hcju/isabelleps huggingface dataset:
Special_Solutions_in_Solutions: fixes x :: "nat list \<times> nat list" assumes "x \<in> Special_Solutions" shows "x \<in> local.Solutions"
1: shows "boruvka_outer_invariant f g \<and> - forest_components f \<sqinter> g = bot"
ProblemIk_axioms: shows "ProblemIk TE_wtFsym wtPsym TE_arOf TE_resOf parOf tPB infTp"
surjective: fixes r :: "('a, 'b) web_scheme" shows "r = \<lparr>edge = edge r, weight = weight r, A = A r, B = B r, \<dots> = web.more r\<rparr>"
Cd_Above: fixes d :: "'b"   and X :: "'a set" shows "Pds'.Cd d X = Above (Pd_shuffle_to_top ds' Y d) (X \<inter> Field (Pd_shuffle_to_top ds' Y d)) \<inter> X"
span_insert_idI: fixes r :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "r \<in> pmdl B" shows "pmdl (insert r B) = pmdl B"
hom_carrier: fixes h :: "'f \<Rightarrow> 'g"   and G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" assumes "h \<in> hom G H" shows "h ` carrier G \<subseteq> carrier H"
Red_F_\<G>_q_def: fixes q :: "'a literal multiset set"   and N :: "('a literal multiset \<times> label) set" shows "FL.Red_F_\<G>_q q N = {C. \<forall>D\<in>\<G>_F (fst C). D \<in> G.Red_F (\<Union>CL\<in>N. \<G>_F (fst CL)) \<or> (\<exists>E\<in>N. E \<sqsubset> C \<and> D \<in> \<G>_F (fst E))}"
assms_2: shows " t \<longrightarrow>\<^sub>1 t2"
listDiffIff: fixes x :: "'a"   and a :: "'a list"   and b :: "'a list" shows "(x \<in> set a \<and> x \<notin> set b) = (x \<in> set (list_diff a b))"
sort_key_def: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "sort_key f xs = foldr (insort_key f) xs []"
r_pos: fixes n :: "nat" assumes "0 < n" shows "0 < r n"
intro: fixes x :: "'c \<Longrightarrow> 'b" assumes "\<And>\<sigma> v. put\<^bsub>x\<^esub> \<sigma> v = \<sigma>" shows "ief_lens_axioms x"
dagger_slide: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "(x ; y)\<^sup>* ; x = x ; (y ; x)\<^sup>*"
prv_imp_trans2: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp \<phi> \<chi>) (imp (imp \<chi> \<psi>) (imp \<phi> \<psi>)))"
Trie_impl_of: fixes t :: "('c, 'd) trie" shows "Trie2.trie.Trie (impl_of t) = t"
max_bag_card_in_bag_cards: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set" assumes "TreeDecomposition G T bag"   and "V\<^bsub>G\<^esub> \<noteq> {}" shows "TreeDecomposition.max_bag_card T bag \<in> {card (bag t) |t. t \<in> V\<^bsub>T\<^esub>}"
Suc_3: shows "KD_Tree.size_kdt kdt = 2 ^ min_height kdt"
subst_img_comp_subset: fixes \<theta>1 :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<theta>2 :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "range_vars (\<theta>1 \<circ>\<^sub>s \<theta>2) \<subseteq> range_vars \<theta>1 \<union> range_vars \<theta>2"
parts_crypt_2: fixes K :: "key"   and X :: "msg" shows "insert (Crypt K X) (parts {X}) \<subseteq> parts {Crypt K X}"
r1: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "x \<noteq> y" shows "P (x # xs) (y # ys)"
2: shows "x\<^sup>T ; r ; x\<^sup>T \<cdot> r\<^sup>T \<le> x\<^sup>T\<^sup>\<star>"
lexLessState_def: fixes F0 :: "'a"   and decisionVars :: "nat set" shows "lexLessState F0 decisionVars \<equiv> {(stateA, stateB). (getM stateA, getM stateB) \<in> lexLessRestricted (vars F0 \<union> decisionVars)}"
measurable_hld: fixes t :: "'c set"   and M :: "'c measure" assumes "t \<in> sets M" shows "Measurable.pred (stream_space M) (HLD t)"
PLemmas_9: fixes m :: "'b \<Rightarrow> 'c option" shows "dom m = {a. m a \<noteq> \<bottom>}"
