hcju/isabelleps huggingface dataset:
rbt_sorted_entries_right_unique: fixes k :: "'a"   and v :: "'c"   and t :: "('a, 'c) rbt"   and v' :: "'c" assumes "(k, v) \<in> set (entries t)"   and "(k, v') \<in> set (entries t)"   and "rbt_sorted t" shows "v = v'"
psi_disjuncts: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "distinct3 p q r" shows "psi p q r \<or> psi p r q \<or> psi q r p \<or> psi q p r \<or> psi r p q \<or> psi r q p"
indep_in_subI_subset: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E>' :: "'b set"   and \<E> :: "'b set"   and X :: "'b set" assumes "indep_system carrier indep"   and "\<E>' \<subseteq> \<E>"   and "indep_system.indep_in indep \<E>' X" shows "indep_system.indep_in indep \<E> X"
FA_and_OC_and_TRel_impl_SRel_bisimulation: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "operational_complete TRel"   and "operational_sound TRel"   and "sym TRel"   and "trans TRel"   and "weak_reduction_bisimulation TRel Target" shows "weak_reduction_bisimulation SRel Source"
r_lr: fixes R :: "('a \<times> 'a) set"   and A :: "'a multiset" assumes "refl R" shows "locally_refl R A"
Collect_subset: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "{x \<in> A. P x} \<subseteq> A"
assms: shows "cfg1 \<turnstile> m \<mapsto> cfg2"
dghm_tdghm_comp_is_tdghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<HH> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN> : \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
combSlice_refl: 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 S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "n \<in> S" shows "n \<in> SDG.combine_SDG_slices sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S"
lNactLe: fixes active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "dynamic_component.lNAct active c t n \<le> n"
irreducible_not_unit: fixes p :: "'a" assumes "irreducible p" shows "\<not> is_unit p"
add_diff_less: fixes k :: "nat"   and n :: "nat"   and m :: "nat" assumes "k < n"   and "0 < m" shows "m + k - n < m"
gap_Suc_3: fixes b :: "nat"   and i :: "nat"   and t :: "nat"   and k :: "nat" assumes "b = block i t"   and "k = gap i t"   and "change_on_neither b k" shows "gap i (Suc t) = k"
subcls1_def: fixes G :: "prog" shows "subcls1 G = {(C, D). C \<noteq> Object \<and> (? c:class G C: super c = D)}"
hom_det: fixes A :: "'a mat" shows "det (mat\<^sub>h A) = hom (det A)"
add_cf_pol_coeff: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "pol_coeff S d" shows "pol_coeff S (add_cf S c d)"
int_one_le_iff_zero_less: fixes z :: "int" shows "(1 \<le> z) = (0 < z)"
mem_imp_elem: fixes S :: "('c, 'd) eq_object_scheme"   and x :: "'c"   and A :: "'c set" assumes "equivalence S"   and "x \<in> carrier S"   and "x \<in> A" shows "x .\<in>\<^bsub>S\<^esub> A"
ereal_openE: fixes A :: "ereal set" assumes "open A" obtains x :: "real" and y :: "real" where "open (ereal -` A)" and "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A" and "- \<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A"
ring_simprules_6: shows "\<one>\<^bsub>P\<^esub> \<in> carrier P"
IH_2: shows "s = Fun g ts \<cdot> \<tau>"
pair_prob_space_def: fixes M1 :: "'a measure"   and M2 :: "'b measure" shows "pair_prob_space M1 M2 \<equiv> pair_sigma_finite M1 M2 \<and> prob_space M1 \<and> prob_space M2"
b_3: shows "set_mset K \<subseteq> r \<down>s (set_mset J \<union> set_mset P)"
