hcju/isabelleps huggingface dataset:
1: shows "a \<in> allAllocations ({seller} \<union> N) (set \<Omega>)"
ex: fixes c :: "real"   and d :: "real" assumes "a \<le> c"   and "c \<le> b"   and "g2 c \<le> d"   and "d \<le> g1 c" shows "\<exists>z\<in>unit_cube. (d, c) = (case z of (y, x) \<Rightarrow> (g2 (a - x * a + x * b) - y * g2 (a - x * a + x * b) + y * g1 (a - x * a + x * b), a - x * a + x * b))"
lifting_wf_correct_state_d: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "lifting_wf JVM_final (mexecd P) (\<lambda>t (xcp, frs) h. \<Phi> \<turnstile> t:(xcp, h, frs) \<surd>)"
nc_vec_par_is_meet: fixes x :: "'a"   and y :: "'a" assumes "\<And>x y z. (d x \<sqinter> d y) \<cdot> z = d x \<cdot> z \<sqinter> d y \<cdot> z" shows "(x \<sqinter> nc) \<cdot> nc \<parallel> ((y \<sqinter> nc) \<cdot> nc) = (x \<sqinter> nc) \<cdot> nc \<sqinter> (y \<sqinter> nc) \<cdot> nc"
assms: shows "\<forall>x\<in>X. f x = g x"
subformulas\<^sub>\<mu>\<^sub>\<nu>_subfrmlsn: fixes \<phi> :: "'a ltln" shows "subformulas\<^sub>\<mu> \<phi> \<union> subformulas\<^sub>\<nu> \<phi> \<subseteq> subfrmlsn \<phi>"
class_leq_trans: fixes x :: "String.literal"   and y :: "String.literal"   and z :: "String.literal" assumes "class_leq cs x y"   and "class_leq cs y z" shows "class_leq cs x z"
card_QuadRes_set_prime: fixes p :: "int" assumes "prime p"   and "odd p" shows "card {x. QuadRes p x \<and> x \<in> {0..<p}} = nat (p + 1) div 2"
fps_X_mult_nth: fixes f :: "'a fps"   and n :: "nat" shows "(fps_X * f) $ n = (if n = 0 then 0::'a else f $ (n - 1))"
Lindelof_space_def: fixes X :: "'b topology" shows "Lindelof_space X \<equiv> \<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> \<Union> \<U> = topspace X \<longrightarrow> (\<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \<U> \<and> \<Union> \<V> = topspace X)"
assms: shows "{x \<in> space M. f x \<in> A x} \<in> sets M"
lmeasurable_box: fixes a :: "'b"   and b :: "'b" shows "box a b \<in> lmeasurable"
paths_not_empty: fixes t :: "'b ttree" shows "(paths t = {}) = False"
uminus_floatarith_def: fixes f :: "floatarith" shows "- f = floatarith.Minus f"
filter_congs_shorten1: fixes f :: "face"   and v :: "nat"   and a :: "nat"   and vs :: "nat list" assumes "distinct (verticesFrom f v)"   and "filter (\<lambda>v. v = a \<or> v \<in> set vs) (verticesFrom f v) = a # vs" shows "filter (\<lambda>v. v \<in> set vs) (verticesFrom f v) = vs"
eq: shows "q.eq_m f (Polynomial.smult (lead_coeff f) (prod_list gs))"
roifex_Trueif_unique: fixes b :: "'b ifex" assumes "ro_ifex b"   and "\<forall>ass. val_ifex b ass" shows "b = Trueif"
local_decreasing_extended_imp_decreasing: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and r :: "('b \<times> 'b) set"   and s :: "'a"   and \<beta>_step :: "'b \<times> 'a"   and \<alpha>_step :: "'b \<times> 'a"   and \<gamma>1 :: "'a \<times> (bool \<times> 'b \<times> 'a) list"   and \<gamma>2 :: "'a \<times> ('b \<times> 'a) list"   and \<gamma>3 :: "'a \<times> (bool \<times> 'b \<times> 'a) list"   and \<upsilon>_step :: "('b \<times> 'a) list"   and \<rho>_step :: "('b \<times> 'a) list" assumes "LT1 ars r (s, [\<beta>_step]) (s, [\<alpha>_step]) \<gamma>1 \<gamma>2 \<gamma>3"   and "trans r"   and "irrefl r"   and "\<And>y. \<lbrakk>(y, (s, [\<beta>_step] @ \<upsilon>_step), s, [\<alpha>_step] @ \<rho>_step) \<in> pex r; peak ars y\<rbrakk> \<Longrightarrow> \<exists>\<sigma>' \<tau>'. DD ars r (fst y, snd y, \<sigma>', \<tau>')" shows "\<exists>\<sigma>1 \<sigma>2 \<sigma>3' \<gamma>1'''. ({\<sigma>1, \<sigma>2, \<sigma>3', \<gamma>1'''} \<subseteq> seq ars \<and> set (labels \<sigma>1) \<subseteq> r \<down>s {fst \<beta>_step} \<and> length (labels \<sigma>2) \<le> 1 \<and> set (labels \<sigma>2) \<subseteq> {fst \<alpha>_step} \<and> set (labels \<sigma>3') \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step}) \<and> set (labels \<gamma>1''') \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step} \<and> snd \<beta>_step = fst \<sigma>1 \<and> lst \<sigma>1 = fst \<sigma>2 \<and> lst \<sigma>2 = fst \<sigma>3' \<and> lst \<sigma>3' = lst \<gamma>1''' \<and> fst \<gamma>1''' = fst \<gamma>3"
sorted_list_of_multiset_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b multiset" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_multiset less_eq M = fold_mset (linorder.insort_key less_eq (\<lambda>x. x)) [] M"
compact_diff: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "open T" shows "topological_space.compact open (S - T)"
closed: shows "(\<lambda>i. \<ominus> p i) \<in> carrier P"
fmeasurable_cbox: fixes a :: "'a"   and b :: "'a" shows "cbox a b \<in> fmeasurable lborel"
ensures_def: fixes A :: "'d set"   and B :: "'d set" shows "A ensures B \<equiv> (A - B co A \<union> B) \<inter> transient (A - B)"
simps_4: fixes a :: "'a"   and b :: "'a" shows "(dRETURN a \<le> dRETURN b) = (a = b)"
empty_rep: fixes q :: "('c, 'd) BinomialHeap" shows "(q = local.empty) = (Rep_BinomialHeap q = [])"
dg_Set_Dom_app: fixes \<alpha> :: "V"   and x :: "V" assumes "arr_Set \<alpha> x" shows "dg_Set \<alpha>\<lparr>Dom\<rparr>\<lparr>x\<rparr> = x\<lparr>ArrDom\<rparr>"
iso: shows "\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y"
strict_implies_not_eq: fixes b :: "'a"   and a :: "'a" assumes "b < a" shows "a \<noteq> b"
Diff_Compl: fixes A :: "'b set"   and B :: "'b set" shows "A - - B = A \<inter> B"
counit_simps_3: shows "cod r.\<epsilon> = trg (tab\<^sub>0 r)"
yun_relD_4: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly" assumes "yun_rel F c f" shows "F \<noteq> 0"
congruent_triangle_sss_14: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "angle a1 b1 c1 = angle a2 b2 c2"
Cons_2: shows "sub2_wit us (w # ws) = (wits, vs)"
float_add_r0: fixes x :: "real"   and e :: "int" shows "x + float (0, e) = x"
minus_0_l: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> x = \<ominus>\<^bsub>R\<^esub> x"
opnet_maintains_dom: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and ns :: "'e net_state"   and a :: "'f node_action"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and ns' :: "'e net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p :: "net_tree" assumes "((\<sigma>, ns), a, \<sigma>', ns') \<in> automaton.trans (opnet np p)" shows "net_ips ns = net_ips ns'"
equivalence_map_def: fixes f :: "('a S.E.term, 'a) S.arr" shows "S.equivalence_map f \<equiv> \<exists>g \<eta> \<epsilon>. S.internal_equivalence f g \<eta> \<epsilon>"
ta_remap_def: fixes f :: "'c \<Rightarrow> 'a"   and TA :: "('c, 'b, 'd) tree_automaton_rec_scheme" shows "ta_remap f TA \<equiv> \<lparr>ta_initial = f ` ta_initial TA, ta_rules = remap_rule f ` ta_rules TA\<rparr>"
