hcju/isabelleps huggingface dataset:
combine_valid_Nodes_aux: fixes a :: "'a tree"   and z :: "'a tree"   and va :: "'a"   and vz :: "'a" assumes "0 < l a"   and "a \<noteq> Leaf"   and "z \<noteq> Leaf"   and "l z = Suc (r a)"   and "well_shaped a"   and "well_shaped z"   and "local.well_valued0 a"   and "val a = Some va"   and "local.well_valued0 z"   and "val z = Some vz" shows "va + vz = fold (+) (rev (map ((!) as) [l a - Suc 0..<r z - Suc 0])) (as ! (r z - Suc 0))"
simps_4: shows "OUTfromCh data4 = {data2}"
assms: shows "eq_\<I>_gpv (BNF_Def.vimage2p f f' A) (map_\<I> g h \<I>) gpv1 gpv2"
splitFace_f12_oldF_neq: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF newVs"   and "minGraphProps g"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs" shows "oldF \<noteq> f12"
entails_ex: fixes P :: "'b \<Rightarrow> assn"   and Q :: "assn" shows "(\<not> (\<exists>\<^sub>Ax. P x \<Longrightarrow>\<^sub>A Q)) = (\<exists>x. \<not> (P x \<Longrightarrow>\<^sub>A Q))"
mapl_F_def: fixes l1 :: "'p \<Rightarrow> 'z"   and l2 :: "'q \<Rightarrow> 'aa"   and l3 :: "'r \<Rightarrow> 'ab" shows "mapl_F l1 l2 l3 = map_F l1 l2 l3 id id id id id id"
image_mset_is_empty_iff: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" shows "(f `# M = {#}) = (M = {#})"
rbt_lookup_rbt_insertw: fixes t :: "('a, 'c) rbt"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and v :: "'c" assumes "is_rbt t" shows "rbt_lookup (rbt_insert_with f k v t) = rbt_lookup t(k \<mapsto> if k \<in> dom (rbt_lookup t) then f (the (rbt_lookup t k)) v else v)"
bounded_incs_def: fixes cond :: "nat list \<Rightarrow> nat \<Rightarrow> bool"   and B :: "nat" shows "bounded_incs cond B \<equiv> \<forall>x xs s. B < x \<longrightarrow> cond (x # xs) s = False"
euclidean_all_zero_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and x :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(\<forall>u\<in>Basis. inner x u = 0) = (x = zero)"
card_of_card_order_on: fixes A :: "'a set" shows "card_order_on A |A|"
sup1_def: fixes a :: "'a"   and b :: "'a" shows "a \<squnion>1 b = ((a l\<rightarrow> b) r\<rightarrow> b) \<sqinter> ((b l\<rightarrow> a) r\<rightarrow> a)"
brc_initial_abs: fixes \<delta> :: "('a, 'b) ta_rule dlist" assumes "ls.invar \<delta>" shows "brc_\<alpha> (brc_initial \<delta>) \<in> br'_initial (ls.\<alpha> \<delta>)"
less_def: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
some_eq_ex: fixes P :: "'d \<Rightarrow> bool" shows "P (SOME x. P x) = (\<exists>x. P x)"
get_add_eq: fixes e :: "'env"   and x :: "'key"   and v :: "'val" shows "get (add e x v) x = Some v"
vector_space_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "vector_space scale \<equiv> ((\<forall>a x y. scale a (x + y) = scale a x + scale a y) \<and> (\<forall>a b x. scale (a + b) x = scale a x + scale b x)) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'a) x = x)"
Sample_L_i1_buffer: fixes req :: "nat \<Rightarrow> reqType list"   and a :: "nat \<Rightarrow> 'a list"   and stop :: "nat \<Rightarrow> stopType list"   and a1 :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and t :: "nat"   and d :: "nat"   and dt :: "nat \<Rightarrow> ECall_Info list"   and st :: "nat \<Rightarrow> GatewayStatus"   and buffer :: "nat \<Rightarrow> ECall_Info list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" assumes "msg (Suc 0) req"   and "msg (Suc 0) a"   and "msg (Suc 0) stop"   and "msg (Suc 0) a1"   and "ts lose"   and "ack t = [connection_ok]"   and "req (Suc t) = [send]"   and "\<forall>k\<le>Suc d. lose (t + k) = [False]"   and "Sample_L req dt a1 stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st buffer" shows "i1 (Suc t) = buffer t"
eigenvector_def: fixes A :: "'a mat"   and v :: "'a vec"   and k :: "'a" shows "eigenvector A v k = (v \<in> carrier_vec (dim_row A) \<and> v \<noteq> 0\<^sub>v (dim_row A) \<and> A *\<^sub>v v = k \<cdot>\<^sub>v v)"
