hcju/isabelleps huggingface dataset:
divide_le_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a b) zero = (less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b)"
Sigma_mset_empty_iff: fixes I :: "'a multiset"   and X :: "'a \<Rightarrow> 'b multiset" shows "(Sigma_mset I X = {#}) = (\<forall>i\<in>#I. X i = {#})"
infsetsum_cong_neutral: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = (0::'d)"   and "\<And>x. x \<in> B - A \<Longrightarrow> g x = (0::'d)"   and "\<And>x. x \<in> A \<inter> B \<Longrightarrow> f x = g x" shows "infsetsum f A = infsetsum g B"
SN_on_relto_Un: fixes S :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and X :: "'a set" assumes "(S\<^sup>* O (R \<union> R') O S\<^sup>*) `` X \<subseteq> X" shows "SN_on (S\<^sup>* O (R \<union> R') O S\<^sup>*) X = (SN_on ((R' \<union> S)\<^sup>* O R O (R' \<union> S)\<^sup>*) X \<and> SN_on (S\<^sup>* O R' O S\<^sup>*) X)"
neutr_eq_iff: fixes a :: "'a"   and b :: "'a" shows "(bot = sup a b) = (a = bot \<and> b = bot)"
pt_def: fixes D :: "'a \<times> 'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "pt D = lfp (dgr D)"
matrix_mul_diag_diag: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" shows "diag_mat f ** diag_mat g = diag_mat (\<lambda>i. f i * g i)"
assms_2: shows "I (\<alpha> m) \<sigma>0"
preserves_ide: fixes a :: "'a" assumes "A.ide a" shows "B.ide (G a)"
rel_pmf_antisym: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf" assumes "rel_pmf R p q"   and "rel_pmf R q p"   and "reflp R"   and "transp R"   and "antisymp R" shows "p = q"
simps_38: fixes f1 :: "nat \<Rightarrow> nat list \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat list \<Rightarrow> 'a"   and f3 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f4 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f5 :: "form \<Rightarrow> 'a"   and f6 :: "form \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "nat list" shows "(case NAtom x21 x22 of PAtom x xa \<Rightarrow> f1 x xa | NAtom x xa \<Rightarrow> f2 x xa | FConj x xa \<Rightarrow> f3 x xa | FDisj x xa \<Rightarrow> f4 x xa | FAll x \<Rightarrow> f5 x | FEx x \<Rightarrow> f6 x) = f2 x21 x22"
subgroup_def: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" shows "subgroup H G \<equiv> (H \<subseteq> carrier G \<and> (\<forall>x y. x \<in> H \<longrightarrow> y \<in> H \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H)) \<and> \<one>\<^bsub>G\<^esub> \<in> H \<and> (\<forall>x. x \<in> H \<longrightarrow> inv\<^bsub>G\<^esub> x \<in> H)"
that: fixes u' :: "value"   and env :: "(name, value) fmap"   and pat :: "Term_Class.term"   and rhs :: "sterm" assumes "fmlookup \<Gamma> n = Some u'"   and "vfind_match cs u' = Some (env, pat, rhs)"   and "\<Gamma> ++\<^sub>f env \<turnstile>\<^sub>v rhs \<down> v" shows "thesis"
rsquarefree_def: fixes p :: "'a poly" shows "rsquarefree p = (p \<noteq> 0 \<and> (\<forall>a. Polynomial.order a p = 0 \<or> Polynomial.order a p = 1))"
simps_3: fixes g :: "char list \<Rightarrow> bool"   and h :: "'b fm \<Rightarrow> bool"   and p :: "'b fm"   and q :: "'b fm" shows "eval g h (p \<^bold>\<or> q) = (eval g h p \<or> eval g h q)"
emeasure_point_measure_finite2: fixes X :: "'a set"   and A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "X \<subseteq> A"   and "finite X" shows "emeasure (point_measure A f) X = sum f X"
