hcju/isabelleps huggingface dataset:
uniformity_dist: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter" assumes "class.uniformity_dist dist uniformity" shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
mset_pos_neg_dual: fixes a :: "'a zmultiset"   and b :: "'a zmultiset" shows "mset_pos a + mset_pos b + (mset_neg a - mset_pos b) + (mset_neg b - mset_pos a) = mset_neg a + mset_neg b + (mset_pos a - mset_neg b) + (mset_pos b - mset_neg a)"
callee_invariant: fixes y :: "'a"   and s' :: "('b, 'a) resource"   and s :: "('b, 'a) resource"   and x :: "'b"   and \<I> :: "('b, 'a) \<I>" assumes "(y, s') \<in> set_spmf (run_resource s x)"   and "\<I> \<turnstile>res s \<surd>"   and "x \<in> outs_\<I> \<I>" shows "\<I> \<turnstile>res s' \<surd>"
inv_a_2: shows "inv a \<otimes> a = \<one>"
simps: fixes a :: "'b \<Rightarrow> bool list option" shows "invariant a = (\<exists>\<sigma>. a = \<sigma> \<and> finite (dom \<sigma>) \<and> length ` ran \<sigma> \<subseteq> {len})"
commute: fixes a :: "'c"   and b :: "'c" shows "gcd a b = gcd b a"
pos_less_divideC_eq: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> '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 c :: "complex"   and a :: "'b"   and b :: "'b" assumes "class.ordered_complex_vector scaleR scaleC plus zero minus uminus less_eq less"   and "0 < c" shows "less a (scaleC (inverse c) b) = less (scaleC c a) b"
l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes 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"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool" shows "l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs create_element known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_element known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr"
Blap: shows "B \<inter> \<Union> S \<noteq> {}"
HFinite_inverse2: fixes x :: "'b star" assumes "x \<in> HFinite - Infinitesimal" shows "inverse x \<in> HFinite"
null_eqI: fixes n :: "'b" assumes "\<And>f. n \<cdot>\<^sub>B f = n \<and> f \<cdot>\<^sub>B n = n" shows "n = B.null"
det_col_sum: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and a :: "'e \<Rightarrow> ('c, 'd) vec"   and S :: "'e set" shows "Square_Matrix.det (upd_col M i (sum a S)) = (\<Sum>s\<in>S. Square_Matrix.det (upd_col M i (a s)))"
domh: shows "dom h = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k"
one_cinner_a_scaleC_one: fixes a :: "'b" shows "\<langle>1::'b, a\<rangle> *\<^sub>C (1::'b) = a"
inf_llist_raw_def: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "inf_llist_raw f n = Yield (f n) (Suc n)"
image_constant: fixes x :: "'a"   and A :: "'a set"   and c :: "'b" assumes "x \<in> A" shows "(\<lambda>x. c) ` A = {c}"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "P \<langle>\<rangle>"   and "\<And>l x r. \<lbrakk>P l; P r\<rbrakk> \<Longrightarrow> P \<langle>l, x, r\<rangle>" shows "P a0"
rk_ax_singleton: fixes P :: "Points" shows "1 \<le> rk {P}"
weakBisimWeakCoinduct: 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><X> Q"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto><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> Q"
REF_R_BRK: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R"   and "it \<noteq> {}"   and "it' \<noteq> {}"   and "\<not> c \<sigma>"   and "\<not> c' \<sigma>'" shows "(\<sigma>, \<sigma>') \<in> R'"
assms: fixes x :: "'a" assumes "x \<in> keys t" shows "poly_deg (f x) = 0"
prime_elem_imp_not_one: fixes x :: "'a" assumes "ASSUMPTION (prime_elem x)" shows "x \<noteq> (1::'a)"
inf_compl: fixes x :: "'a" shows "x \<sqinter> - x = \<bottom>"