r_inv_ex: fixes x :: "'c" assumes "x \<in> carrier M" shows "\<exists>y\<in>carrier M. x \<oplus>\<^bsub>M\<^esub> y = \<zero>\<^bsub>M\<^esub>"
INF_le_iff: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and x :: "'c" shows "(Inf (f ` A) \<le> x) = (\<forall>y>x. \<exists>i\<in>A. f i < y)"
less_imp_le: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.preorder less_eq less"   and "less x y" shows "less_eq x y"
While_1: fixes P :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P" shows "\<turnstile>\<^sub>p {wlp S P} S {P}"
AtSuc: fixes x :: "'a"   and n :: "nat" assumes "At x n" shows "\<exists>x'. At x' (Suc n) \<and> Follows x' x"
exp_m1_ge_quarter: shows "1 / 4 \<le> exp (- 1)"
sigma_finite_measure_axioms: shows "sigma_finite_measure (count_space ((SIGMA x:UNIV. set_pmf (J x))\<^sup>* `` {x}))"
sig_red_sing_reg_cases: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "sig_red (\<preceq>\<^sub>t) top_tail F p q = (sig_red (=) top_tail F p q \<or> sig_red (\<prec>\<^sub>t) top_tail F p q)"
simps: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and uu :: "Mode \<Rightarrow> Eg1.var set"   and uv :: "Eg1.var \<Rightarrow> nat"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and uw :: "Mode \<Rightarrow> var\<^sub>C set"   and ux :: "var\<^sub>C \<Rightarrow> nat" shows "abs_steps \<langle>c\<^sub>A, uu, uv\<rangle>\<^sub>A \<langle>c\<^sub>C, uw, ux\<rangle>\<^sub>C = abs_steps' c\<^sub>A c\<^sub>C"
tree_rank_estimate_lower: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.tree_invar (SkewBinomialHeapStruc.Node e a r ts)" shows "2 ^ r \<le> size (SkewBinomialHeapStruc.tree_to_multiset (SkewBinomialHeapStruc.Node e a r ts))"
eqvt_at_apply: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and p :: "perm" assumes "eqvt_at f x" shows "(p \<bullet> f) x = f x"
eval_wp_Seq: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" shows "wp (a ;; b) P = (wp a \<circ> wp b) P"
finite_row: fixes p :: "nat"   and k :: "nat" assumes "prime p" shows "finite {i. 1 < i \<and> real (p ^ (i + k)) \<le> x}"
exec_gpv_fixp_induct_strong: fixes P :: "(('a, 'b, 'c) gpv \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'd) spmf) \<Rightarrow> bool"   and callee :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf" assumes "spmf.admissible (\<lambda>f. P (\<lambda>c s. f (c, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>exec_gpv'. \<lbrakk>\<And>c s. ord_spmf (=) (exec_gpv' c s) (exec_gpv callee c s); P exec_gpv'\<rbrakk> \<Longrightarrow> P (\<lambda>c s. bind_spmf (the_gpv c) (case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. bind_spmf (callee s out) (\<lambda>(x, y). exec_gpv' (c x) y))))" shows "P (exec_gpv callee)"
inject_1: fixes x1 :: "('b, 'c, 'd, 'e) Tree set['b]"   and y1 :: "('b, 'c, 'd, 'e) Tree set['b]" shows "(tConj x1 = tConj y1) = (x1 = y1)"
otherwith_actionD: fixes Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'e set"   and P :: "('e \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd"   and a :: "'f" assumes "otherwith Q I P \<sigma> \<sigma>' a" shows "P \<sigma> a"
j: shows "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
lborel_eq_real: shows "lborel = interval_measure (\<lambda>x. x)"
monom_vars_list_def: shows "monom_vars_list \<equiv> map_fun Rep_monom id (map fst)"
degree_sum_pdevs_le: fixes f :: "'b \<Rightarrow> 'a pdevs"   and X :: "'b set" shows "degree (sum_pdevs f X) \<le> Max (degree ` f ` X)"
uniqListImpliesUniqPrefix: fixes p :: "'b list"   and l :: "'b list" assumes "isPrefix p l"   and "uniq l" shows "uniq p"
of_nat_less_imp_less: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)" shows "m < n"
ample_invisible: fixes q :: "'state"   and A :: "'action set" assumes "ample_set q A"   and "A \<subset> {a. en a q}" shows "A \<subseteq> invisible"
