C = C, \<dots> = more\<rparr> = C"
reachable_state_induct: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and sh :: "('g, 'h, 'i, 'j, 'k) system_state"   and P :: "('g, 'h, 'i, 'j, 'k) system_state \<Rightarrow> bool" assumes "reachable_state sys sh"   and "\<And>s. initial_state sys s \<Longrightarrow> P \<lparr>GST = s, HST = []\<rparr>"   and "\<And>sh ls' p. \<lbrakk>reachable_state sys sh; P sh; GST sh p \<rightarrow>\<^bsub>\<tau>\<^esub> ls'\<rbrakk> \<Longrightarrow> P \<lparr>GST = (GST sh)(p := ls'), HST = HST sh\<rparr>"   and "\<And>sh ls1' ls2' p1 p2 \<alpha> \<beta>. \<lbrakk>reachable_state sys sh; P sh; GST sh p1 \<rightarrow>\<^bsub>\<guillemotleft>\<alpha>, \<beta>\<guillemotright>\<^esub> ls1'; GST sh p2 \<rightarrow>\<^bsub>\<guillemotright>\<alpha>, \<beta>\<guillemotleft>\<^esub> ls2'; p1 \<noteq> p2\<rbrakk> \<Longrightarrow> P \<lparr>GST = (GST sh)(p1 := ls1', p2 := ls2'), HST = HST sh @ [(\<alpha>, \<beta>)]\<rparr>" shows "P sh"
simps_3: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<phi> :: "fml" shows "usubstappf \<sigma> U (! \<phi>) = Noto (usubstappf \<sigma> U \<phi>)"
cong_ex_bigtheta: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f2 :: "'a \<Rightarrow> 'b"   and g1 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> \<Theta>[F](g1)) = (f2 \<in> \<Theta>[F](g2))"
prime_elem_multiplicity_eq_zero_iff: fixes p :: "'a"   and x :: "'a" assumes "prime_elem p"   and "x \<noteq> (0::'a)" shows "(multiplicity p x = 0) = (\<not> p dvd x)"
bv_extend_norm_unsigned: fixes w :: "bit list" shows "bv_extend (length w) \<zero> (norm_unsigned w) = w"
carrierE: fixes x :: "'b" assumes "x \<in> a.dest.carrier" obtains ca :: "('b, 'b) chart" where "ca \<in> manifold_eucl.dest.charts_submanifold (codomain c)" and "x \<in> domain ca"
cases: assumes "wf_residual_network" obtains "\<And>x y. (x, y) \<in> \<^bold>E \<Longrightarrow> (y, x) \<notin> \<^bold>E" and "(source \<Delta>, sink \<Delta>) \<notin> \<^bold>E"
A: fixes i :: "'i" assumes "i \<in> I" shows "countable (A i)"
mask_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "mask 0 = zero"
qFreshAbs_imp_ex_qAFreshAbs1: fixes V :: "'d set"   and A :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and zs :: "'varSort"   and z :: "'var" assumes "finite V"   and "qGoodAbs A"   and "qFreshAbs zs z A" shows "\<exists>A'. qGoodAbs A' \<and> A $= A' \<and> qAFreshAbs zs z A'"
set_bset: fixes x :: "'a set['b]" shows "set_bset x \<in> {A. |A| <o natLeq +c |UNIV|}"
assms_1: fixes n :: "nat" assumes "N \<le> n" shows "X n \<le> Y n"
boundedI: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "c \<le> a" shows "max b c \<le> a"
same_prefix_prefix: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"
MOST_coeff_eq_0: fixes p :: "'a poly" shows "\<forall>\<^sub>\<infinity>n. coeff p n = (0::'a)"
InstanceOf1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and T :: "ty" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>e instanceof T,s\<rangle> -ta\<rightarrow> \<langle>e' instanceof T,s'\<rangle>"
normal_exponent_bounds_int_1: fixes x :: "('a, 'b) IEEE.float" assumes "is_normal x" shows "2 - 2 ^ (LENGTH('a) - 1) - int LENGTH('b) \<le> normal_exponent x"
foundation9': fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> not x" shows "\<tau> |\<noteq> x"
finpref_fin: fixes x :: "'b llist"   and A :: "'b set"   and s :: "'b llist" assumes "x \<in> finpref A s" shows "x \<in> A\<^sup>\<star>"
add_order_preserving_right: fixes a :: "'a"   and b :: "'a"   and v :: "'a" assumes "a \<le> b" shows "a + v \<le> b + v"
subset1: shows "(\<lambda>(y, d). Fun.swap x y d) ` {(y, d). y \<in> S \<and> y \<noteq> x \<and> d \<in> derangements (S - {x, y})} \<subseteq> derangements S"
ord0_order_less_le: fixes x :: "ord0"   and y :: "ord0" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
assms_1: fixes j :: "nat" assumes "j < n" shows "d j < base"
equiv_liftn: fixes s :: "dB"   and t :: "dB"   and n :: "nat"   and k :: "nat" assumes "s \<leftrightarrow> t" shows "liftn n s k \<leftrightarrow> liftn n t k"
ZObisT_matchC_ZOC: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01T t" shows "matchC_ZOC ZObisT s t"
uminus_add_conv_diff_mset: fixes a :: "'b zmultiset"   and b :: "'b zmultiset" shows "- a + b = b - a"