adjoint_transpose_left_4: fixes u :: "'a"   and v :: "'a"   and \<nu> :: "'a" assumes "ide u"   and "ide v"   and "src f = trg v"   and "src g = trg u"   and "\<guillemotleft>\<nu> : v \<Rightarrow> g \<star> u\<guillemotright>" shows "trnl\<^sub>\<eta> v (trnl\<^sub>\<epsilon> u \<nu>) = \<nu>"
unit_closed: fixes G :: "('c, 'd) Group_scheme" assumes "Group G" shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
run_state_set: fixes s' :: "'c"   and s :: "'c" shows "run_state (local.set s') s = ((), s')"
that_1: fixes k :: "real" shows "piecewise_continuous_on 0 k D f"
card_UNIV_list_def: shows "card_UNIV_class.card_UNIV = Phantom('a list) 0"
nummul: fixes bs :: "real list"   and t :: "MIR.num"   and i :: "int" shows "Inum bs (nummul t i) = Inum bs (Mul i t)"
intercept_upf_def: fixes k :: "'prf_key"   and b :: "bool" shows "intercept_upf k b = plus_intercept (intercept_upf_enc k b) intercept_upf_dec"
AR_closed_Un: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T"   and "AR S"   and "AR T"   and "AR (S \<inter> T)" shows "AR (S \<union> T)"
simps_2: fixes uu :: "'a" shows "(extT.Top \<le> extT.Val uu) = False"
ordIso_natLeq_on_imp_finite: fixes A :: "'a set"   and n :: "nat" assumes "|A| =o natLeq_on n" shows "finite A"
is_red_uminus: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_red F (- p) = is_red F p"
ldistinct_ltake: fixes xs :: "'c llist"   and n :: "enat" assumes "ldistinct xs" shows "ldistinct (ltake n xs)"
pdevs_val_domain_cong: fixes b :: "'b pdevs"   and d :: "'b pdevs"   and a :: "nat \<Rightarrow> real"   and c :: "nat \<Rightarrow> real" assumes "b = d"   and "\<And>i. i \<in> pdevs_domain b \<Longrightarrow> a i = c i" shows "pdevs_val a b = pdevs_val c d"
arg_div_real_positive: fixes k :: "real"   and z :: "complex" assumes "0 < k" shows "arg (z / cor k) = arg z"
intro: fixes remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "remove_shadow_root = l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root get_child_nodes get_shadow_root set_shadow_root"   and "remove_shadow_root_locs = l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root_locs set_shadow_root_locs" shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_shadow_root set_shadow_root set_shadow_root_locs remove_shadow_root remove_shadow_root_locs"
outside: fixes e :: "'v \<times> 'v" assumes "e \<notin> \<^bold>E" shows "f e = 0"
stable_on_allocation: fixes d :: "'b"   and ds :: "'b set"   and X :: "'a set" assumes "Singleton_for_d.stable_on d ds X" shows "inj_on Xd X"
vector_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.vector f = (\<forall>i j k. f (i, j) = f (i, k))"
cancels_to_append2: fixes a :: "(bool \<times> 'c) list"   and a' :: "(bool \<times> 'c) list"   and b :: "(bool \<times> 'c) list"   and b' :: "(bool \<times> 'c) list" assumes "cancels_to a a'"   and "cancels_to b b'" shows "cancels_to (a @ b) (a' @ b')"
lenext_irrefl: fixes gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "\<not> gts xs xs" shows "\<not> lenext gts xs xs"
square_free_factorization_prod_list: fixes p :: "'c poly"   and c :: "'c"   and bs :: "('c poly \<times> nat) list" assumes "square_free_factorization p (c, bs)" shows "p = smult c (\<Prod>(a, i)\<leftarrow>bs. a ^ Suc i)"