force_def: fixes xs :: "'a llist" shows "force xs = (case xs of LNil \<Rightarrow> None | LCons x ys \<Rightarrow> Some (x, ys))"
rel_fundefs_Some1: fixes f :: "'k \<Rightarrow> ('b, 'l, 'm, 'n, 'e) Inca.instr fundef option"   and g :: "'k \<Rightarrow> ('b, 'l, 'm, 'n, 'e, 'g, 'h, 'i) Ubx.instr fundef option"   and x :: "'k"   and y :: "('b, 'l, 'm, 'n, 'e) Inca.instr fundef" assumes "rel_fundefs f g"   and "f x = Some y" shows "\<exists>z. g x = Some z \<and> rel_fundef (\<lambda>x y. x = norm_instr y) y z"
1: shows "subst_domain (rm_vars (UNIV - X) \<I>) = X"
Min_le: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "Min A \<le> x"
hbase_hd': fixes n :: "C"   and ns :: "C list"   and b :: "nat" assumes "C (n # ns) \<in> hbase b" shows "n \<in> hbase b"
slice2_is_SDG_path_slice2: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'"   and "n'' \<in> sum_SDG_slice2 n" shows "n'' \<in> sum_SDG_slice2 n'"
assms: shows "\<forall>y\<in>vars lp. y \<noteq> x \<longrightarrow> v1 y = v2 y"
execute_bind_1: fixes f :: "'b Heap"   and h :: "heap"   and x :: "'b"   and h' :: "heap"   and g :: "'b \<Rightarrow> 'c Heap" assumes "execute f h = Some (x, h')" shows "execute (f \<bind> g) h = execute (g x) h'"
size_list_conv_sum_list: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> nat"   and xs :: "'b list" assumes "class.monoid_add plus zero" shows "size_list f xs = sum_list (map f xs) + length xs"
manipulable_profile_def: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" shows "lowered.manipulable_profile R i Ri' = lowered R \<prec>[SD (R i)] lowered (R(i := Ri'))"
alw_wait_holds_cong: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and Q :: "'b \<Rightarrow> bool" assumes "alw (holds P) xs" shows "wait (holds (\<lambda>x. P x \<and> Q x)) xs = wait (holds Q) xs"
Suc: shows "geom_rand_walk u d v n w = lgeom_proc u d v (stake n w)"
freshEnv_def: fixes xs :: "'b"   and x :: "'c"   and rho :: "'b \<Rightarrow> 'c \<Rightarrow> ('d, 'e, 'b, 'c, 'f) qTerm set option" shows "freshEnv xs x rho \<equiv> rho xs x = None \<and> (\<forall>ys. liftAll (fresh xs x) (rho ys))"
env_def: fixes w :: "'c el_step list" shows "env w \<equiv> filter (\<lambda>e. case e of LOC a \<Rightarrow> False | ENV a \<Rightarrow> True) w"
af_prefix_true: fixes i :: "nat"   and j :: "nat"   and \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "i \<le> j"   and "af \<phi> (prefix i w) \<sim>\<^sub>P true\<^sub>n" shows "af \<phi> (prefix j w) \<sim>\<^sub>P true\<^sub>n"
or_dfa_def: shows "or_dfa = binop_dfa (\<or>)"
exhaust_sel: fixes mrexp3 :: "'b mrexp3" obtains "mrexp3 = Zero3"   | "mrexp3 = One3"   | "mrexp3 = Atom3 (un_Atom31 mrexp3) (un_Atom32 mrexp3)"   | "mrexp3 = Plus3 (un_Plus31 mrexp3) (un_Plus32 mrexp3) (fin1 mrexp3) (nul mrexp3)"   | "mrexp3 = Times3 (un_Times31 mrexp3) (un_Times32 mrexp3) (fin1 mrexp3) (nul mrexp3)"   | "mrexp3 = Star3 (un_Star31 mrexp3) (fin1 mrexp3)"
right_adjoint_simps_1: fixes f :: "'a" assumes "is_left_adjoint f" shows "ide f\<^sup>*"
elims_2: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "pred x xa" obtains f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = \<Up>fa__" and "xa = cf__ # cfs__"   | Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = (Q__)\<^sub>\<surd>" and "xa = cfa__ # cfsa__" and "Q__ (fst cfa__)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and r :: "'ret" and p :: "'pname" and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Qa__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs" and "xa = cfb__ # cfsb__" and "Qa__ (fst cfb__, r)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and p :: "'pname" and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "Q cf \<and> cfs \<noteq> []"
