hcju/isabelleps huggingface dataset:
SUP_subset_mono: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
crel_vs_update: fixes param :: "'k" shows "state_dp_consistency.crel_vs (=) () (update_st param (dp param))"
scale_minus_left: fixes a :: "'a"   and x :: "'a" shows "- a \<cdot> x = - (a \<cdot> x)"
primitive_extractor_correct_1: fixes m :: "'d match_expr"   and disc :: "'d \<Rightarrow> bool"   and sel :: "'d \<Rightarrow> 'e"   and C :: "'e \<Rightarrow> 'd"   and as :: "'e negation_type list"   and ms :: "'d match_expr"   and \<gamma> :: "('d \<Rightarrow> 'f \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'f \<Rightarrow> bool)"   and a :: "action"   and p :: "'f" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "(matches \<gamma> (alist_and (NegPos_map C as)) a p \<and> matches \<gamma> ms a p) = matches \<gamma> m a p"
l_to_tree_order_def: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_to_tree_order to_tree_order \<equiv> \<forall>ptr h. pure (to_tree_order ptr) h"
associatedE1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" obtains u :: "'a" where "dvd.dvd times u one" and "a = times u b"
authTicket_crypt_authK: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbV" shows "authK \<in> authKeys evs"
abs_eq: fixes x :: "'c \<Rightarrow> 'd option"   and xa :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "eq_onp (\<lambda>m. finite (dom m)) x x" shows "fmpred xa (Abs_fmap x) = map_pred xa x"
simps_3: fixes uv :: "'b"   and a :: "('a, 'b) Node"   and b :: "('a, 'b) Node"   and c :: "('a, 'b) Node" shows "FingerTreeStruc.nodeToList (Node3 uv a b c) = FingerTreeStruc.nodeToList a @ FingerTreeStruc.nodeToList b @ FingerTreeStruc.nodeToList c"
finite_permutations: fixes S :: "'b set" assumes "finite S" shows "finite {p. p permutes S}"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> B"   and "y \<in> B" shows "x \<otimes> y \<in> B"
sup_left_commute: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<union># (y \<union># z) = y \<union># (x \<union># z)"
cont_right_sync: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process"   and A :: "'c set" assumes "chain Y" shows "(S \<lbrakk>A\<rbrakk> (\<Squnion>i. Y i)) = (\<Squnion>i. S \<lbrakk>A\<rbrakk> Y i)"
mult_Suc_right: fixes m :: "nat"   and n :: "nat" shows "m \<cdot> Suc n = m + m \<cdot> n"
Iset_star_n: fixes X :: "nat \<Rightarrow> 'b"   and A :: "nat \<Rightarrow> 'b set" shows "(star_n X \<in> Iset (star_n A)) = (\<forall>\<^sub>F n in \<U>. X n \<in> A n)"
defined_def: fixes x :: "'b" shows "defined x = (x \<noteq> \<bottom>)"
r_prefixes_aux_recfn: shows "recfn 3 r_prefixes_aux"
all_formulas_entailed: fixes N2 :: "'a set"   and N1 :: "'a set" assumes "\<forall>C\<in>N2. N1 \<Turnstile> {C}" shows "N1 \<Turnstile> N2"
indep_vars_def: fixes M' :: "'c \<Rightarrow> 'd measure"   and X :: "'c \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'd"   and I :: "'c set" shows "indep_vars M' X I = ((\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> indep_sets (\<lambda>i. sigma_sets (space (Pi\<^sub>M A M)) {X i -` Aa \<inter> space (Pi\<^sub>M A M) |Aa. Aa \<in> sets (M' i)}) I)"
sign_changes_distrib: fixes p :: "real poly"   and x :: "real"   and ps\<^sub>1 :: "real poly list"   and ps\<^sub>2 :: "real poly list" assumes "poly p x \<noteq> 0" shows "sign_changes (ps\<^sub>1 @ [p] @ ps\<^sub>2) x = sign_changes (ps\<^sub>1 @ [p]) x + sign_changes ([p] @ ps\<^sub>2) x"
LeftDerivationIntro_is_sentence: fixes \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationIntro \<alpha> i r ix D j \<gamma>" shows "is_sentence \<alpha> \<and> is_sentence \<gamma>"
ballb: shows "\<forall>n\<ge>nob. \<bar>b n - y\<bar> < e"