rel_funD: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j"   and x :: "'g"   and y :: "'h" assumes "(A ===> B) f g"   and "A x y" shows "B (f x) (g y)"
exec_step_ind_Getstatic_Done: fixes D' :: "char list"   and b :: "staticb"   and t :: "ty"   and P :: "jvm_prog"   and D :: "char list"   and F :: "char list"   and C :: "char list"   and ics :: "init_call_status"   and sh :: "sheap"   and sfs :: "sfields"   and v :: "val"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "(D', b, t) = field P D F"   and "P \<turnstile> C has F,Static:t in D"   and "\<forall>Cs. ics \<noteq> Called Cs"   and "sh D' = \<lfloor>(sfs, Done)\<rfloor>"   and "v = the (sfs F)" shows "exec_step_ind (StepI (Getstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (v # stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
all_countable_subset_image: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and P :: "'b set \<Rightarrow> bool" shows "(\<forall>T. countable T \<and> T \<subseteq> f ` S \<longrightarrow> P T) = (\<forall>T. countable T \<and> T \<subseteq> S \<longrightarrow> P (f ` T))"
Suc_1: fixes i :: "nat" assumes "k = j - i"   and "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i < j"   and "j < length l"   and "x \<in> lvars (\<T> (l ! i))"   and "x \<in> rvars (\<T> (l ! j))" shows "\<exists>k\<ge>i. k < j \<and> k + 1 < length l \<and> x \<in> lvars (\<T> (l ! k)) \<and> x \<in> rvars (\<T> (l ! (k + 1)))"
Rel_def: shows "Rel = indRelSTEQ SRel TRel - _Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q)"
mrw_values_WriteMemD: fixes ad :: "'a"   and al :: "addr_loc"   and v' :: "'a val"   and obs :: "('a, 'b) obs_event action list"   and P :: "'c prog"   and vs0 :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option" assumes "NormalAction (WriteMem ad al v') \<in> set obs" shows "\<exists>v. mrw_values P vs0 obs (ad, al) = \<lfloor>(v, True)\<rfloor>"
generate_ring_min_subring1: fixes H :: "'a set"   and E :: "'a set" assumes "H \<subseteq> carrier R"   and "subring E R"   and "H \<subseteq> E" shows "generate_ring R H \<subseteq> E"
atLeastAtMost_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and u :: "'a" shows "ord.atLeastAtMost less_eq l u \<equiv> ord.atLeast less_eq l \<inter> ord.atMost less_eq u"
tendsto_mult_right_zero: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and c :: "'c" assumes "(f \<longlongrightarrow> (0::'c)) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> (0::'c)) F"
s_def: shows "s = {(x, y). (f x, f y) \<in> r2 \<and> (f x = f y \<longrightarrow> (x, y) \<in> s2)}"
min_enat_simps_1: fixes m :: "nat"   and n :: "nat" shows "min (enat m) (enat n) = enat (min m n)"
i_eq: shows "to_nat (from_nat i) = to_nat (from_nat i)"
H1: fixes x1b :: "nat set"   and x2a :: "('a, int mpoly) fmap"   and b :: "int mpoly multiset"   and xa :: "int mpoly" assumes "vars spec \<subseteq> x1b"   and "\<Union> (vars ` set_mset (ran_m A)) \<subseteq> x1b"   and "\<Union> (vars ` set_mset (ran_m x2a)) \<subseteq> x1b"   and "restricted_ideal_to\<^sub>I x1b b \<subseteq> restricted_ideal_to\<^sub>I x1b (ran_m x2a)"   and "xa \<in> restricted_ideal_to\<^sub>I (\<Union> (vars ` set_mset (ran_m A)) \<union> vars spec) b" shows "xa \<in> restricted_ideal_to\<^sub>I (\<Union> (vars ` set_mset (ran_m A)) \<union> vars spec) (ran_m x2a)"
mat_map_def: fixes f :: "'c \<Rightarrow> 'c" shows "mat_map f = map (vec_map f)"