assms: shows "\<not> occs (case v of (x, xa) \<Rightarrow> Fv x xa) t"
l2_inv2_init: shows "init l2 \<subseteq> l2_inv2"
prems_12: shows "InvariantVarsM (getM state') F0 Vbl"
cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "class.linorder less_eq less" obtains "xa__ = []"   | x :: "'b" and ys :: "'b list" where "xa__ = x # ys"
alphaAbs_qAbs_iff_alphaAbs_ex_distinct_qAFresh: fixes xs :: "'e"   and x :: "'f"   and X :: "('c, 'd, 'e, 'f, 'g) qTerm"   and xs' :: "'e"   and x' :: "'f"   and X' :: "('c, 'd, 'e, 'f, 'g) qTerm" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_ex_distinct_qAFresh xs x X xs' x' X'"
Keys_Macaulay_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "Keys (set (Macaulay_list ps)) \<subseteq> Keys (set ps)"
assms: shows "h \<turnstile> local.a_set_tag_name element_ptr tag \<rightarrow>\<^sub>h h'"
vars_monom_keys: fixes a :: "'a"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "a \<noteq> (0::'a)" shows "vars (MPoly_Type.monom m a) = keys m"
Nml_VcompNml_3: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Dom t = Cod u" shows "Cod (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Cod t"
dense_closed_top: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus top = bot"
rbt_sorted_RBT_Impl_diag: fixes lt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt"   and leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "ord.rbt_sorted lt t" shows "ord.rbt_sorted (less_prod leq lt lt) (RBT_Impl_diag t)"
succs: fixes g' :: "'b"   and succs :: "'b \<Rightarrow> 'b list"   and g :: "'b"   and g'' :: "'b" assumes "g' \<in> set (succs g)"   and "g' [succs]\<rightarrow>* g''" shows "g [succs]\<rightarrow>* g''"
stateIsDecreasedByValidTransitions: fixes stateA :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set"   and stateB :: "State"   and learnL :: "(Literal list list \<times> Literal list list) set" assumes "invariantsHoldInState stateA F0 decisionVars"   and "transition stateA stateB decisionVars"   and "appliedLearn stateA stateB \<longrightarrow> (getF stateB, getF stateA) \<in> learnL" shows "(stateB, stateA) \<in> terminationLess F0 decisionVars learnL"
vars_term_subst: fixes t :: "('b, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('b, 'a) Term.term" shows "fv (t \<cdot> \<sigma>) = \<Union> (fv ` \<sigma> ` fv t)"
codomain_unique: fixes b :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and b' :: "'a \<times> 'a" assumes "b \<in> VV.codomains f"   and "b' \<in> VV.codomains f" shows "b = b'"
equiintegrable_on_sing: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" shows "({f} equiintegrable_on cbox a b) = (f integrable_on cbox a b)"
insertE: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" assumes "a \<in> insert b A" obtains "a = b"   | "a \<in> A"
assms: shows "aa \<subseteq> N \<times> (Pow G - {{}})"
adopt_node_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> adopt_node_locs parent owner_document document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
comm_neutral: fixes a :: "'g" shows "a + (0::'g) = a"
