hcju/isabelleps huggingface dataset:
pred_prod_split: fixes P :: "bool \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool"   and xy :: "'a \<times> 'b" shows "P (pred_prod Q R xy) = (\<forall>x y. xy = (x, y) \<longrightarrow> P (Q x \<and> R y))"
Parallel: fixes as :: "(('d, 'e, 'f) ann \<times> 'd set \<times> 'd set) list"   and cs :: "('d, 'e, 'f) com list"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and Q :: "'d set"   and A :: "'d set" assumes "length as = length cs"   and "\<forall>i<length cs. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (fst (as ! i)) (cs ! i) (fst (snd (as ! i))), (snd (snd (as ! i)))"   and "interfree \<Gamma> \<Theta> F as cs"   and "\<Inter> (set (map (\<lambda>a. fst (snd a)) as)) \<subseteq> Q"   and "\<Union> (set (map (\<lambda>a. snd (snd a)) as)) \<subseteq> A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnPar as) (Parallel cs) Q, A"
Q_2: fixes X :: "complex multiset" assumes "X \<in> Roots_ms" shows "Polynomial.content (Q X) = 1"
hyps: shows "\<forall>ys. length ys < length ts \<longrightarrow> consistent\<^sub>F ys \<longrightarrow> height\<^sub>F ys = 0 \<longrightarrow> sortedByWeight ys \<longrightarrow> ys \<noteq> [] \<longrightarrow> optimum (huffman ys)"
le_supI: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x"   and "b \<le> x" shows "a \<squnion> b \<le> x"
add_le_imp_le_right: fixes a :: "complex"   and c :: "complex"   and b :: "complex" assumes "a + c \<le>\<^sub>\<complex> b + c" shows "a \<le>\<^sub>\<complex> b"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "inj_idom_hom hom" shows "map_poly_inj_idom_hom hom"
worklist_tree_aux_rule: fixes succs :: "'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and ws :: "'c list"   and s :: "'d"   and ws' :: "'c list"   and s' :: "'d"   and I :: "'c \<Rightarrow> bool"   and P :: "'c list \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" assumes "worklist_tree_aux succs f (ws, s) = Some (ws', s')"   and "invariant I succs"   and "\<forall>x\<in>set ws. I x"   and "\<And>s. P [] s s"   and "\<And>r x ws s. \<lbrakk>I x; \<forall>x\<in>set ws. I x; P ws (f x s) r\<rbrakk> \<Longrightarrow> P (x # ws) s r" shows "\<exists>rs. set rs = (Rel succs)\<^sup>* `` set ws \<and> P rs s s'"
set_subset_Cons: fixes xs :: "'a list"   and x :: "'a" shows "list.set xs \<subseteq> list.set (x # xs)"
I: fixes x :: "'a"   and S :: "'b" assumes "\<not> memb x S"   and "is_node x"   and "invariant S"   and "I S" shows "I (ins x S)"
mat_assoc_test_2: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "adjoint (A * adjoint B) * C = B * (adjoint A * C)"
ord_eq_le_subst: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and b :: "'c"   and c :: "'c" assumes "a = f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
emeasure_count_space_finite: fixes X :: "'d set"   and A :: "'d set" assumes "X \<subseteq> A"   and "finite X" shows "emeasure (count_space A) X = of_nat (card X)"
plus_u_strict_1: fixes x :: "'a\<^sub>\<bottom>" shows "x + \<bottom> = \<bottom>"
locally_path_connected_space_open_path_components: fixes X :: "'b topology" shows "locally_path_connected_space X = (\<forall>U c. openin X U \<and> c \<in> path_components_of (subtopology X U) \<longrightarrow> openin X c)"
grow_memory: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and n :: "nat"   and c :: "i32"   and mem' :: "mem"   and vs :: "v list" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "mem_size m = n"   and "mem_grow m (nat_of_int c) = mem'" shows "\<lparr>s;vs;[$C ConstInt32 c, $Grow_memory]\<rparr> \<leadsto>_ i \<lparr>s\<lparr>s.mem := (s.mem s)[j := mem']\<rparr>;vs;[$C ConstInt32 (int_of_nat n)]\<rparr>"
natural_transformation_axioms: shows "natural_transformation J (\<cdot>) \<chi>f.A.map D' (D'.cones_map f \<tau>o\<chi>.map)"
valid_msg_set_def: shows "abstract_com.valid_msg_set = Collect valid_msg"