sasp_op_to_strips_effect_consistent_1: fixes op :: "('c \<times> 'd) strips_operator"   and \<Psi> :: "('c, 'd) sas_plus_problem"   and op' :: "('c, 'd) sas_plus_operator"   and v :: "'c"   and a :: "'d" assumes "op = \<phi>\<^sub>O \<Psi> op'"   and "op' \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "is_valid_operator_sas_plus \<Psi> op'" shows "(v, a) \<in> set (add_effects_of op) \<longrightarrow> (v, a) \<notin> set (delete_effects_of op)"
normalize_det: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and expr :: "'a expr"   and expr\<^sub>1 :: "'a expr"   and expr\<^sub>2 :: "'a expr" assumes "\<Gamma> \<turnstile> expr \<Rrightarrow> expr\<^sub>1"   and "\<Gamma> \<turnstile> expr \<Rrightarrow> expr\<^sub>2" shows "expr\<^sub>1 = expr\<^sub>2"
simps_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "monoidal_language C" shows "monoidal_language.Ide C \<^bold>\<langle>f\<^bold>\<rangle> = partial_magma.ide C f"
assms_3: shows "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'"
varset_action_def: fixes a :: "('c, bool) fmap \<times> ('c, bool) fmap"   and varset :: "'c set" shows "varset_action a varset \<equiv> fmdom' (snd a) \<subseteq> varset"
weakSimI: fixes Q :: "ccs"   and P :: "ccs"   and Rel :: "(ccs \<times> ccs) set" assumes "\<And>\<alpha> Q'. Q \<longmapsto>\<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto><Rel> Q"
hsize_apps: fixes t :: "('a, 'b) tm"   and ts :: "('a, 'b) tm list" shows "hsize (apps t ts) = hsize t + sum_list (map hsize ts)"
singleton_conv: fixes a :: "'d" shows "{x. x = a} = {a}"
neg_0_le_iff_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (uminus a) = less_eq a zero"
A5: fixes x :: "i"   and y :: "i"   and z :: "i" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
l1_def: shows "l1 = fst (case z' of (t, x) \<Rightarrow> (t mod \<bar>k\<bar>, int x mod \<bar>k\<bar>))"
nxt_monotone: fixes b :: "nat"   and b' :: "nat" assumes "nxt b \<down>= b'" shows "e_length b < e_length b'"
assms_2: shows "{a, b} \<subseteq> arcs G"
distinct_1: fixes x21 :: "'a"   and x22 :: "'a msllist" shows "MSLNil \<noteq> MSLCons x21 x22"
subseqs_of_length_Suc_Cons: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "{ys. subseq_of_length (Suc n) (x # xs) ys} = (#) x ` {ys. subseq_of_length n xs ys} \<union> {ys. subseq_of_length (Suc n) xs ys}"
lookup_pair_eq_value: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and v :: "'c"   and k :: "'a" assumes "oalist_inv_raw x xs"   and "v \<noteq> (0::'c)" shows "(lookup_pair x xs k = v) = ((k, v) \<in> set xs)"
that: fixes b :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> A" shows "thesis"
invertible_matrix_inv: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "invertible A" shows "invertible (matrix_inv A)"
another_point: fixes A :: "'p" shows "\<exists>B. A \<noteq> B"
view_partition_def: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" shows "view_partition P D R \<equiv> \<forall>u\<in>range D. equiv (traces P) (R u)"
cancel_at_def: fixes i :: "nat"   and l :: "(bool \<times> 'b) list" shows "cancel_at i l = take i l @ drop (2 + i) l"
nf_ACI_D_1: fixes \<phi> :: "('a, 'b) aformula" assumes "nf_ACI \<phi>" shows "sorted (disjuncts_list \<phi>)"
member_rec_2: fixes y :: "'x" shows "List.member [] y = False"
simps_22: fixes x61 :: "nat"   and x62 :: "recf" shows "Mn x61 x62 \<noteq> S"
Ord_vsingleton_in_Vset_succI: fixes \<alpha> :: "V"   and a :: "V" assumes "Ord \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>" shows "set {a} \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
diff_lem: fixes A :: "'b set option"   and A' :: "'b set option"   and b :: "'b" assumes "A \<sqsubseteq> A'" shows "A \<ominus> b \<sqsubseteq> A' \<ominus> b"
le_SUP_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 bot :: "'b"   and top :: "'b"   and x :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq x (Sup (f ` A)) = (\<forall>y. less y x \<longrightarrow> (\<exists>i\<in>A. less y (f i)))"