conc_star_star: fixes A :: "'a list set" shows "A\<star> \<cdot> A\<star> = A\<star>"
abstract_def: fixes disc :: "'a common_primitive negation_type \<Rightarrow> bool" shows "abstract disc \<equiv> optimize_matches (abstract_primitive disc)"
weakenBisimE_2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx>\<^sub>w Q" shows "\<Psi> \<rhd> P \<leadsto>\<^sub>w<weakenBisim> Q"
finsum_insert: fixes F :: "'d set"   and a :: "'d"   and f :: "'d \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier R"   and "f a \<in> carrier R" shows "finsum R f (insert a F) = f a \<oplus> finsum R f F"
antisym_lexord: fixes R :: "('a \<times> 'a) set" assumes "antisym R"   and "irrefl R" shows "antisym (lexord R)"
simps: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and q :: "'c"   and n :: "nat" shows "semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n = {x. x \<le> n \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = q}"
distinct_29: shows "ToLowerCaseOp \<noteq> ToRealOp"
mono_det_FD: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<le> P'"   and "S \<le> S'" shows "(P [+] S) \<le> (P' [+] S')"
residual_unique: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "\<exists>!g. residuated_pair f g"
L2_Controller_Zero: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l t = Zero"   and "300 < hd (y (Suc t))" shows "z (Suc t) = []"
length_concat: fixes xss :: "'c list list" shows "length (concat xss) = sum_list (map length xss)"
solves_ode_vderivD: fixes y :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set" assumes "(y solves_ode f) T X" shows "(y has_vderiv_on (\<lambda>t. f t (y t))) T"
asymp_less: shows "asymp (\<lambda>a b. cmp a b = Less)"
ALG_P'_def: fixes Strat :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and i :: "nat"   and x :: "'c"   and y :: "'c" shows "ALG_P' Strat qs init i x y = ALG_P (snd (Strat ! i)) x y (steps' init qs Strat i)"
missing_cs_contradicts: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and k :: "nat"   and \<sigma>' :: "'b \<Rightarrow> 'c"   and n :: "nat"   and n' :: "nat" assumes "IFC prob"   and "\<nexists>k'. IFC_def.cs prob (IFC_def.path prob \<sigma>) k = IFC_def.cs prob (IFC_def.path prob \<sigma>') k'"   and "k < n"   and "IFC_def.cs prob (IFC_def.path prob \<sigma>) n = IFC_def.cs prob (IFC_def.path prob \<sigma>') n'" shows "\<exists>j'. IFC_def.contradicts prob (\<sigma>', j') (\<sigma>, k)"
mult_def: fixes r :: "('b \<times> 'b) set" shows "mult r = (mult1 r)\<^sup>+"
db_defs_3: fixes inp :: "state \<Rightarrow> 'b"   and mid :: "state \<Rightarrow> 'b"   and out :: "state \<Rightarrow> 'b"   and q1 :: "state \<Rightarrow> 'b list"   and q2 :: "state \<Rightarrow> 'b list"   and ic :: "state \<Rightarrow> 'c"   and q :: "state \<Rightarrow> 'c list"   and oc :: "state \<Rightarrow> 'c" assumes "DBuffer inp mid out q1 q2" shows "Deq ic q oc \<equiv> #0 < length<$q> \<and> oc$ = hd<$q> \<and> q$ = tl<$q> \<and> ic$ = $ic"
is_inf_def: fixes x :: "'b"   and y :: "'b"   and inf :: "'b" shows "is_inf x y inf = (inf \<sqsubseteq> x \<and> inf \<sqsubseteq> y \<and> (\<forall>z. z \<sqsubseteq> x \<and> z \<sqsubseteq> y \<longrightarrow> z \<sqsubseteq> inf))"
inside_arc_empty: fixes \<gamma> :: "real \<Rightarrow> 'b" assumes "arc \<gamma>" shows "inside (path_image \<gamma>) = {}"