Translations_E: fixes f :: "'a \<Rightarrow> 'a" assumes "f \<in> Translations" obtains x :: "'a" where "x \<in> G" and "f = (x)\<^sub>R"
cod_simps_2: shows "src.cod.leg0 = src.leg0"
A_def: shows "A = {x \<in> space M. ennreal e \<le> f x}"
wf_stateful_policy_is_wf_graph: fixes \<T> :: "'a stateful_policy" assumes "wf_stateful_policy \<T>" shows "wf_graph \<lparr>nodes = hosts \<T>, edges = all_flows \<T>\<rparr>"
eSuc_Inf: fixes A :: "enat set" shows "eSuc (\<Sqinter> A) = \<Sqinter> (eSuc ` A)"
fds_abs_converges_altdef: fixes f :: "'a fds"   and s :: "'a" shows "fds_abs_converges f s = ((\<lambda>n. fds_nth f n / nat_power n s) abs_summable_on {1..})"
reaches_Reverse_iff: fixes fsm :: "'b fsm"   and st :: "hf"   and xs :: "'b list"   and st' :: "hf" shows "reaches (Reverse fsm) st (rev xs) st' = reaches fsm st' xs st"
taut_test_ifex_def: fixes t :: "'d ifex" shows "taut_test_ifex t = taut_test_rec t Mapping.empty"
simps_4: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "merge_guards (Seq c\<^sub>1 c\<^sub>2) = Seq (merge_guards c\<^sub>1) (merge_guards c\<^sub>2)"
aeX3: shows "AE y in T \<Otimes>\<^sub>M P. \<integral>\<^sup>+ x. ennreal (Pxz (x, snd y)) \<partial>S = ennreal (Pz (snd y))"
isomorphic_implies_hpar_4: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.isomorphic f f'" shows "Span.trg f = Span.trg f'"
assms_3: fixes i :: "'c" assumes "i \<in> B - A" shows "g i = \<one>"
preserves_iso: fixes f :: "'a" assumes "A.iso f" shows "B.iso (G f)"
While_aux: fixes b :: "'a set"   and i :: "'a set"   and v :: "'a \<Rightarrow> nat"   and c :: "'a com"   and s :: "'a"   and s' :: "'a"   and I :: "'a set" assumes "Sem (While b i v c) s s'"   and "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> I \<and> s \<in> b \<longrightarrow> s' \<in> I"   and "s \<in> I" shows "s' \<in> I \<and> s' \<notin> b"
pred_of_succ_def: fixes s :: "'c \<Rightarrow> 'd set" shows "pred_of_succ s \<equiv> \<lambda>a b. b \<in> s a"
ex_inverse: fixes e :: "nat"   and P :: "nat"   and Q :: "nat" assumes "coprime e ((P - 1) * (Q - 1))"   and "prime P"   and "prime Q"   and "P \<noteq> Q" shows "\<exists>d. [e * d = 1] (mod P - 1) \<and> d \<noteq> 0"
assms_1: fixes x :: "'a" shows "1' + x ; rtc x \<le> rtc x"
quasi_inversesE: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.quasi_inverses V H \<a> \<i> src trg f g" obtains "partial_magma.ide V f" and "partial_magma.ide V g" and "category.isomorphic V (src f) (H g f)" and "category.isomorphic V (H f g) (trg f)"
min_add_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus (ord.min less_eq x y) z = ord.min less_eq (plus x z) (plus y z)"
executable: fixes p :: "'state"   and a :: "'action" assumes "p \<in> nodes"   and "en a p" shows "a \<in> executable"
HStartBallot_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2c s"   and "HStartBallot s s' p"   and "Inv2a s" shows "Inv2c s'"
crel_vs_iterate_and_compute: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and g :: "'f \<Rightarrow> 'g"   and f :: "'f \<Rightarrow> ('b, 'h) state"   and x :: "'f"   and xs :: "'f list" assumes "((=) ===>\<^sub>T R) g f" shows "crel_vs R (g x) (iterate_state f xs \<bind> (\<lambda>_. f x))"
SetDC_finite_insert: fixes S :: "'a set"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'a" assumes "finite S"   and "S \<noteq> {}" shows "SetDC a (\<lambda>_. insert x S) = a x \<Sqinter> SetDC a (\<lambda>_. S)"
sets_bind': fixes A :: "'c measure"   and M :: "'c measure"   and B :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure" assumes "A \<in> space (prob_algebra M)"   and "B \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "sets (A \<bind> B) = sets N"
iteration_step_invariant: fixes ES :: "('b list set \<times> 'b trm set) set"   and X :: "'b list set"   and xrhs :: "'b trm set" assumes "invariant ES"   and "(X, xrhs) \<in> ES"   and "Cond ES" shows "invariant (Iter X ES)"