inv_f_f: fixes f :: "'h \<Rightarrow> 'i"   and x :: "'h" assumes "inj f" shows "inv f (f x) = x"
WTrt_binop_simps_4: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>LessOrEqual\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Boolean)"
nrcons_invmeas: fixes t :: "('a, 'b) FingerTreeStruc"   and nd :: "('a, 'b) Node" assumes "FingerTreeStruc.is_measured_ftree t"   and "FingerTreeStruc.is_measured_node nd" shows "FingerTreeStruc.is_measured_ftree (FingerTreeStruc.nrcons t nd)"
tfr\<^sub>s\<^sub>t_if_comp_tfr\<^sub>s\<^sub>t: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, ('a, 'b) Term.term \<times> nat) Term.term \<Rightarrow> ('a, ('a, 'b) Term.term \<times> nat) Term.term list \<times> ('a, ('a, 'b) Term.term \<times> nat) Term.term list"   and \<Gamma> :: "('a, ('a, 'b) Term.term \<times> nat) Term.term \<Rightarrow> ('a, 'b) Term.term"   and M :: "('a, ('a, 'b) Term.term \<times> nat) Term.term list"   and S :: "('a, ('a, 'b) Term.term \<times> nat) strand_step list" assumes "typed_model' arity public Ana \<Gamma>"   and "comp_tfr\<^sub>s\<^sub>t arity Ana \<Gamma> M S" shows "typed_model.tfr\<^sub>s\<^sub>t arity Ana \<Gamma> S"
PDG_path_ddep: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and control_dependence :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g"   and V :: "'i"   and n' :: "'g" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence"   and "PDG.PDG_edge sourcenode targetnode valid_edge Def Use control_dependence n (Some V) n'" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n n'"
trans_greater: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "compare cmp a b = Greater"   and "compare cmp b c = Greater" shows "compare cmp a c = Greater"
tm_dghm_ObjMap_in_Vset: shows "\<FF>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
awalk_ends: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "awalk u p v"   and "awalk u' p v'" shows "p \<noteq> [] \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
continuous_map_fst: fixes X :: "'d topology"   and Y :: "'e topology" shows "continuous_map (prod_topology X Y) X fst"
qAFresh_imp_qFresh: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" assumes "qAFresh xs x X" shows "qFresh xs x X"
calc_ideal_points_on_unit_circle: fixes H :: "circline" shows "\<forall>z\<in>calc_ideal_points H. z \<in> circline_set unit_circle"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "pre (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = ($ 1 \<and>* (\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then pre C\<^sub>1 Q (ps, n) else pre C\<^sub>2 Q (ps, n))))"
map_le_map_add: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "f \<subseteq>\<^sub>m Map.map_add g f"
bind_eq_ErrorI_1: fixes x :: "('a, 'b) result"   and z :: "'a"   and f :: "'b \<Rightarrow> ('a, 'c) result" assumes "x = Error z" shows "x \<bind> f = Error z"
intros_2: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> Suc 0 \<Down> s(x := aval a s)"
pos_lit_in_atms_of: fixes A :: "'a"   and C :: "'a literal multiset" assumes "Pos A \<in># C" shows "A \<in> atms_of C"
\<A>_models_\<Phi>\<^sub>I: shows "\<A> \<Turnstile> \<Phi>\<^sub>I \<Pi>"
supd_assn_supdI: fixes Q :: "'b assn"   and Y :: "vals"   and s :: "state"   and Z :: "'b"   and f :: "state \<Rightarrow> state" assumes "Q Y s Z" shows "(f .; Q ;. f) Y s Z"