sams_nos__nts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and J :: "'p" assumes "SAMS A B C C B J"   and "\<not> B C OS A J" shows "\<not> A B TS C J"
CFG_SSA_wf_defs_7: fixes g :: "'g"   and v :: "'val" shows "defNode g v \<equiv> hd (filter (\<lambda>n. v \<in> allDefs g n) (\<alpha>n g))"
fully_prappend_signed_list: fixes xs :: "('a \<times> bool) list" shows "prappend_signed_list (rev (map flip_signed xs)) xs = []"
rid: shows "\<forall>f. f \<approx>> \<iota> \<longrightarrow> f ;; \<iota> = f"
i0_iless_eSuc: fixes n :: "enat" shows "0 < eSuc n"
ag_a_in_ar_cos: fixes H :: "'a set"   and a :: "'a" assumes "M +> H"   and "a \<in> carrier M" shows "a \<in> a \<uplus>\<^bsub>M\<^esub> H"
AR_imp_ANR: fixes S :: "'a set" assumes "AR S" shows "ANR S"
distr_PiM_component: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and i :: "'b" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"   and "i \<in> I" shows "distr (Pi\<^sub>M I M) (M i) (\<lambda>\<omega>. \<omega> i) = M i"
get_tag_name_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "ShadowRootClass.type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_tag_name element_ptr"
bis_imp: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c1 :: "('c, 'd, 'a) cmd"   and c2 :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "(c1, c2) \<in> PL_Indis.Sbis aval tval cval indis" shows "(c1, c2) \<in> PL_Indis.ZObis aval tval cval indis"
Amin_mem: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat" assumes "f \<in> {i. i \<le> n} \<rightarrow> Z\<^sub>\<infinity>" shows "Amin n f \<in> Z\<^sub>\<infinity>"
iso: fixes p :: "int" shows "relative_homology_group p X S \<cong> relative_homology_group p Y T"
sep_implI: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'a"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "\<And>h'. \<lbrakk>sep_disj h h'; P h'\<rbrakk> \<Longrightarrow> Q (plus h h')" shows "sep_algebra.sep_impl plus sep_disj P Q h"
lesprop: shows "\<exists>y'>k. \<forall>(d, e, f)\<in>set b. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0"
eq_refl: fixes x :: "'t"   and y :: "'t" assumes "x = y" shows "x \<preceq>\<^sub>t y"
add_poly_p_empty_l: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset" shows "add_poly_p\<^sup>*\<^sup>* (p, q, r) ({#}, q, p + r)"
PSimQ: shows "P \<leadsto>\<^sub>l[Rel] Q"
empty_sets: shows "{} \<in> sigma_sets \<Omega> G"
dim_unique: fixes B :: "('a, 'c) vec set"   and V :: "('a, 'c) vec set"   and n :: "nat" assumes "B \<subseteq> V"   and "V \<subseteq> vec.span B"   and "vec.independent B"   and "card B = n" shows "vec.dim V = n"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "(('fsym, 'tp) efsym, 'psym) atm" shows "satL \<xi> (Pos a) = satA \<xi> a"
prems_9: shows "InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
FaultProp: fixes c :: "('a, 'b, 'c) com"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and f :: "'c" assumes "c \<noteq> Skip"   and "redex c = c" shows "\<Gamma>\<turnstile> (c, Fault f) \<rightarrow> (Skip, Fault f)"
r1: shows "\<rho>\<^sub>1 \<in> carrier_mat d d"
C_eq_RS2: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "wellformed_policy1_strong p" shows "C (list2FWpolicy (removeShadowRules2 p)) = C (list2FWpolicy p)"
fixp_above_Kleene_iter_ex: fixes a :: "'a" assumes "\<forall>M\<in>Chains leq. finite M" obtains k :: "nat" where "fixp_above a = (f ^^ k) a"
cases: fixes x :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" obtains bs :: "('t \<Rightarrow>\<^sub>0 'b) list" and ss :: "'t list" and z :: "nat" where "x = ((bs__, ss__, []), z__)"   | bs :: "('t \<Rightarrow>\<^sub>0 'b) list" and ss :: "'t list" and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" and z :: "nat" where "x = ((bs, ss, p # ps), z)"
weakenBisimWeakCoinduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox>\<^sub>w<X> Q"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto>\<^sub>w<X> Q"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<approx>\<^sub>w Q"
explicit_info_set_def: fixes X0 :: "'d set" shows "explicit_info_set X0 = isEmpty_spec X0 \<bind> (\<lambda>e. WHILE\<^bsup>\<lambda>(e, X, Y). (e \<longrightarrow> X = {}) \<and> X0 = X \<union> (\<Union>(sctn, IS)\<in>Y. IS)\<^esup> (\<lambda>(e, X, Y). \<not> e) (\<lambda>(e, X, Y). split_with_info X \<bind> (\<lambda>(sctn, S, X'). isEmpty_spec X' \<bind> (\<lambda>e. RETURN (e, X', insert (sctn, S) Y)))) (e, X0, {}) \<bind> (\<lambda>(uu_, uu_, Xis). RETURN Xis))"
GreenThm_typeII_twoChain: shows "two_chain_integral two_chain (partial_vector_derivative (\<lambda>a. F a \<bullet> j) i) = one_chain_line_integral F {j} (two_chain_boundary two_chain)"
infinite_super: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
nat_of_bool: fixes P :: "bool" shows "nat (of_bool P) = of_bool P"
sup_loc_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and LT :: "ty err list"   and LT' :: "ty err list" shows "P \<turnstile> LT [\<le>\<^sub>\<top>] LT' \<equiv> Listn.le (sup_ty_opt P) LT LT'"
subseq_Cons2_iff: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "subseq (x # xs) (y # ys) = (if x = y then subseq xs ys else subseq (x # xs) ys)"
disjunctive_def: shows "disjunctive.disjunctive (\<squnion>) (\<squnion>) (*) = {x. \<forall>y z. x * (y \<squnion> z) = x * y \<squnion> x * z}"
winding_number_homotopic_paths_null_explicit_eq: fixes p :: "real \<Rightarrow> complex"   and \<zeta> :: "complex" assumes "path p"   and "\<zeta> \<notin> path_image p" shows "(winding_number p \<zeta> = 0) = homotopic_paths (- {\<zeta>}) p (linepath (pathstart p) (pathstart p))"
point_in_one_group: fixes x :: "'a \<times> 'b"   and G1 :: "('a \<times> 'b) set"   and G2 :: "('a \<times> 'b) set" assumes "x \<in> G1"   and "G1 \<in> {{x} \<times> I |x. x \<in> bl}"   and "G2 \<in> {{x} \<times> I |x. x \<in> bl}"   and "G1 \<noteq> G2" shows "x \<notin> G2"
consistent_roots: fixes prb :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb" shows "fst (root (red prb)) = init (black prb)"
dup_simps_2: fixes f :: "'a" assumes "arr f" shows "local.dom (dup f) = local.dom f"
sym_iff_eq_refl: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "antisymmetric A less_eq"   and "x \<in> A"   and "y \<in> A" shows "sympartp less_eq x y = (x = y \<and> less_eq y y)"
float_defs_12: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fle a b = (fcompare a b = Lt \<or> fcompare a b = Eq)"
cfg_on_valid: fixes k :: "'a \<Rightarrow> nat"   and v :: "'a \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'a"   and A :: "('b \<times> ('a, real) cconstraint \<times> ('a set \<times> 'b) pmf) set \<times> ('b \<Rightarrow> ('a, real) cconstraint)"   and \<phi> :: "'b \<times> ('a \<Rightarrow> real) \<Rightarrow> bool"   and \<psi> :: "'b \<times> ('a \<Rightarrow> real) \<Rightarrow> bool"   and s :: "'b \<times> ('a \<Rightarrow> real)"   and cfg :: "('b \<times> ('a \<Rightarrow> real)) cfg" assumes "Probabilistic_Timed_Automaton_Regions_Reachability k v n not_in_X A \<phi> \<psi> s"   and "cfg \<in> Probabilistic_Timed_Automaton_Regions.cfg_on_div A s" shows "cfg \<in> Markov_Decision_Process_Invariant.valid_cfg (Probabilistic_Timed_Automaton.K A) (Probabilistic_Timed_Automaton.S A)"
apaths_finite_triple: shows "finite {(u, p, v). apath u p v}"
inject_2: fixes x2 :: "ltkey"   and y2 :: "ltkey" shows "(ltK x2 = ltK y2) = (x2 = y2)"