simps_2: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b tree"   and a :: "'b"   and r :: "'b tree" shows "bst_wrt P \<langle>l, a, r\<rangle> = ((\<forall>x\<in>set_tree l. P x a) \<and> Ball (set_tree r) (P a) \<and> bst_wrt P l \<and> bst_wrt P r)"
unit_vec_def: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i = tensor_from_lookup [n] (\<lambda>x. if x = [i] then 1::'b else (0::'b))"
braun_indices1: fixes t :: "'a tree" shows "0 \<notin> braun_indices t"
image_image: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'i \<Rightarrow> 'h"   and A :: "'i set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
Class_def: shows "e1.Class = (\<lambda>a\<in>Sigma I \<FF>1. {b \<in> Sigma I \<FF>1. (b, a) \<in> {(x, y). stk1.rel x y}})"
col_swaprows_2: fixes i :: "nat"   and k :: "nat"   and l :: "nat"   and n :: "nat" assumes "i \<noteq> k"   and "i \<noteq> l"   and "i < n" shows "col (swaprows_mat n k l) i = unit_vec n i"
runit_naturality: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "\<mu> \<cdot>\<^sub>B B.runit (B.dom \<mu>) = B.runit (B.cod \<mu>) \<cdot>\<^sub>B \<mu> \<star>\<^sub>B src\<^sub>B \<mu>"
div_dvd_div: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b"   and "dvd.dvd times a c" shows "dvd.dvd times (divide b a) (divide c a) = dvd.dvd times b c"
ring_hom_restrict: fixes f :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c"   and S :: "('c, 'd) ring_scheme"   and g :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c" assumes "f \<in> ring_hom P S"   and "\<And>r. r \<in> carrier P \<Longrightarrow> f r = g r" shows "g \<in> ring_hom P S"
simps_2: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "qsort (x # xs) l = qsort (filter (\<lambda>y. \<not> smaller x y l) xs) l @ [x] @ qsort (filter (\<lambda>y. smaller x y l) xs) l"
Call\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and a :: "nat"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and es :: "nat exp list"   and vs :: "val list"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>2 :: "val list"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list"   and ls\<^sub>2' :: "val list"   and e' :: "nat exp"   and h\<^sub>3 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>3 :: "val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, ls\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = body in D"   and "length vs = length Ts"   and "ls\<^sub>2' = Addr a # vs @ replicate (max_vars body) undefined"   and "P \<turnstile>\<^sub>1 \<langle>body,(h\<^sub>2, ls\<^sub>2')\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, ls\<^sub>3)\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, ls\<^sub>2)\<rangle>"
distinct_remove1_removeAll: fixes xs :: "'d list"   and x :: "'d" assumes "distinct xs" shows "remove1 x xs = removeAll x xs"
induced_automorphism_unique_automorphism: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and s :: "'b \<Rightarrow> 'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "ChamberComplexAutomorphism X s"   and "s \<noteq> id"   and "fixespointwise s (C0 \<inter> g ` C0)" shows "s = OpposedThinChamberComplexFoldings.induced_automorphism X f g"
strict_part_mono_by_steps: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "strict_part_mono {..n} f = (n \<noteq> 0 \<longrightarrow> f (n - 1) < f n \<and> strict_part_mono {..n - 1} f)"
has_overrider_wf_mdecl: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and M :: "char list"   and m :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs' :: "char list list" assumes "wf_prog wf_md P"   and "P \<turnstile> (C, Cs) has overrider M = m via Cs'" shows "wf_mdecl wf_md P (last Cs') (M, m)"
vars0_def: fixes vars1 :: "nat set"   and vars2 :: "nat set" assumes "partial_state2 vars1 vars2" shows "partial_state2.vars0 vars1 vars2 = vars1 \<union> vars2"