mul_closed_set_sub: fixes S :: "'a set" assumes "mul_closed_set R S" shows "S \<subseteq> carrier R"
closed_cball: fixes x :: "'a"   and e :: "real" shows "closed (cball x e)"
dg_Cod_app_in_Obj: fixes \<alpha> :: "V"   and \<CC> :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "\<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
strict_total_ordering_axioms: shows "strict_total_ordering UNIV (\<le>) (<)"
POR_strict_appI: fixes xs :: "Three \<Rightarrow> ValD"   and fs :: "Three \<Rightarrow> ValD \<rightarrow> ValD" assumes "xs \<in> unlr POR.delta"   and "\<And>xs. xs \<in> unlr POR.delta \<Longrightarrow> (\<lambda>j. fs j\<cdot>(xs j)) \<in> unlr POR.delta" shows "(\<lambda>j. strictify\<cdot>(fs j)\<cdot>(xs j)) \<in> unlr POR.delta"
44: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> - y) \<sqinter> y = x \<sqinter> y"
card_Int_Un_le_Sum_card_Int: fixes S :: "'a set set"   and A :: "'a set" assumes "finite S" shows "card (A \<inter> \<Union> S) \<le> (\<Sum>B\<in>S. card (A \<inter> B))"
measure_restrict_space: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "\<Omega> \<inter> space M \<in> sets M"   and "A \<subseteq> \<Omega>" shows "Sigma_Algebra.measure (restrict_space M \<Omega>) A = Sigma_Algebra.measure M A"
not_is_red_empty: fixes f :: "'a \<Rightarrow>\<^sub>0 'c" shows "\<not> punit.is_red {} f"
HD: fixes G :: "'a set"   and a :: "'a"   and b :: "'a" assumes "geodesic_segment_between G a b" shows "hausdorff_distance (f ` G) {f a--f b} \<le> K"
linear_spans_image: fixes f :: "'b \<Rightarrow> 'c"   and V :: "'b set"   and B :: "'b set" assumes "linear (*a) (*b) f"   and "V \<subseteq> vs1.span B" shows "f ` V \<subseteq> vs2.span (f ` B)"
permute_eqvt: fixes p :: "perm"   and q :: "perm"   and x :: "'e" shows "p \<bullet> q \<bullet> x = (p \<bullet> q) \<bullet> p \<bullet> x"
compact_closed_sums: fixes S :: "'a set"   and T :: "'a set" assumes "compact S"   and "closed T" shows "closed (\<Union>x\<in>S. \<Union>y\<in>T. {x + y})"
2: fixes g :: "'a \<Rightarrow> 'b" assumes "sup_continuous g" shows "sup_continuous (f g)"
1: shows "proj (resOf f) (I_intF f (map (I.int \<xi>) Tl)) = I_intF f (map (I.int \<xi>) Tl)"
B_Axiom: shows "universal (\<lambda>x y. x r y \<longrightarrow> y r x)"
l_set_disconnected_nodes_get_parent\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<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 :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_parent\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs get_parent get_parent_locs \<equiv> l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs \<and> l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs"
mem_imp_elem: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> carrier S"   and "x \<in> A" shows "x .\<in> A"
Carmichael_closed_formula: fixes n :: "nat" shows "Carmichael n = (LCM p\<in>prime_factors n. let k = multiplicity p n in if p = 2 \<and> 2 < k then 2 ^ (k - 2) else p ^ (k - 1) * (p - 1))"
y: shows "c\<^sup>2 + 3 * d\<^sup>2 = x * y"
less_eq_aexp_def: fixes e1 :: "'a aexp"   and e2 :: "'a aexp" shows "e1 \<le> e2 \<equiv> e1 < e2 \<or> e1 = e2"
l1_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "l1_step2 Rb A B gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> guessed_frame Rb xgnx = Some gnx \<and> guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny))) \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xgny, xgnx, xsk}), signalsInit := if can_signal s A B then addSignal (signalsInit s) (Running A B (Exp gnx (NonceF (Rb $ ny)))) else signalsInit s\<rparr>}"
terminal_char1: fixes t :: "'t" shows "S'.terminal t = (S'.ide t \<and> (\<exists>!x. x \<in> S'.set t))"
fbox_diff_inv: fixes I :: "'b \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real" shows "(I \<le> |x\<acute>=f & G on U S @ t\<^sub>0] I) = diff_invariant I f U S t\<^sub>0 G"