lincomb_list_index: fixes i :: "nat"   and Xs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "i < n"   and "set Xs \<subseteq> carrier_vec n" shows "lincomb_list c Xs $ i = (\<Sum>j = 0..<length Xs. c j * Xs ! j $ i)"
inject: fixes x1 :: "'a fset"   and x2 :: "'a"   and x3 :: "'b set"   and y1 :: "'a fset"   and y2 :: "'a"   and y3 :: "'b set" shows "(Antecedent x1 x2 x3 = Antecedent y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
has_finite_products: fixes I :: "nat set" assumes "finite I" shows "has_products I"
assms_1: fixes x :: "'a ttree" shows "paths (f x) = \<Union> (t ` paths x)"
add_new_ts_mmsaux_def: fixes args :: "Monitor.args"   and nt :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'b option list set) queue \<times> (nat \<times> 'b option list set) queue \<times> ('b option list, nat) mapping \<times> ('b option list, nat) mapping" shows "add_new_ts_mmsaux args nt aux = add_new_ts_mmsaux' args nt (shift_end args nt aux)"
hyps_3: shows "aux = compl gs bs (ps -- sps) sps (snd data)"
ResTau: fixes x :: "name"   and P :: "pi" shows "<\<nu>x>\<tau>.P \<equiv>\<^sub>e \<tau>.(<\<nu>x>P)"
map_closed: fixes a :: "'a" assumes "a \<in> S" shows "\<alpha> a \<in> T"
gram_schmidt_fs_lin_indpt_axioms: shows "gram_schmidt_fs_lin_indpt n (map of_int_hom.vec_hom fs)"
tauPres: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "\<tau>.P \<sim> \<tau>.Q"
assms_2: shows "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<M>"
in_OAlist_tc_sorted_domain_iff_lookup: fixes k :: "'b"   and xs :: "('b, 'c) oalist_tc" shows "(k \<in> set (OAlist_tc_sorted_domain xs)) = (OAlist_tc_lookup xs k \<noteq> (0::'c))"
join_above2: fixes x :: "'b"   and y :: "'b" assumes "compatible x y" shows "y \<sqsubseteq> x \<squnion> y"
AG_step: fixes y :: "'a"   and z :: "'a"   and s :: "'a set" assumes "y \<rightarrow>\<^sub>i z"   and "y \<in> AG s" shows "z \<in> AG s"
respection: shows "rel_respects_binary_pred (indRelSTEQ SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
sim_rI: fixes M :: "('a, 'd, 'b) KripkeStructure"   and f :: "'b \<Rightarrow> 'c"   and M' :: "('a, 'd, 'c) KripkeStructure" assumes "\<And>a u v'. \<lbrakk>u \<in> worlds M; (f u, v') \<in> relations M' a\<rbrakk> \<Longrightarrow> \<exists>v. (u, v) \<in> relations M a \<and> f v = v'" shows "sim_r M M' f"
less_Suc_eq_le: fixes m :: "nat"   and n :: "nat" shows "(m < Suc n) = (m \<le> n)"
Diagonalize_Diag: fixes t :: "'c local.term" assumes "Diag t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = t"
set_insort_key: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and f :: "'l \<Rightarrow> 'k"   and x :: "'l"   and xs :: "'l list" assumes "class.linorder less_eq less" shows "set (linorder.insort_key less_eq f x xs) = insert x (set xs)"
axioms_3: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "ring R addition multiplication zero unit" shows "ring_axioms R addition multiplication"
projection_def: fixes M :: "'a set" shows "projection M \<equiv> SOME \<pi>. is_projection_on \<pi> M"
assms_2: shows "\<turnstile> B \<longrightarrow> C"
A_2: fixes p :: "real poly"   and x :: "real" assumes "p \<in> ps"   and "p \<noteq> 0"   and "poly p x = 0" shows "x < u"
fv_simps_1: fixes x1 :: "'a" shows "fv (Var x1) = {x1}"
embed_measure_count_space': fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "embed_measure (count_space A) f = count_space (f ` A)"
prod_ord_pair_alt: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and P :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool" assumes "oalist_inv_raw x xs"   and "oalist_inv_raw x ys" shows "prod_ord_pair x P xs ys = (\<forall>k\<in>fst ` set xs \<union> fst ` set ys. P k (lookup_pair x xs k) (lookup_pair x ys k))"
Units_l_inv: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
get_IFS_def: fixes M :: "'a SecurityInvariant list" shows "TopoS_Composition_Theory_impl.get_IFS M \<equiv> filter implc_isIFS M"