upper_sup: fixes A :: "'b interval"   and B :: "'b interval" shows "upper (sup A B) = sup (upper A) (upper B)"
sets_pair_measure_cong: fixes M1 :: "'a measure"   and M1' :: "'a measure"   and M2 :: "'b measure"   and M2' :: "'b measure" assumes "sets M1 = sets M1'"   and "sets M2 = sets M2'" shows "sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
assms: fixes i :: "'a" shows "spmf p i = spmf q i"
elims_2: fixes x :: "('c, 'd) SkewBinomialTree list" assumes "rank_invar x" obtains "x = []"   | t :: "('c, 'd) SkewBinomialTree" where "x = [t__]"   | t :: "('c, 'd) SkewBinomialTree" and t' :: "('c, 'd) SkewBinomialTree" and bq :: "('c, 'd) SkewBinomialTree list" where "x = t # t' # bq" and "rank t < rank t' \<and> rank_invar (t' # bq)"
arr_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "A_B.arr f = (f \<noteq> A_B.Null \<and> A_B.Dom f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> A_B.Cod f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> A_B.Map f \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Dom f) (A_B.Cod f)))"
red_Lam: fixes x :: "name"   and t :: "trm"   and r :: "trm" assumes "\<Lambda> x . t \<mapsto> r" shows "(\<exists>t'. r = \<Lambda> x . t' \<and> t \<mapsto> t') \<or> t = App r (Var x) \<and> x \<sharp> r"
abc_2: shows "b \<noteq> c \<and> \<not> p b < p c"
aezfun_decomp_aezdeltafun: fixes c :: "('c, 'd) aezfun" shows "\<exists>ras. set (map snd ras) = supp (aezfun c) \<and> c = (\<Sum>(r, a)\<leftarrow>ras. r \<delta>\<delta> a)"
pp_of_term_def: fixes v :: "'a" shows "punit.pp_of_term v = fst (to_pair_unit v)"
Arccos_in_Reals: fixes z :: "complex" assumes "z \<in> \<real>"   and "\<bar>Re z\<bar> \<le> 1" shows "Arccos z \<in> \<real>"
induct: fixes x :: "State"   and P :: "State \<Rightarrow> bool" assumes "exhaustiveUnitPropagate_dom x"   and "\<And>state. \<lbrakk>\<lbrakk>\<not> getConflictFlag state; getQ state \<noteq> []\<rbrakk> \<Longrightarrow> exhaustiveUnitPropagate_dom (applyUnitPropagate state); \<lbrakk>\<not> getConflictFlag state; getQ state \<noteq> []\<rbrakk> \<Longrightarrow> P (applyUnitPropagate state)\<rbrakk> \<Longrightarrow> P state" shows "P x"
star_inductr_var_equiv: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c" assumes "class.kleene_algebra_zerol plus times one zero less_eq less star" shows "less_eq (times y x) y = less_eq (times y (star x)) y"
length_bvxor_bound: fixes a :: "nat"   and l :: "bit list"   and l2 :: "bit list" assumes "a \<le> length l" shows "a \<le> length (bvxor l l2)"
aux0_0: fixes x :: "('b, 'c) Combinators list" assumes "singleCombinators x" shows "\<nexists>a b. a \<oplus> b \<in> set x"
zero_spaceN_iff': fixes x :: "'c"   and N :: "'c quasinorm" shows "(x \<in> zero_space\<^sub>N N) = (x \<in> space\<^sub>N N \<and> Norm N x = 0)"
cgenideal_minimal: fixes J :: "'a set"   and a :: "'a" assumes "ideal J R"   and "a \<in> J" shows "PIdl a \<subseteq> J"
strict_mono_on_imp_mono_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "Fun.strict_mono_on f A" shows "Fun.mono_on f A"
cases: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do a" obtains s :: "('a, 'b, 'c) state" where "a = [s__]"   | s' :: "('a, 'b, 'c) state" and s :: "('a, 'b, 'c) state" and sl :: "('a, 'b, 'c) state list" where "a = s # s' # sl" and "s' \<in> GM_sec_model.\<delta> do s" and "GM_sec_model.wffp do (s' # sl)"