pos_bounded: shows "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
pwd_owner_iii_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_owner_iii n)"
map_closed: fixes \<alpha> :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'e set"   and a :: "'d" assumes "Set_Theory.map \<alpha> S T"   and "a \<in> S" shows "\<alpha> a \<in> T"
simps_5: fixes uu :: "nat"   and uv :: "nat"   and v :: "variable"   and va :: "typ" shows "incr_bv uu uv (Fv v va) = Fv v va"
r_inv_ex: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "\<exists>y\<in>carrier (K [X]). x \<oplus>\<^bsub>K [X]\<^esub> y = \<zero>\<^bsub>K [X]\<^esub>"
r_inconsist: fixes i :: "nat"   and e :: "nat" shows "eval r_inconsist [i, e] = inconsist i e"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_on_left_moving_norm (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> ml + mr = Suc m \<and> 0 < mr \<and> (if lm1 = [] then l = Oc \<up> ml @ Bk # Bk # ires else l = Oc \<up> ml @ Bk # <rev lm1> @ Bk # Bk # ires) \<and> (r = Oc \<up> mr @ Bk # <lm2> @ Bk \<up> rn \<or> lm2 = [] \<and> r = Oc \<up> mr))"
trans_onode_comp: fixes i :: "nat"   and S :: "((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and R :: "nat set" shows "automaton.trans (\<langle>i : S : R\<rangle>\<^sub>o) = onode_sos (automaton.trans S)"
ii: fixes a :: "nat"   and aa :: "int"   and b :: "nat" assumes "(a, aa, b) \<in> set L" shows "uGraph_from_list_\<alpha>_weight L (Upair a b) = aa"
assms_1: shows "(li, l) \<in> \<langle>A\<rangle>list_set_rel"
si: shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
continuous_map_preimage_topspace: fixes T1 :: "'c topology"   and T2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map T1 T2 f" shows "f -` topspace T2 \<inter> topspace T1 = topspace T1"
P_mod_mult1: fixes I :: "'a set"   and J :: "'a set"   and p :: "'a"   and q :: "'a" assumes "ideal S I"   and "ideal S J"   and "p \<in> carrier R"   and "q \<in> carrier R"   and "P_mod R S X I p"   and "P_mod R S X J q" shows "P_mod R S X (I \<diamondsuit>\<^sub>r\<^bsub>S\<^esub> J) (p \<cdot>\<^sub>r q)"
adjust_nonzero: fixes us :: "'a vec list"   and w :: "'a vec" assumes "set us \<subseteq> carrier_vec n"   and "distinct us"   and "w \<in> carrier_vec n"   and "w \<notin> local.span (set us)" shows "adjuster n w us + w \<noteq> 0\<^sub>v n"
wf_dbtm_imp_is_tm: fixes x :: "dbtm" assumes "wf_dbtm x" shows "\<exists>t. x = trans_tm [] t"
cong_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b zero a = dvd.dvd times a b"
mgu_wt_if_same_type: fixes s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term"   and \<sigma> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "mgu s t = Some \<sigma>"   and "wf\<^sub>t\<^sub>r\<^sub>m s"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<Gamma> s = \<Gamma> t" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>"
Rep_eint: fixes x :: "eint" shows "Rep_eint x \<in> UNIV"
tri_three_paths: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "\<triangle> a b c" shows "\<exists>ab bc caa. path ab a b \<and> path bc b c \<and> path caa c a \<and> ab \<noteq> bc \<and> ab \<noteq> caa \<and> bc \<noteq> caa"
X_def: shows "X = {y. (y, x) \<in> (support_flow (h_diff i))\<^sup>*} - {x}"
bij_betw_linorders_on': fixes A :: "'b set" assumes "finite A" shows "bij_betw (\<lambda>R. sorted_wrt_list_of_set R A) (linorders_on A) (permutations_of_set A)"
inter_rel_internal: fixes AA :: "('c \<times> 'e set) set"   and BB :: "('d \<times> 'e set) set" shows "inter_rel AA BB = {((a, b), A \<inter> B) |a b A B. (a, A) \<in> AA \<and> (b, B) \<in> BB}"