set_update_distinct: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "distinct xs"   and "n < length xs" shows "set (xs[n := x]) = insert x (set xs - {xs ! n})"
ereal_minus_2: fixes r :: "real" shows "- \<infinity> - ereal r = - \<infinity>"
append: shows "rev [0..<Suc n] = rev [1..<Suc n] @ [0]"
abs_ereal_one: shows "\<bar>1\<bar> = 1"
Or_append_syntactic: fixes xs :: "'a ltl list"   and ys :: "'a ltl list" assumes "xs \<noteq> []" shows "Or (xs @ ys) = Or (Or xs # ys)"
assms_3: fixes x :: "'a" assumes "P x"   and "Q x" shows "R x"
ff_dghm_op_dghm_is_ff_dghm: shows "op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
A_3: shows "properfactor (Ring_Divisibility.mult_of R) b a"
exhaust: fixes y :: "'a textbuffer" obtains x1 :: "'a list" and x2 :: "'a list" where "y = BUF x1 x2"
CollectionPartsItemTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and x :: "'a collection_literal_part_expr"   and y :: "'a collection_literal_part_expr"   and xs :: "'a collection_literal_part_expr list"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>C x # y # xs : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "\<tau> = \<tau>' \<squnion> \<sigma>" and "\<Gamma> \<turnstile>\<^sub>P x : \<tau>'" and "\<Gamma> \<turnstile>\<^sub>C y # xs : \<sigma>"
inf_commute: fixes inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
ordermap_surj: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "elts (ordertype A r) \<subseteq> ordermap A r ` A"
magQ_def: fixes x :: "'c['d, 'e]" shows "\<lbrakk>x\<rbrakk>\<^sub>Q = mag (fromQ x)"
Ando_undef: fixes \<phi> :: "fml option"   and \<psi> :: "fml option" shows "(Ando \<phi> \<psi> = undeff) = (\<phi> = undeff \<or> \<psi> = undeff)"
weakPsiCongSymI: fixes Prop :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d"   and Q :: "'d"   and \<Psi> :: "'b"   and C :: "'d \<Rightarrow> ('a, 'b, 'c) psi" assumes "Prop P Q"   and "\<And>P Q. Prop P Q \<Longrightarrow> Prop Q P"   and "\<And>P Q. Prop P Q \<Longrightarrow> \<Psi> \<rhd> C P \<approx> C Q"   and "\<And>P Q. Prop P Q \<Longrightarrow> \<Psi> \<rhd> C P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> C Q" shows "\<Psi> \<rhd> C P \<doteq> C Q"
separation_openin_Un_gen: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "separatedin X S T = (S \<subseteq> topspace X \<and> T \<subseteq> topspace X \<and> disjnt S T \<and> openin (subtopology X (S \<union> T)) S \<and> openin (subtopology X (S \<union> T)) T)"
natural_transformation_axioms: shows "natural_transformation J C D D' \<tau>"
add_pos_nonneg: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "{#} \<subset># a"   and "{#} \<subseteq># b" shows "{#} \<subset># a + b"
nAct_Null: fixes n :: "enat"   and t :: "cnf llist"   and c :: "'id" assumes "n \<le> llength t"   and "\<langle>c #\<^bsub>n\<^esub>t\<rangle> = 0" shows "\<forall>i. enat i < n \<longrightarrow> \<not> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"
cyclic_rotation: fixes cs :: "'b list"   and n :: "nat" assumes "cycle cs" shows "map (cycle_of_list cs ^^ n) cs = rotate n cs"
bin_all_nemp: fixes ws :: "'a list list" assumes "ws \<in> lists {u\<^sub>0, u\<^sub>1}"   and "concat ws = \<epsilon>" shows "ws = \<epsilon>"
cancel_semigroup_axioms: shows "cancel_semigroup f\<^sub>b"
sandwich_apply: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "sandwich A *\<^sub>V B = A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L A*"