maddux_3_13: fixes 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 bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.pd_algebra inf less_eq less sup bot top uminus" shows "inf (sup x y) (uminus x) = inf y (uminus x)"
sconf_type_ok_def: fixes ET :: "(String.literal \<Rightarrow> ty option) \<times> ty"   and t :: "'thread_id"   and ex :: "'addr expr \<times> (String.literal \<Rightarrow> 'addr val option)"   and h :: "'heap" shows "sconf_type_ok ET t ex h \<equiv> fst ET \<turnstile> (h, snd ex) \<surd> \<and> type_ok P ET (fst ex) h \<and> P,h \<turnstile> t \<surd>t"
connected_append_edge: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and v :: "nat"   and w :: "nat" assumes "Graph.connected c u v"   and "(v, w) \<in> Graph.E c" shows "Graph.connected c u w"
assms_2: shows "p \<in> hom_deg_set z A"
jordan_block_def: fixes n :: "nat"   and a :: "'b" shows "jordan_block n a = mat n n (\<lambda>(i, j). if i = j then a else if Suc i = j then 1::'b else (0::'b))"
assms: shows "\<nexists>k. D = k\<^sup>2"
bind_ask2: fixes m :: "'m"   and f :: "'a multiset \<Rightarrow> 'b \<Rightarrow> 'm" shows "bind m (\<lambda>x. ask (f x)) = ask (\<lambda>r. bind m (\<lambda>x. f x r))"
less_eq_num1_def: fixes a :: "num1"   and b :: "num1" shows "(a \<le> b) = (Rep_num1 a \<le> Rep_num1 b)"
prod_cases3: fixes y :: "'e \<times> 'f \<times> 'g" obtains a :: "'e" and b :: "'f" and c :: "'g" where "y = (a, b, c)"
bounds_2: fixes n :: "nat" shows "(\<Sum>j = 0..<n. (- 1) ^ j * (1 / real (2 * (0 + j) + 1)) * real_of_float (y ^ j)) \<le> real_of_float (ub_arctan_horner prec n (2 * 0 + 1) y)"
integral_minus: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" shows "LINT x|M. - f x = - integral\<^sup>L M f"
Abs_JF1_inverse: fixes y :: "((bd_type_F + bd_type_F) list set \<times> ((bd_type_F + bd_type_F) list \<Rightarrow> ('d, bd_type_F, bd_type_F) F1 + ('d, bd_type_F, bd_type_F) F2)) set" assumes "y \<in> car_final1" shows "Rep_JF1 (Abs_JF1 y) = y"
analz_insertI: fixes c :: "msg"   and G :: "msg set"   and a :: "msg" assumes "c \<in> analz G" shows "c \<in> analz (insert a G)"
connected_connected_component: fixes S :: "'b set"   and x :: "'b" shows "connected (connected_component_set S x)"
isomorphic_def: fixes a :: "'c"   and a' :: "'c" shows "C.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>C a'\<guillemotright> \<and> C.iso f)"
do_def: fixes e :: "'d \<Rightarrow> 'c" shows "do e \<equiv> (\<lambda>(A, A'). tr A = tr A' \<and> e (alpha_rp.more A) \<notin> ref A') \<triangleleft> wait \<circ> snd \<triangleright>  (\<lambda>(A, A'). tr A' = tr A @ [e (alpha_rp.more A)])"
domain_pdevs_def: fixes xs :: "'a pdevs list" shows "domain_pdevs xs = \<Union> (pdevs_domain ` set xs)"
distinct_9: fixes x1 :: "'b"   and x61 :: "'b formula"   and x62 :: "'b formula" shows "Atom (x1::'b) \<noteq> (x61::'b formula) \<^bold>\<rightarrow> (x62::'b formula)"
take_lt: fixes n :: "nat"   and al :: "'a list" assumes "n < length al" shows "take n al < al"
not_S_Var: fixes x :: "'a"   and t :: "('f, 'a) Term.term" shows "\<not> S (Var x) t"
primeness_condition_monoid_axioms: shows "primeness_condition_monoid G"
analz_Un_implInsecSet_aux_3: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (PairSet Tags (PairSet AgentSet (PairSet AgentSet G)) \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
Join_project_constrains: fixes F :: "'a program"   and C :: "'c set"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" shows "(F \<squnion> project h C G \<in> A co B) = (extend h F \<squnion> G \<in> C \<inter> extend_set h A co extend_set h B \<and> F \<in> A co B)"