get_tag_name_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "DocumentClass.type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_tag_name element_ptr"
abs_supp_3: fixes a :: "tyvrs"   and x :: "'a" shows "supp ([a].x) = supp x - {a}"
convex_on_domain: fixes f :: "'b \<Rightarrow> ereal" shows "Lower_Semicontinuous.convex_on (domain f) f = Lower_Semicontinuous.convex_on UNIV f"
finsum_scalar_prod_sum: fixes f :: "'b \<Rightarrow> 'a vec"   and U :: "'b set"   and w :: "'a vec" assumes "f \<in> U \<rightarrow> carrier_vec n"   and "w \<in> carrier_vec n" shows "finsum V f U \<bullet> w = (\<Sum>u\<in>U. f u \<bullet> w)"
get_cont_def: fixes get :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> ('d, 'e) contT" shows "get_cont get f = ContT (\<lambda>k. get (\<lambda>s. run_cont (f s) k))"
SPEC_rule: fixes \<Phi> :: "'a \<Rightarrow> bool"   and \<Phi>' :: "'a \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Phi>' x" shows "SPEC \<Phi> \<le> SPEC \<Phi>'"
card_un_add: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<longrightarrow> |i \<squnion> j| = |i| + |j|"
simps_3: fixes a :: "'b"   and b :: "'b" shows "(Num a \<le> Num b) = (a \<le> b)"
leq_lit_def: fixes L :: "('c, 'd) Term.term literal"   and K :: "('c, 'd) Term.term literal" shows "leq_lit L K = (case (K, L) of (Pos x, Pos a) \<Rightarrow> leq_head (atm_of L) (atm_of K) | (Pos x, Neg xa) \<Rightarrow> False | (Neg x, Pos xa) \<Rightarrow> True | (Neg x, Neg a) \<Rightarrow> leq_head (atm_of L) (atm_of K))"
infsuff_finpref: fixes t :: "'b llist"   and A :: "'b set"   and r :: "'b llist" assumes "t \<in> infsuff A r"   and "r \<in> A\<^sup>\<star>" shows "r \<in> finpref A t"
configuration_step_non_empty: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q' :: "'d"   and n :: "nat"   and q :: "'d" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.configuration \<delta> q\<^sub>0 w q' n \<noteq> {}"   and "q = \<delta> q' (w n)" shows "semi_mojmir_def.configuration \<delta> q\<^sub>0 w q (Suc n) \<noteq> {}"
hyps: shows "support (preList upds C1) \<subseteq> lesvars upds"
BinOpSetGroup_axioms: shows "BinOpSetGroup G binop e"
ih: fixes m :: "nat"   and as :: "nat list"   and bs :: "nat list" assumes "xs \<in> set ((enumAppend nmax ^^ m) (enumBase nmax))"   and "as @ bs = xs" shows "\<forall>x\<in>set as. \<forall>xa\<in>set bs. x \<le> xa"
vimage_fst_transfer_h: fixes A :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'e \<times> 'f" shows "pred_prod (Domainp A) (Domainp B) x = (x \<in> Collect (Domainp A) \<times> Collect (Domainp B))"
dom_char: fixes f :: "('obj, 'obj) concrete_category.arr" shows "local.dom f = (if arr f then f else null)"
the_ext_prop_2: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and x :: "'d"   and y :: "'c"   and e :: "'d"   and y' :: "'c"   and g :: "'d" assumes "initial_arrow_to_functor C D G x y e"   and "arrow_to_functor C D G x y' g" shows "g = D (G (initial_arrow_to_functor.the_ext C D G y e y' g)) e"
Ds_Vals: fixes vs :: "val list"   and A :: "'a set option" shows "\<D>s (map Val vs) A"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "\<And>x y zs. \<lbrakk>x = y \<Longrightarrow> P (y # zs); x \<noteq> y \<Longrightarrow> P (y # zs)\<rbrakk> \<Longrightarrow> P (x # y # zs)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