sim: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F"   and "F \<triangleright> \<Gamma> \<union> AX10 \<turnstile>\<^sub>H G" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H G"
c_fst_def: fixes u :: "nat" shows "c_fst u = u - sf (c_sum u)"
weakBisimCoinductAux: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto><(X \<union> weakBisim)> Q \<and> (Q, P) \<in> X \<union> weakBisim" shows "P \<approx> Q"
take_bit_eq_0_iff: fixes n :: "nat"   and a :: "'a" shows "(take_bit n a = (0::'a)) = ((2::'a) ^ n dvd a)"
le_sup_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y \<le> z) = (x \<le> z \<and> y \<le> z)"
zip_last: fixes r1 :: "'d list"   and r2 :: "'e list" assumes "0 < length r1"   and "length r1 = length r2" shows "last (r1 || r2) = (last r1, last r2)"
a0n_inv1_niagreeD: fixes d :: "'b"   and x :: "'b a0n_state"   and h :: "agent list" assumes "d \<notin> corrupted x"   and "0 < signals x (Commit h d)"   and "x \<in> a0n_inv1_niagree"   and "set h \<subseteq> good" shows "0 < signals x (Running h d)"
vdomain_vempty: shows "\<D>\<^sub>\<circ> 0 = 0"
constructive_security_obsf_sim_cong: fixes real_resource :: "('y + 'z, 'aa + 'ab) resource"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real :: "('y, 'aa) \<I>"   and \<I>_ideal :: "('ac, 'ad) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and \<A> :: "('y + 'z, 'aa + 'ab) distinguisher_obsf"   and adv :: "real"   and sim' :: "('y, 'aa, 'ac, 'ad) converter" assumes "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common \<A> adv"   and "\<I>_real,\<I>_ideal \<turnstile>\<^sub>C sim \<sim> sim'"   and "pfinite_converter \<I>_real \<I>_ideal sim'" shows "constructive_security_obsf real_resource ideal_resource sim' \<I>_real \<I>_ideal \<I>_common \<A> adv"
Q_cases: shows "Q \<in> insert_into_member elem P ` P \<or> Q = insert {elem} P"
new_syz_sigs_alt_spp: fixes ss :: "'t list"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" shows "new_syz_sigs ss bs p = new_syz_sigs_spp ss (map spp_of bs) (app_pair spp_of p)"
sep_disj_addD1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x ## y + z"   and "y ## z" shows "x ## y"
poly_tendsto_2: fixes p :: "real poly"   and x :: "real" shows "(poly p \<longlongrightarrow> poly p x) (at_left x)"
sum_is_subspace: fixes W1 :: "'a vec set"   and W2 :: "'a vec set" assumes "subspace class_ring W1 V"   and "subspace class_ring W2 V" shows "subspace class_ring (subspace_sum W1 W2) V"
NCSaux3: fixes p :: "('b, 'c) Combinators list"   and a :: "'b"   and b :: "'b"   and l :: "'b set list"   and w :: "'c"   and u :: "'b"   and v :: "'b" assumes "noDenyAll p"   and "{a, b} \<in> set l"   and "all_in_list p l"   and "singleCombinators p"   and "FWNormalisationCore.sorted (AllowPortFromTo a b w # p) l"   and "{a, b} \<noteq> firstList p"   and "DenyAllFromTo u v \<in> set p" shows "{a, b} \<noteq> {u, v}"
lbisimulation_refl: shows "lbisimulation Id ga_automaton ga_automaton"
list_product_iff: fixes x :: "'b list"   and A :: "'b list set"   and B :: "'b list set" shows "(x \<in> A \<otimes>\<^sub>l B) = (\<exists>a b. a \<in> A \<and> b \<in> B \<and> x = a @ b)"
has_bochner_integral_diff: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c"   and g :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "has_bochner_integral M f x"   and "has_bochner_integral M g y" shows "has_bochner_integral M (\<lambda>x. f x - g x) (x - y)"
Y_nonzero: fixes y :: "(real, 'n) Finite_Cartesian_Product.vec"   and i :: "'n" assumes "y \<in> Y" shows "y $h i \<noteq> 0"