nth_map: fixes n :: "nat"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "n < \<^bold>|xs\<^bold>|" shows "map f xs ! n = f (xs ! n)"
normalize'_def: fixes p :: "((int \<times> int) set set, int) Combinators" shows "normalize' p = (removeAllDuplicates \<circ> insertDenies \<circ> separate \<circ> sort' (Nets_List p) \<circ> removeShadowRules2 \<circ> remdups \<circ> rm_MT_rules C \<circ> insertDeny \<circ> removeShadowRules1 \<circ> policy2list) p"
Bang: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "P \<parallel> !P \<longmapsto>\<alpha> \<prec> P'" shows "!P \<longmapsto>\<alpha> \<prec> P'"
bigstep_progress: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and p :: "nat"   and t :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> p \<Down> t" shows "0 < p"
class_of_def: fixes U :: "'a set"   and s :: "'b" shows "\<lfloor>U, s\<rfloor> \<equiv> Class (U, s)"
rightUniqueTrivialCartes: fixes Y :: "'c set"   and X :: "'d set" assumes "trivial Y" shows "runiq (X \<times> Y)"
Nil_is_append_conv: fixes xs :: "'g list"   and ys :: "'g list" shows "([] = xs @ ys) = (xs = [] \<and> ys = [])"
transitions_nil_eq: fixes root :: "(att \<times> char list, att, nat) env"   and root' :: "(att \<times> char list, att, nat) env" shows "root \<Midarrow>[]\<Rightarrow> root' = (root = root')"
select_convs_1: fixes carrier :: "'e set"   and more :: "'f" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
update_Gamma_nomatch: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and a :: "action"   and rs :: "'c rule list"   and rs' :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>(chain \<mapsto> Rule m a # rs),\<gamma>,p\<turnstile> \<langle>rs', s\<rangle> \<Rightarrow> t = \<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>rs', s\<rangle> \<Rightarrow> t"
af_G_respectfulness_2: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl"   and \<nu> :: "'b set" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "af_G_letter \<phi> \<nu> \<equiv>\<^sub>P af_G_letter \<psi> \<nu>"
cri_asym1: fixes x :: "'loc \<times> 't"   and y :: "'loc \<times> 't" shows "x<\<^sub>py \<longrightarrow> \<not> y<\<^sub>px"
simps_4: fixes C :: "char list"   and F :: "char list"   and D :: "char list" shows "is_relevant_class (Putstatic C F D) = (\<lambda>P C. True)"
UINF_mem_UNIV: fixes P :: "'d \<Rightarrow> ('b, 'c) uexpr" shows "(\<Sqinter> x \<in> UNIV \<bullet> P x) = (\<Sqinter> x \<bullet> P x)"
oreachable_local: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a \<times> 'b"   and a :: "'c"   and s' :: "'a \<times> 'b" assumes "oreachablep A S U s"   and "(s, a, s') \<in> automaton.trans A"   and "S (fst s) (fst s') a" shows "oreachablep A S U s'"
g_def: fixes v :: "('a, 'n) vec" shows "g v = f (THE i. X $ i = v)"
preserves_dom_def: fixes G :: "('a, 'b, 'd, 'e) functor" shows "preserves_dom G \<equiv> \<forall>f\<in>Ar. G \<^bsub>\<o>\<^esub> (Dom f) = Dom\<^bsub>BB\<^esub> G \<^bsub>\<a>\<^esub> f"
language_def: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" shows "language idx \<phi> = L idx {\<AA>. sat \<AA> \<phi> \<and> #\<^sub>V \<AA> = idx}"
equal_neg_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(a = uminus a) = (a = zero)"
st_proj1_continuous: shows "continuous_on (UNIV - {top_sphere}) st_proj1"
distinct_adj_Cons: fixes x :: "'b"   and xs :: "'b list" shows "distinct_adj (x # xs) = (xs = \<epsilon> \<or> x \<noteq> hd xs \<and> distinct_adj xs)"
simps_3: fixes ll :: "'b tree"   and a :: "'b"   and lr :: "'b tree"   and b :: "'b"   and r :: "'b tree" shows "T_dm \<langle>\<langle>ll, a, lr\<rangle>, b, r\<rangle> = (if ll = \<langle>\<rangle> then 1 else T_dm ll + 1)"
irreducible_altdef: fixes x :: "'c" shows "irreducible x = (x \<noteq> (0::'c) \<and> \<not> is_unit x \<and> (\<forall>b. b dvd x \<longrightarrow> x dvd b \<or> is_unit b))"