finite_S1: fixes n :: "nat" shows "finite ({..<n} \<times> {..<n})"
skip: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[], t\<rangle> \<Rightarrow>\<^sub>s t"
faithful_functors_compose: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c" assumes "faithful_functor A B F"   and "faithful_functor B C G" shows "faithful_functor A C (G \<circ> F)"
rel_star_contl: fixes X :: "('b \<times> 'c) set"   and Y :: "('c \<times> 'c) set" shows "X O Y\<^sup>* = (\<Union>n. X O rel_dioid.power Y n)"
Carmichael_nonzero: fixes n :: "nat" shows "Carmichael n \<noteq> 0"
unucleus_def: fixes f :: "'b \<Rightarrow> 'b" shows "unucleus f = (nucleus f \<and> (1::'b) \<le> f (1::'b))"
pathstart_linepath: fixes a :: "'a"   and b :: "'a" shows "pathstart (linepath a b) = a"
simps: fixes k :: "nat" shows "e' k = (if k = 0 then 1 else if n < k then 0 else - (\<Sum>i = 1..k. (- 1) ^ i * e' (k - i) * f i) / of_nat k)"
gen_bg_impl_rel_sv: fixes Rm :: "('a \<times> 'b) set"   and Rf :: "('c \<times> 'd set) set" assumes "single_valued Rm"   and "single_valued Rf" shows "single_valued (\<langle>Rm, Rf\<rangle>gen_bg_impl_rel_eext)"
matches_def: fixes t :: "'a"   and u :: "'a" shows "(t \<lesssim> u) = (\<exists>env. subst t env = u)"
last_message_hold_init_nth: fixes i :: "nat"   and xs :: "'b message_af list"   and m :: "'b message_af" assumes "i < length xs" shows "last_message_hold_init xs m ! i = last_message (m # xs \<down> Suc i)"
p1_def: shows "p1 = quadratic_part_1 var a b d (Eq p)"
eta_postponement': fixes s :: "dB"   and t :: "dB"   and u :: "dB" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* t"   and "t => u" shows "\<exists>t'. s => t' \<and> t' \<rightarrow>\<^sub>\<eta>\<^sup>* u"
tail_add_arc: fixes a :: "'b" shows "tail (add_arc a) = tail G"
ideal_prod_r_distr: fixes I :: "'a set"   and J :: "'a set"   and K :: "'a set" assumes "ideal I R"   and "ideal J R"   and "ideal K R" shows "I \<cdot> (J <+> K) = I \<cdot> J <+> I \<cdot> K"
product_of_first_summand: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CEM"   and "O x y"   and "\<not> O x z" shows "P (x \<otimes> (y \<oplus> z)) (x \<otimes> y)"
\<sigma>_valid: shows "\<forall>j. \<sigma> j \<in> qs \<and> (\<forall>\<psi>\<in>old (\<sigma> j). suffix j \<xi> \<Turnstile>\<^sub>r \<psi>) \<and> (\<forall>\<psi>\<in>next (\<sigma> j). suffix j \<xi> \<Turnstile>\<^sub>r X\<^sub>r \<psi>) \<and> {\<eta>. \<exists>\<mu>. \<mu> U\<^sub>r \<eta> \<in> old (\<sigma> j) \<and> suffix j \<xi> \<Turnstile>\<^sub>r \<eta>} \<subseteq> old (\<sigma> j)"
tree_def: fixes g :: "'a ugraph" shows "tree g \<equiv> Undirected_Graph.connected g \<and> cycle_free g"
af_G_letter_opt_sat_core: fixes \<G> :: "'a ltl set"   and \<phi> :: "'a ltl"   and \<nu> :: "'a set" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P \<phi>" shows "\<G> \<Turnstile>\<^sub>P af_G_letter_opt \<phi> \<nu>"
ZfunI: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>r. 0 < r \<Longrightarrow> \<forall>\<^sub>F x in F. norm (f x) < r" shows "Zfun f F"
domain: fixes a :: "'a" shows "Domainp r a = (a \<in> A)"
iftrue: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<iota> :: "nat"   and c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom" assumes "BMap (E b m) = True" shows "\<langle>if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi,m\<rangle> \<rightarrow>\<lhd>[]\<rhd> \<langle>Some c1,m\<rangle>"
distinct_9: shows "hp_IdleInit \<noteq> hp_InitMark"
perm_1: fixes pi :: "(name \<times> name) list"   and x1 :: "'d" shows "pi \<bullet> \<langle>\<epsilon>, x1\<rangle> = \<langle>\<epsilon>, pi \<bullet> x1\<rangle>"
poincare_line_x_axis: fixes x :: "complex_homo"   and y :: "complex_homo" assumes "x \<in> unit_disc"   and "y \<in> unit_disc"   and "x \<in> circline_set x_axis"   and "y \<in> circline_set x_axis"   and "x \<noteq> y" shows "poincare_line x y = x_axis"