product_set_def: fixes X :: "'b fset set set" shows "\<Otimes> X = Finite_Set.fold (\<otimes>) {{||}} X"
inverse_arrowsI: fixes g :: "'a1"   and f :: "'a1" assumes "C1.ide (g \<cdot>\<^sub>1 f)"   and "C1.ide (f \<cdot>\<^sub>1 g)" shows "C1.inverse_arrows f g"
simps_2: fixes var :: "nat"   and p :: "real mpoly"   and A :: "real mpoly list"   and L :: "atom list"   and F :: "atom fm list" shows "qe_eq_repeat_helper var (p # A) L F = Or (And (Neg (split_p var p)) (elimVar var (map Eq (p # A) @ L) F (Eq p))) (And (split_p var p) (qe_eq_repeat_helper var A L F))"
h1: shows "monomials p1 \<inter> monomials p2 = {}"
\<R>_sym: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option" shows "sym (\<R> \<Gamma>)"
neq: shows "obs \<sigma> k1 \<noteq> obs \<sigma>' k1"
red_lcl_add: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and l\<^sub>0 :: "char list \<Rightarrow> val option" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "P \<turnstile> \<langle>e,(h, l\<^sub>0 ++ l)\<rangle> \<rightarrow> \<langle>e',(h', l\<^sub>0 ++ l')\<rangle>"
strictly_subsumes_has_minimum: fixes CC :: "'a literal multiset set" assumes "CC \<noteq> {}" shows "\<exists>C\<in>CC. \<forall>D\<in>CC. \<not> strictly_subsumes D C"
vertical_chop_empty_down: shows "\<forall>v. \<exists>u. v=v--u"
trancl_rbt_impl_def: shows "trancl_rbt_impl = trancl_impl rm_succ (\<lambda>as bs. rs.union bs (rs.from_list as)) rs.memb (rs.empty ())"
rel_includes_TRel_modulo_preorder: fixes TRel :: "('a \<times> 'a) set"   and Rel :: "(('b, 'a) Proc \<times> ('b, 'a) Proc) set" assumes "trans TRel" shows "((\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+)) = (TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel))"
d_subid: fixes x :: "('a \<times> 'a set) set" assumes "mrel_c_lattice.d x = x" shows "x \<subseteq> 1\<^sub>\<sigma>"
V_subset_W: fixes i :: "nat" assumes "i \<in> N" shows "closure (V i) \<subseteq> W i"
A: shows "sinks I D (D x) (x # xs) = insert (D x) (sinks I D (D x) xs)"
zero_left: fixes b :: "'b" shows "blinfun_apply 0 b = (0::'a)"
\<beta>_simps_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<iota> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and w' :: "'c"   and \<theta>' :: "'c"   and \<beta> :: "'c" assumes "uw\<theta>w'\<theta>'\<beta> V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>' \<beta>" shows "partial_magma.arr V \<beta>"
index_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "i < nr"   and "j < nc" shows "mat nr nc f $$ (i, j) = f (i, j)"
simps_5: fixes c1 :: "('a, 'atom) com"   and c2 :: "('a, 'atom) com" shows "SC_discr (c1 | c2) = (SC_discr c1 \<and> SC_discr c2)"
other: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg"   and \<sigma>' :: "nat \<Rightarrow> state" assumes "P \<sigma> m"   and "other Q {i} \<sigma> \<sigma>'" shows "P \<sigma>' m"
cipher_queriesE: fixes x :: "'a"   and ose :: "'b \<Rightarrow> ('c \<times> 'd \<times> 'a list) option" assumes "x \<in> ind_cca2.cipher_queries ose" obtains n :: "'b" and ek :: "'c" and dk :: "'d" and ciphers :: "'a list" where "ose n = Some (ek, dk, ciphers)" and "x \<in> set ciphers"
lift_Suc_mono_less_iff: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat"   and m :: "nat" assumes "class.order less_eq less"   and "\<And>n. less (f n) (f (Suc n))" shows "less (f n) (f m) = (n < m)"