evaln_elim_cases_28: fixes G :: "prog"   and s :: "st"   and accC :: "qtname"   and statDeclC :: "qtname"   and stat :: "bool"   and e :: "expr"   and fn :: "vname"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In2 ({accC,statDeclC,stat}e..fn)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains s1 :: "state" and a :: "val" and s2 :: "state" and va :: "vvar" and s2' :: "state" where "v = In2 va" and "s' = check_field_access G accC statDeclC fn stat a s2'" and "G\<turnstile>Norm s \<midarrow>Init statDeclC\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>e-\<succ>a\<midarrow>n\<rightarrow> s2" and "(va, s2') = fvar statDeclC stat fn a s2"
run_definitions_4: fixes s :: "nat \<Rightarrow> 'a" shows "is_fair_run s \<equiv> \<exists>l. is_lrun l s \<and> is_lfair l s"
r_neg1: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> x \<oplus>\<^bsub>K [X]\<^esub> (x \<oplus>\<^bsub>K [X]\<^esub> y) = y"
\<delta>_states_cart: fixes q :: "'a \<times> 'b"   and \<delta>1 :: "('a, 'c) ta_rule set"   and \<delta>2 :: "('b, 'c) ta_rule set" assumes "q \<in> \<delta>_states (\<delta>_prod \<delta>1 \<delta>2)" shows "q \<in> \<delta>_states \<delta>1 \<times> \<delta>_states \<delta>2"
dom_in_domains: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.domains f \<noteq> {}" shows "A1xA2.dom f \<in> A1xA2.domains f"
statesNames_def: fixes s :: "(integer \<times> edge list) iarray" shows "statesNames s = procVarDeclName ` statesDecls s"
eventually_at: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and S :: "'d set" shows "eventually P (at a within S) = (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
satA_transE: fixes at :: "('fsym, 'psym) atm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wtA at"   and "I.satA \<xi> at" shows "Ik.satA (transE \<xi>) at"
vertices_def: fixes labeled_graph :: "('f, 'e) labeled_graph" shows "vertices labeled_graph = (case labeled_graph of LG x1 x2 \<Rightarrow> x2)"
irrefl_f: shows "\<forall>x\<in>A. \<not> gt (f x) (f x)"
pcr_cr_eq: shows "pcr_view\<^sub>m = cr_view\<^sub>m"
rep_eq: fixes x :: "('c, 'd) fmap" shows "lookup (Pm_fmap x) = lookup0 x"
nres_rel_def: fixes R :: "('f \<times> 'g) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
i: shows "(g has_vector_derivative vector_derivative g (at ((v - u) * x + u))) (at ((v - u) * x + u))"
add_iso_r: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and z :: "'b"   and x :: "'b" assumes "class.dioid_one_zero plus times one zero less_eq less"   and "less_eq y z" shows "less_eq (plus x y) (plus x z)"
simps: fixes a :: "'trm" shows "(a \<in> atrm) = ((\<exists>n. a = n \<and> n \<in> num) \<or> (\<exists>x. a = Var x \<and> x \<in> var) \<or> (\<exists>t. a = suc t \<and> t \<in> atrm) \<or> (\<exists>t t'. a = pls t t' \<and> t \<in> atrm \<and> t' \<in> atrm) \<or> (\<exists>t t'. a = tms t t' \<and> t \<in> atrm \<and> t' \<in> atrm))"
in_set_zip: fixes p :: "'b \<times> 'c"   and xs :: "'b list"   and ys :: "'c list" shows "(p \<in> ran\<^sub>l (zip xs ys)) = (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n < length xs \<and> n < length ys)"
pnop_vdomain: shows "\<D>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A ^\<^sub>\<times> 2\<^sub>\<nat>"
bigo_def: fixes f :: "'a \<Rightarrow> 'b" shows "O(f) = {h. \<exists>c. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>}"
big_e: fixes a :: "nat list"   and b :: "nat list"   and xs :: "nat list"   and j :: "nat" shows "set (big_e a b xs j) = hlde_ops.Ej a b j xs"
Obj_implies_Ide: fixes t :: "'a E.term" assumes "E.Obj t" shows "E.Ide t"