preallocated_def: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "preallocated h \<equiv> \<forall>C\<in>sys_xcpts. \<exists>fs. h (addr_of_sys_xcpt C) = \<lfloor>(C, fs)\<rfloor>"
ide_hcomp: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "C.ide \<nu>"   and "C.ide \<mu>"   and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" shows "C.ide (\<nu> \<star>\<^sub>C \<mu>)"
simps_1: fixes d :: "'b"   and c :: "'b" shows "injval (Atom d) c Void = Atm d"
bhvks1: fixes t :: "nat \<Rightarrow> cnf"   and p :: "PROB"   and kId :: "'kid"   and t' :: "nat \<Rightarrow> 'KS"   and P :: "PROB set" assumes "t \<in> arch"   and "p = prob kId" shows "sb.eval kId t t' 0 (\<box>\<^sub>b[\<lambda>ks. sub P = ksrp ks]\<^sub>b \<and>\<^sup>b (\<forall>\<^sub>bq. sb.pred (q \<in> P) \<longrightarrow>\<^sup>b \<diamond>\<^sub>b[\<lambda>ks. (q, solve q) \<in> kscs ks]\<^sub>b) \<longrightarrow>\<^sup>b \<diamond>\<^sub>b[\<lambda>ks. (p, solve p) = ksns ks]\<^sub>b)"
arcosh_eq_iff: fixes x :: "real"   and y :: "real" assumes "1 \<le> x"   and "1 \<le> y" shows "(arcosh x = arcosh y) = (x = y)"
proj_ik_union_is_unlabel_ik: fixes A :: "('c strand_label \<times> ('a, 'b) strand_step) list" shows "ik\<^sub>s\<^sub>t (unlabel A) = (\<Union>l. ik\<^sub>s\<^sub>t (proj_unl l A))"
integrable_measure_pmf_finite: fixes M :: "'a pmf"   and f :: "'a \<Rightarrow> 'b" assumes "finite (set_pmf M)" shows "integrable (measure_pmf M) f"
uminus_bit1_def: fixes x :: "'b bit1" shows "- x = Abs_bit1' (- Rep_bit1 x)"
Diff_insert0: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "x \<notin> A" shows "A - insert x B = A - B"
IfNE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a expr"   and b :: "'a expr"   and c :: "'a expr"   and d :: "'a expr" assumes "\<Gamma> \<turnstile> expr.If a b c \<Rrightarrow> d" obtains a\<^sub>2 :: "'a expr" and b\<^sub>2 :: "'a expr" and c\<^sub>2 :: "'a expr" where "d = expr.If a\<^sub>2 b\<^sub>2 c\<^sub>2" and "\<Gamma> \<turnstile> a \<Rrightarrow> a\<^sub>2" and "\<Gamma> \<turnstile> b \<Rrightarrow> b\<^sub>2" and "\<Gamma> \<turnstile> c \<Rrightarrow> c\<^sub>2"
simps_7: fixes I :: "('b, 'c, 'a) interp"   and c :: "'c"   and \<phi> :: "('b, 'c, 'a) formula" shows "fml_sem I (InContext c \<phi>) = Contexts I c (fml_sem I \<phi>)"
range_J: shows "range (justifying_exec \<circ> J) \<subseteq> \<E>"
vp_split_Cons: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>\<surd>* n'"   and "as \<noteq> []" obtains a' :: "'edge" and as' :: "'edge list" where "as = a' # as'" and "n = sourcenode a'" and "valid_edge a'" and "targetnode a' -as'\<rightarrow>\<^sub>\<surd>* n'"
arith_extra_simps_5: fixes a :: "'c" shows "(0::'c) + a = a"
euler_trail_app_isoI: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "wf_digraph G"   and "pre_digraph.euler_trail G u p v"   and "pre_digraph.digraph_isomorphism G hom" shows "pre_digraph.euler_trail (app_iso hom G) (iso_verts hom u) (map (iso_arcs hom) p) (iso_verts hom v)"
nsum_add_nm: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j \<in> carrier A"   and "\<forall>j\<le>m. g j \<in> carrier A" shows "\<Sigma>\<^sub>e A jointfun n f m g Suc (n + m) = \<Sigma>\<^sub>e A f n \<plusminus> \<Sigma>\<^sub>e A g m"
lex_list_wfP: fixes order :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wfP order" shows "wfP (lexp order)"
zero_eq_lcm_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(zero = lcm a b) = (a = zero \<or> b = zero)"
wf_subset: fixes r :: "('c \<times> 'c) set"   and p :: "('c \<times> 'c) set" assumes "wf r"   and "p \<subseteq> r" shows "wf p"
subspace_span: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale" shows "module.subspace scale (module.span scale S)"