sbintrunc_eq_take_bit: fixes n :: "nat"   and k :: "int" shows "signed_take_bit n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n"
cases: fixes x :: "('a \<times> 'b \<times> 'a) list option" obtains "x = None"   | p :: "('a \<times> 'b \<times> 'a) list" where "x = Some p"
intra_SDG_path_matched: 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"   and ns :: "'b SDG_node 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.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" shows "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"
Int_stable_def: fixes M :: "'c set set" shows "Int_stable M = (\<forall>a\<in>M. \<forall>b\<in>M. a \<inter> b \<in> M)"
dfa2wf_nfa: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat" assumes "wf_dfa M n" shows "wf_nfa (nfa_of_dfa M) n"
swap_Trans_Snapshot: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isTrans ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
Hole_def: shows "\<box> \<equiv> Abs_ctx Hole_Rep"
Mp_Mp: fixes f :: "int poly" shows "pl.eq_m (pl.Mp f) f"
homI: fixes f :: "'a"   and A :: "'b"   and B :: "'b" assumes "f \<in> Ar"   and "Dom f = A"   and "Cod f = B" shows "f \<in> Hom A B"
assms_2: fixes j :: "nat" assumes "i = Suc j" shows "(\<lambda>x. g x j) \<in> M \<rightarrow>\<^sub>M N"
fact_asymp_equiv_aux: shows "fact \<sim>[sequentially] (\<lambda>x. exp c * sqrt (real x) * (real x / exp 1) powr real x)"
same_at_least_as_equal: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and z :: "'a"   and y :: "'a" assumes "rational_preference carrier relation"   and "z \<approx>[relation] y" shows "at_least_as_good z carrier relation = at_least_as_good y carrier relation"
cycle_free_diag_equiv: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd" shows "(\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs) = cycle_free M n"
assms_2: shows "\<forall>x\<in>U\<^sub>2. f x \<in> U"
normalize_left_idem: fixes a :: "'a"   and b :: "'a" shows "normalize a \<^bold>* b = a \<^bold>* b"
xt1_7: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y < x"   and "z \<le> y" shows "z < x"
op_concat_listset_def: fixes xs :: "'a list set list" shows "op_concat_listset xs = concat ` listset xs"
edge_kind_cases: fixes et :: "('a, 'b, 'c, 'd) edge_kind" obtains "intra_kind et"   | Q :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool" and r :: "'c" and p :: "'d" and fs :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list" where "et = Q__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs"   | Q :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool" and p :: "'d" and f :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" where "et = Q\<hookleftarrow>\<^bsub>p\<^esub>f"
map_of_zip_is_Some: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a" assumes "length xs = length ys" shows "(x \<in> set xs) = (\<exists>y. map_of (zip xs ys) x = Some y)"
fold_simps_2: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list"   and s :: "'c" shows "fold f (x # xs) s = fold f xs (f x s)"
perm_inv: fixes f :: "atom \<Rightarrow> atom" assumes "f \<in> perm" shows "inv f \<in> perm"
in_sub_out12: shows "inside (path_image c1 \<union> path_image c) \<subseteq> outside (path_image c2 \<union> path_image c)"
da_elim_cases_3: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Expr e\<rangle>\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
pmat_extension_def: fixes m :: "'b mat" shows "pmat_extension m = ptensor_mat m (1\<^sub>m d2)"
matrix_vs: fixes nr :: "nat"   and nc :: "nat" shows "vectorspace class_ring (module_mat TYPE('b) nr nc)"
right_move_to_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq (plus a (uminus c)) b" shows "less_eq a (plus b c)"