simple_extension_subring_incl: fixes K' :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "subring K' R"   and "K \<subseteq> K'"   and "x \<in> K'" shows "simple_extension K x \<subseteq> K'"
some_non_bottom_element: shows "some_non_bottom_element \<noteq> \<bottom>"
m: fixes p :: "'a"   and q :: "'a" shows "((p, q) \<in> m) = p \<parallel> q"
assms_1: shows "s\<^sub>o - s\<^sub>e \<le> distance0"
mwb_lens_axioms_def: fixes x :: "'b \<Longrightarrow> 'c" shows "mwb_lens_axioms x \<equiv> \<forall>\<sigma> v u. put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u"
supp_2: fixes x1 :: "name" shows "supp (BoundOutputS x1) = supp x1"
assms_2: shows "gen P I \<inter> gen P J \<noteq> {}"
fieldE_1: fixes R :: "('e, 'f) ring_scheme" assumes "field R" shows "cring R"
assms_1: shows "(c, s) \<rightarrow>c (c', s')"
approximable_on_division: fixes e :: "real"   and d :: "'a set set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "0 \<le> e"   and "d division_of cbox a b"   and "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i" obtains g :: "'a \<Rightarrow> 'b" where "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" and "g integrable_on cbox a b"
ext: fixes g :: "'a1"   and f :: "'a1" assumes "g \<cdot>\<^sub>1 f \<noteq> C1.null" shows "C1.seq g f"
rad_ideal_def: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" shows "rad_ideal R I = {a \<in> carrier R. nilpotent (R /\<^sub>r I) (pj R I a)}"
to_int_mod_ring_rebase: fixes x :: "'p mod_ring" shows "to_int_mod_ring (@x) = to_int_mod_ring x"
utp_sym_eval_def: fixes \<Gamma> :: "'a \<Rightarrow> 'a"   and P :: "'a hrel" shows "\<Gamma> \<Turnstile> P = \<langle>\<Gamma>\<rangle>\<^sub>a ;; P"
B: fixes ys' :: "'a list"   and zs' :: "'a list" shows "xs \<cong> {ys', zs', P} \<longrightarrow> filter Q xs \<cong> {filter Q ys', filter Q zs', P}"
hashcode_iarray_def: fixes a :: "'a iarray" shows "hashcode a = foldl (\<lambda>h v. h * 33 + hashcode v) 0 (IArray.list_of a)"
UNIV_4: shows "UNIV = insert 1 (insert 2 (insert 3 (skip 4)))"
totient_primorial_less: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "\<forall>\<^sub>F x in at_top. real (totient (primorial x)) < (1 + \<epsilon>) * third_mertens_const * real (primorial x) / ln (ln (real (primorial x)))"
seq_nil_left: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice" assumes "sequential (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (a::'b::refinement_lattice) = a"
eq_lifted': fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> {{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}"   and "B \<in> {{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}"   and "(SOME y. y \<in> A, SOME y. y \<in> B) \<in> r"   and "(SOME y. y \<in> B, SOME y. y \<in> A) \<in> r" shows "A = B"
step_11: shows "InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
disj_matrices_commute: fixes A :: "'a matrix"   and B :: "'a matrix" shows "disj_matrices A B = disj_matrices B A"
ide_def: fixes a :: "'a" shows "B.ide a \<equiv> a \<cdot> a \<noteq> B.null \<and> (\<forall>f. (f \<cdot> a \<noteq> B.null \<longrightarrow> f \<cdot> a = f) \<and> (a \<cdot> f \<noteq> B.null \<longrightarrow> a \<cdot> f = f))"
Kronecker_delta_def: fixes i :: "nat"   and j :: "nat" shows "\<delta>\<^bsub>i j\<^esub> = (if i = j then 1 else 0)"
that_2: fixes new :: "qtname \<times> methd" assumes "methd G C sig = Some new"   and "G \<turnstile> qmdecl sig new overrides\<^sub>S qmdecl sig old" shows "thesis"