bijective_conv_mapping: fixes x :: "'a" shows "bijective x = (coreflexive (x\<^sup>T[[x\<^sup>T]]) \<and> x[[top]] = top)"
Inf_less_eq: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and u :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v \<in> A \<Longrightarrow> less_eq v u"   and "A \<noteq> {}" shows "less_eq (Inf A) u"
prime_number_theorem: shows "\<pi> \<sim>[at_top] (\<lambda>x. x / ln x)"
mat_distr: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'd \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<oplus> g \<otimes> h = (f \<otimes> h) \<oplus> (g \<otimes> h)"
1: fixes x21 :: "'f"   and x22 :: "('f, 'a) Term.term list"   and x21a :: "'f"   and x22a :: "('f, 'a) Term.term list"   and i :: "nat" assumes "vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> weight t \<le> weight s"   and "\<not> weight t < weight s"   and "s = Fun x21 x22"   and "t = Fun x21a x22a"   and "\<not> pr_strict (x21, length x22) (x21a, length x22a)"   and "pr_weak (x21, length x22) (x21a, length x22a)"   and "i < length x22"   and "i < length x22a" shows "kbo'.kbo (x22 ! i) (x22a ! i) = kbo (x22 ! i) (x22a ! i)"
tendsto_0_if_Limsup_eq_0_ennreal: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> ennreal" assumes "Limsup F f = 0" shows "(f \<longlongrightarrow> 0) F"
known_ptrs_is_l_known_ptrs: shows "l_known_ptrs DocumentClass.known_ptr DocumentClass.known_ptrs"
eventually_at_right_less: fixes x :: "'b" shows "eventually ((<) x) (at_right x)"
dests_bigger_than_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:15..PAodv-:19} \<union> {PPkt-:7..PPkt-:11} \<union> {PRreq-:9..PRreq-:13} \<union> {PRreq-:21..PRreq-:25} \<union> {PRrep-:10..PRrep-:14} \<union> {PRerr-:1..PRerr-:5} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> kD (rt \<xi>) \<and> sqn (rt \<xi>) ip \<le> the (dests \<xi> ip)))"
Vfrom_mono: fixes i :: "V"   and a :: "V"   and b :: "V"   and j :: "V" assumes "Ord i"   and "a \<subseteq>\<^sub>\<circ> b"   and "i \<subseteq>\<^sub>\<circ> j" shows "Vfrom a i \<subseteq>\<^sub>\<circ> Vfrom b j"
lists_succ_remove1: fixes xss :: "'b list list"   and xs :: "'b list" assumes "distincts xss"   and "xs \<in> set xss" shows "lists_succ (xs # remove1 xs xss) = lists_succ xss"
octo_product_flexible: fixes x :: "octo"   and y :: "octo" shows "x * y * x = x * (y * x)"
ParTL_MtransC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.MtransT tval aval (c1, s) s'" shows "PL.MtransC tval aval (c1 | c2, s) (c2, s')"
sorted_from_radix_sort_step: fixes xss :: "'a list list"   and n :: "nat"   and i :: "nat" assumes "cols xss n"   and "i < n"   and "sorted (map (drop (i + 1)) xss)" shows "sorted (map (drop i) (sort_col i xss))"
assms: shows "0 < (y - x) \<bullet> (z - x)"
assms: shows "(M, N) \<in> ns_mul_ext Id R"
trimmingD_wave: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "trimming \<Gamma> f g" shows "wave \<Gamma> g"
Infinitesimal_add_approx_self: fixes y :: "'a star"   and x :: "'a star" assumes "y \<in> Infinitesimal" shows "x \<approx> x + y"
simps_2: fixes x21 :: "'c"   and x22 :: "'c llist" shows "LNil \<noteq> LCons x21 x22"
mk_disjoint_insert: fixes a :: "'k"   and A :: "'k set" assumes "a \<in> A" shows "\<exists>B. A = insert a B \<and> a \<notin> B"
comp_apply: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'i \<Rightarrow> 'h"   and x :: "'i" shows "(f \<circ> g) x = f (g x)"