injD: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "inj f"   and "f x = f y" shows "x = y"
open_segment_idem: fixes a :: "'a" shows "open_segment a a = {}"
diff_single_trivial: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<notin># M" shows "M \<ominus> x = M"
Suc: shows "eval r_badblock [n, v] \<down>= Partial_Recursive.list_encode (replicate n 0 @ [1 - v])"
single_diff: fixes k :: "'b"   and a :: "'c"   and b :: "'c" shows "single k (a - b) = single k a - single k b"
midpoint_bet: fixes B :: "'p"   and A :: "'p"   and C :: "'p" assumes "B Midpoint A C" shows "Bet A B C"
eq3: fixes r1 :: "'a"   and r2 :: "'a" assumes "r1 \<in> carrier R"   and "r2 \<in> carrier R" shows "f (r1 \<oplus>\<^bsub>R\<^esub> r2) = g (r1 \<oplus>\<^bsub>R\<^esub> r2)"
S_intersection1_rep_def: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection1_rep p q \<equiv> S_intersection_coeff1 p q *\<^sub>R p + q"
S: fixes i :: "nat" shows "S i \<in> fmeasurable M"
rev_perm: fixes xs :: "'b list"   and ys :: "'b list" shows "(rev xs <~~> ys) = (xs <~~> ys)"
ardenable_set_sub: fixes rhs :: "'b trm set"   and A :: "'b trm set" assumes "ardenable rhs" shows "ardenable (rhs - A)"
vconn_implies_hpar_4: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>D g\<guillemotright>" shows "trg\<^sub>D g = trg\<^sub>D f"
vfiniteD: fixes A :: "V" assumes "vfinite A" shows "\<exists>n\<in>elts \<omega>. n \<approx>\<^sub>\<circ> A"
at_least_at_most_Inf: fixes L :: "('a, 'b) gorder_scheme"   and a :: "'a"   and b :: "'a" assumes "weak_complete_lattice L"   and "a \<in> carrier L"   and "b \<in> carrier L"   and "a \<sqsubseteq>\<^bsub>L\<^esub> b" shows "\<Sqinter>\<^bsub>L\<^esub>\<lbrace>a..b\<rbrace>\<^bsub>L\<^esub> .=\<^bsub>L\<^esub> a"
basis_null_space_transpose_eq_basis_left_null_space: fixes A :: "(('b, 'd) vec, 'c) vec" shows "basis_null_space (Finite_Cartesian_Product.transpose A) = basis_left_null_space A"
exists_fresh_1: fixes x :: "'a" assumes "finite (supp x)" obtains c :: "name" where "c \<sharp> x"
l_new_document_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_document_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"
Rf_indep: fixes N' :: "'a literal multiset set"   and N :: "'a literal multiset set" assumes "N' \<subseteq> Rf N" shows "Rf N \<subseteq> Rf (N - N')"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b \<cdot> (a \<cdot> c) = a \<cdot> (b \<cdot> c)"
partition_on_mset_elem_exists_part: fixes A :: "'b multiset"   and P :: "'b multiset multiset"   and x :: "'b" assumes "partition_on_mset A P"   and "x \<in># A" obtains p :: "'b multiset" where "p \<in># P" and "x \<in># p"
antisymp: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" assumes "merkle_interface h bo m" shows "antisymp bo"
hyps_2: fixes f :: "'c" assumes "t_ = Fault f"   and "f \<notin> F" shows "\<Gamma>\<turnstile> \<langle>strip_guards F c\<^sub>1_,Normal s_\<rangle> =n_\<Rightarrow> Fault f"
\<delta>_states_insert: fixes r :: "('a, 'b) ta_rule"   and \<delta> :: "('a, 'b) ta_rule set" shows "\<delta>_states (insert r \<delta>) = rule_states r \<union> \<delta>_states \<delta>"
Turn_into_distance_axioms: shows "Turn_into_distance extended_predist"