strict_mono_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x < f y) = less x y"
SUP_apply: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'c" shows "Sup (f ` A) x = (SUP y\<in>A. f y x)"
WT_bodiesD: fixes pn :: "pname"   and b :: "com" assumes "WT_bodies"   and "body pn = Some b" shows "WT b"
fun_def: fixes tm :: "('c, 'd) tm" shows "fun tm = (case tm of Hd \<zeta> \<Rightarrow> Hd \<zeta> | App x21 x22 \<Rightarrow> x21)"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> parts H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> parts H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> parts H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P Y"   and "\<And>K X. \<lbrakk>Crypt K X \<in> parts H; P (Crypt K X)\<rbrakk> \<Longrightarrow> P X" shows "P x"
Plus_def: fixes A :: "'c set"   and B :: "'d set" shows "A <+> B = Inl ` A \<union> Inr ` B"
subst_in_cfls: fixes G :: "('a, 'b) cfg"   and P :: "('a + 'b) list fset" shows "subst G P = to_language {w. in_cfls G P w}"
P: shows "(\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> P"
lspace_bounded_measurable: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set"   and g :: "'a \<Rightarrow> real"   and p :: "real" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "g \<in> lspace (lebesgue_on S) (ennreal p)"   and "0 < p"   and "AE x in lebesgue_on S. norm (\<bar>f x\<bar> powr p) \<le> norm (\<bar>g x\<bar> powr p)" shows "f \<in> lspace (lebesgue_on S) (ennreal p)"
valid_SDG_node_cases: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n" shows "n = CFG_node (parent_node n) \<or> SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses (CFG_node (parent_node n)) None None n"
valid_program_history: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "causal_program_history is sb"
Q_def: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2" shows "subresultant_prs_locale.Q F \<delta> f i \<equiv> Polynomial.smult (subresultant_prs_locale.\<alpha> \<delta> f i) (fst (pdivmod (F (i - 2)) (F (i - 1))))"
mat_hom_inj: fixes A :: "int mat"   and B :: "int mat" assumes "of_int_hom.mat_hom A = of_int_hom.mat_hom B" shows "A = B"
get_elements_by_tag_name_def: shows "get_elements_by_tag_name \<equiv> l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_tag_name to_tree_order"
assms: shows "h \<turnstile> ok (f \<bind> g)"
Stuck_at_b: shows "\<turnstile> \<box>[(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2) \<longrightarrow> stable pc1 = #b"
makespan_correct_2: fixes T :: "nat \<Rightarrow> nat" shows "\<exists>x\<in>{1..m}. T x = makespan T"
simps: fixes S :: "'c set set"   and a :: "'c set" shows "generate_topology S a = (a = UNIV \<or> (\<exists>aa b. a = aa \<inter> b \<and> generate_topology S aa \<and> generate_topology S b) \<or> (\<exists>K. a = \<Union> K \<and> (\<forall>x. x \<in> K \<longrightarrow> generate_topology S x)) \<or> (\<exists>s. a = s \<and> s \<in> S))"
Pow_iff: fixes A :: "'b set"   and B :: "'b set" shows "(A \<in> Pow B) = order.greater_eq B A"
finite_measure_Union: fixes A :: "'h set"   and p :: "'h spmf"   and B :: "'h set" assumes "A \<in> sets (measure_spmf p)"   and "B \<in> sets (measure_spmf p)"   and "A \<inter> B = {}" shows "Sigma_Algebra.measure (measure_spmf p) (A \<union> B) = Sigma_Algebra.measure (measure_spmf p) A + Sigma_Algebra.measure (measure_spmf p) B"