odd_pos: fixes n :: "nat" assumes "odd n" shows "0 < n"
irreflexive_axioms: shows "irreflexive A (\<lambda>x y. y \<sqsubset> x)"
Amicable_pair_sym: fixes m :: "nat"   and n :: "nat" assumes "m Amic n" shows "n Amic m"
PB1: shows "P1 * B1 \<in> carrier_mat n n"
d_OUT_split_Edge: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" shows "d_OUT (split f) (Edge x y) = max 0 (f (x, y))"
strand_fv_card_map_fun_eq: fixes S :: "('b, 'a) strand_step list"   and f :: "'b"   and X :: "('b, 'a) Term.term list"   and S' :: "('b, 'a) strand_step list" shows "card (fv\<^sub>s\<^sub>t (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S')) = card (fv\<^sub>s\<^sub>t (S @ map Send X @ S'))"
Plus_def: fixes A :: "'d set"   and B :: "'e set" shows "A <+> B = Inl ` A \<union> Inr ` B"
sel_1: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x4 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x5 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x6 :: "'b \<Rightarrow> 'b"   and x7 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x8 :: "'b \<Rightarrow> 'b"   and x9 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x10 :: "'b \<Rightarrow> 'b"   and x11 :: "'b \<Rightarrow> 'b"   and x12 :: "int \<Rightarrow> 'b"   and x13 :: "'b \<Rightarrow> int"   and x14 :: "'b \<Rightarrow> bool" shows "arith_ops_record.zero (Arith_Ops_Record x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) = x1"
hadamard_def: shows "hadamard = mat 2 2 (\<lambda>(i, j). if i = 0 \<or> j = 0 then 1 / csqrt 2 else complex_of_real (- 1 / sqrt 2))"
hf: fixes v :: "'a" shows "higher_differentiable_on S (\<lambda>x. frechet_derivative f (at x) v) n"
fp_cop_F_doctor_optimal_match: fixes ds :: "'b set" shows "doctor_optimal_match ds (cop ds)"
INTER_UNIV_conv_1: fixes B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "(UNIV = \<Inter> (B ` A)) = (\<forall>x\<in>A. B x = UNIV)"
diff_divide_distrib: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (minus a b) c = minus (divide a c) (divide b c)"
fUnionM_def: shows "fUnionM \<equiv> ffold funionM (return {||})"
join_bottom_2: fixes x :: "'a" shows "\<bottom> \<squnion> x = x"
conversep_conversep: fixes r :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<inverse>\<inverse> = r"
simps_2: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and ys :: "'g list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # ys) = (Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
range_Leaf_subset_sexp: shows "range Leaf \<subseteq> sexp"
\<phi>_zero_iff: fixes l :: "nat"   and i :: "int"   and x :: "real" shows "(\<phi> (l, i) x = 0) = (x \<notin> {real_of_int (i - 1) / 2 ^ (l + 1)<..<real_of_int (i + 1) / 2 ^ (l + 1)})"
Inter_insert: fixes a :: "'c set"   and B :: "'c set set" shows "\<Inter> (insert a B) = a \<inter> \<Inter> B"
LTL_suffix_G: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl"   and i :: "nat" assumes "w \<Turnstile> G \<phi>" shows "suffix i w \<Turnstile> G \<phi>"
predE: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "Measurable.pred M P" shows "{x \<in> space M. P x} \<in> sets M"
add_rat_roots: fixes r :: "rat"   and p :: "int poly"   and x :: "'a" shows "(ipoly (poly_add_rat r p) x = (0::'a)) = (ipoly p (x - of_rat r) = (0::'a))"
ymax_2: shows "\<forall>z\<in>{y. x \<noteq> y \<and> y pd\<rightarrow> x}. z = y \<or> z pd\<rightarrow> y"
invar'_def: fixes m :: "'e\<cdot>('f, 'g) writerT" shows "invar' m = (fmapWT\<cdot>ID\<cdot>m = m)"
find_max_def: fixes xs :: "nat list" shows "find_max xs \<equiv> fold max xs 1"
DIAndCut12Intro_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAndCut12Intro vid1 pid1 pid2 = (([[$\<alpha> vid1]]Pc pid2 \<rightarrow> (Pc pid1 && Pc pid2)) \<rightarrow> ([[$\<alpha> vid1]]Pc pid2) \<rightarrow> [[$\<alpha> vid1]]Pc pid1 && Pc pid2)"