simps_3: fixes t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and ts :: "'a tree list" shows "sortedByWeight (t\<^sub>1 # t\<^sub>2 # ts) = (weight t\<^sub>1 \<le> weight t\<^sub>2 \<and> sortedByWeight (t\<^sub>2 # ts))"
rank_link: fixes t :: "('c, 'd) BinomialTree"   and t' :: "('c, 'd) BinomialTree" assumes "rank t = rank t'" shows "rank (link t t') = rank t + 1"
sides_9: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "dist b1 c1 = dist c2 b2"
unl_lift4: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'a"   and x :: "'f \<Rightarrow> 'b"   and y :: "'f \<Rightarrow> 'c"   and z :: "'f \<Rightarrow> 'd"   and zz :: "'f \<Rightarrow> 'e"   and w :: "'f" shows "(f<x, y, z, zz>) w \<equiv> f (x w) (y w) (z w) (zz w)"
s: shows "{(k, m). k + m \<le> n} \<subseteq> {0..n} \<times> {0..n}"
adt_wgraph_axioms: shows "adt_wgraph G.\<alpha>w G.\<alpha>g G.invar G_adj G_empty G_add_edge"
continuous_on_real: shows "continuous_on (UNIV - {\<infinity>, - \<infinity>}) real_of_ereal"
extend_set_def: fixes h :: "'d \<times> 'e \<Rightarrow> 'c"   and A :: "'d set" shows "extend_set h A = h ` (A \<times> UNIV)"
dist_quat_def: fixes x :: "quat"   and y :: "quat" shows "dist x y = norm (x - y)"
convex_prod: fixes P :: "'c \<Rightarrow> real \<Rightarrow> bool" assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}" shows "convex {x. \<forall>i\<in>Basis. P i (x \<bullet> i)}"
simps: fixes n :: "nat"   and a :: "'a"   and n_as :: "(nat \<times> 'a) list"   and i :: "nat" shows "j_to_jb_index ((n, a) # n_as) i = (if i < n then (0, i) else let rec = j_to_jb_index n_as (i - n) in (Suc (fst rec), snd rec))"
simps_9: fixes v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp One (Plus v va) = True"
z_def: shows "z \<equiv> prime_square_sum_nat_decomp p"
cont_compose: fixes c :: "'d \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'd" assumes "cont c"   and "cont f" shows "cont (\<lambda>x. c (f x))"
F_n_integral_pos: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat" assumes "f \<in> borel_measurable (F n)"   and "\<forall>w. 0 \<le> f w" shows "has_bochner_integral M f (\<Sum>w\<in>range (pseudo_proj_True n). prob (pseudo_proj_True n -` {w} \<inter> space M) * f w)"
supdoes: shows "R (Sup (fst_proj {(A, B). A \<le> lfp f \<and> B \<le> lfp g \<and> R A B})) (Sup (snd_proj {(A, B). A \<le> lfp f \<and> B \<le> lfp g \<and> R A B}))"
older_seniors_Min_coboundedI: fixes a :: "nat"   and x :: "nat"   and n :: "nat" assumes "a \<in> older_seniors x n" shows "Min (older_seniors x n) \<le> a"
connected_local: fixes S :: "'a set" shows "connected S = (\<nexists>e1 e2. openin (top_of_set S) e1 \<and> openin (top_of_set S) e2 \<and> S \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and> e1 \<noteq> {} \<and> e2 \<noteq> {})"
is_OK_returns_result_E: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog" assumes "h \<turnstile> ok f" obtains x :: "'z" where "h \<turnstile> f \<rightarrow>\<^sub>r x"
Min_in: fixes A :: "'c set" assumes "finite A"   and "A \<noteq> {}" shows "Min A \<in> A"
lg: shows "l2norm {a..b} (\<lambda>x. f x - g x) < e / 2"
pos: fixes D :: "'a set" shows "AE x in M. 0 \<le> f x * indicator D x"
neqE: fixes x :: "'b"   and y :: "'b" assumes "x \<noteq> y" obtains "y < x"   | "x < y"
enat_ltl_Suc: fixes n :: "nat"   and xs :: "'c llist" assumes "enat n < llength (ltl xs)" shows "enat (Suc n) < llength xs"
inf_top_left: fixes x :: "'d upred" shows "(true \<and> x) = x"