carrier_vs_is_self: fixes W :: "'a vec set" shows "carrier (AB.NC.vs W) = W"
nonzero_abs_inverse: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "\<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
homogeneous_solution_set_subspace: fixes A :: "(('b, 'c) vec, 'd) vec" shows "vec.subspace (solution_set A 0)"
bl_to_bin_ge2p_aux: fixes w :: "int"   and bs :: "bool list" shows "w * 2 ^ length bs \<le> bl_to_bin_aux bs w"
degen_acc_run_complete0: fixes r :: "nat \<Rightarrow> 'Q"   and m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b" assumes "num_acc = 0"   and "is_acc_run r" shows "b_graph.is_acc_run (degeneralize_ext m) (\<lambda>i. (r i, 0))"
range_composition: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd" shows "range (\<lambda>x. f (g x)) = f ` range g"
bin_rsplit_aux_len_le: fixes n :: "nat"   and ws :: "int list"   and nw :: "nat"   and w :: "int"   and bs :: "int list"   and m :: "nat" assumes "n \<noteq> 0"   and "ws = bin_rsplit_aux n nw w bs" shows "(length ws \<le> m) = (nw + length bs * n \<le> m * n)"
I: fixes t :: "real" shows "(f has_integral I t) {0..t}"
lm041: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "Outside' X P = P || (Domain P - X)"
l_get_host_wf_axioms: shows "l_get_host_wf heap_is_wellformed known_ptr known_ptrs type_wf get_host"
F_STOP: shows "F STOP = {(s, X). s = []}"
euler_err_fas_nth_def: fixes X0 :: "'d"   and h :: "floatarith"   and CX :: "floatarith list"   and i :: "nat" shows "euler_err_fas_nth X0 h CX i = h ^\<^sub>e 2 / floatarith.Num 2 * ode_d_fa_nth 0 CX (ode_fa CX) i"
length_rows: fixes A :: "'a mat" shows "length (rows A) = dim_row A"
Proj_vimage_chart_last_domainP: shows "Proj -` Collect chart_last_domainP = Collect chart_last_domain_nonzeroP"
empty_set_k_words: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option"   and k :: "nat" assumes "source_code M b fi X enc dec"   and "0 < k" shows "source_code.set_of_k_words_length_m M X enc k 0 = {}"
UN_least: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and C :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C" shows "\<Union> (B ` A) \<subseteq> C"
distinct_elim_rev: fixes v :: "'v"   and v1 :: "'v"   and w :: "'w"   and v2 :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_trail v ((v1, w, v2) # ps) v'" shows "((v2, w, v1) \<in> edges (rem_unPath ps G)) = ((v2, w, v1) \<in> E)"
circuit_inI_aux: fixes X :: "'a set" assumes "\<E>.circuit X" shows "circuit_in \<E> X"
constant_rules_def: fixes C :: "'b set" shows "constant_rules C \<equiv> const_exists ` C \<union> const_exists_rev ` C \<union> const_prop ` C \<union> {const_disj c\<^sub>1 c\<^sub>2 |c\<^sub>1 c\<^sub>2. c\<^sub>1 \<in> C \<and> c\<^sub>2 \<in> C \<and> c\<^sub>1 \<noteq> c\<^sub>2}"
OPT2_ub: fixes qs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set qs \<subseteq> {x, y}" shows "T\<^sub>p [x, y] qs (OPT2 qs [x, y]) \<le> length qs"
liset_def: fixes A :: "'b set"   and w :: "'b llist" shows "liset A w \<equiv> {i. enat i < llength w \<and> w ?! i \<in> A}"
assms_1: shows "2 \<cdot> g \<cdot> x = 2 \<cdot> g \<cdot> h + v \<cdot> v"
P': shows "{} \<turnstile> local.P IFF local.H(X0::=\<guillemotleft>local.P\<guillemotright>)"