cases: fixes t :: "'t"   and a :: "'x \<times> 'm" assumes "\<tau>diverge t a" obtains s :: "'x \<times> 'm" and s' :: "'x \<times> 'm" where "a = s" and "silent_move t s s'" and "\<tau>diverge t s'"
deg_uminus: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
simps_8: fixes f :: "'l \<Rightarrow> 'k" shows "map f [] = []"
path_connectedin_path_components_of: fixes C :: "'a set"   and X :: "'a topology" assumes "C \<in> path_components_of X" shows "path_connectedin X C"
src_src: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "src\<^sub>B (src\<^sub>B \<mu>) = src\<^sub>B \<mu>"
gC_\<Phi>_satPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "Ik.satPB \<xi> \<Phi>" shows "GE.satPB \<xi> (gC ` \<Phi>)"
compE1_eq_Val: fixes Vs :: "String.literal list"   and e :: "'a expr"   and v :: "'a val" shows "(compE1 Vs e = Val v) = (e = Val v)"
eq: fixes x :: "'c"   and y :: "'d" shows "{(x, y)} = {x} \<times> {y}"
lookup_def: fixes A :: "'b tensor" shows "lookup A = lookup_base (dims A) (vec A)"
rep_eq: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'d \<Rightarrow>\<^sub>0 'e" assumes "inj f" shows "lookup (Poly_Mapping.map_key f x) = lookup x \<circ> f"
empty_le: fixes A :: "'d multiset" shows "{#} \<subseteq># A"
add_increasing: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "{#} \<subseteq># a"   and "b \<subseteq># c" shows "b \<subseteq># a + c"
ide_implies_can: fixes a :: "'c local.term set" assumes "ide a" shows "can a"
integrable_distr: fixes T :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and M' :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "T \<in> M \<rightarrow>\<^sub>M M'"   and "integrable (distr M M' T) f" shows "integrable M (\<lambda>x. f (T x))"
xor_nonnegative_int_iff: fixes k :: "int"   and l :: "int" shows "(0 \<le> k XOR l) = ((0 \<le> k) = (0 \<le> l))"
fs'_fs: fixes k :: "nat" assumes "k < i - 1" shows "fs' ! k = fs ! k"
ND_sound: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" assumes "\<Gamma> \<turnstile> F" shows "\<Gamma> \<TTurnstile> F"
extended_Ax: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<inter># \<Delta> \<noteq> {#}" shows "\<Gamma> \<Rightarrow> \<Delta>"
scaleR_scaleR: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and x :: "'b" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (scaleR b x) = scaleR (a * b) x"
strictly_maximal_wrt_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and A :: "'b"   and CA :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.strictly_maximal_wrt A CA = (\<forall>B\<in>atms_of CA. B < A)"
degcmc': shows "Reflected_Multivariate_Polynomial.degree (c +\<^sub>p ~\<^sub>p c') = 0"
that: fixes f :: "'a \<Rightarrow> nat"   and n :: "nat" assumes "f ` A = {i. i < n}"   and "inj_on f A" shows "thesis"
max_deg: shows "max (degree q1) (degree q2) \<le> 1"
gen_ML_def: fixes A :: "type_scheme list"   and t :: "typ" shows "gen_ML A t = gen_ML_aux (free_tv_ML A) t"
Sbis_matchC_C: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis"   and "(c, d) \<in> PL_Indis.Sbis tval aval indis" shows "PL_Indis.matchC_C tval aval indis (PL_Indis.Sbis tval aval indis) c d"
sctn_rel_def: fixes A :: "('c \<times> 'd) set" shows "\<langle>A\<rangle>sctn_rel = {(a, b). rel_sctn (in_rel A) a b}"
backward_terminating_iff2: fixes x :: "'a" assumes "path x" shows "backward_terminating x = (x \<le> x\<^sup>T\<^sup>\<star> ; - (x\<^sup>T ; 1))"
simps_2: fixes mc :: "'c monom \<times> 'd"   and p :: "('c monom \<times> 'd) list"   and q :: "('c monom \<times> 'd) list" shows "poly_mult (mc # p) q = poly_add (monom_mult_poly mc q) (poly_mult p q)"
counit_simps_3: shows "cod \<psi> = trg f"
le_max_self_2: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b \<or> b \<le> a" shows "b \<le> max a b"
h1: fixes x :: "'a" shows "h (1, x) = g x"
bit_word_cat_iff: fixes v :: "'c word"   and w :: "'d word"   and n :: "nat" shows "bit (word_cat v w) n = (n < LENGTH('b) \<and> (if n < LENGTH('d) then bit w n else bit v (n - LENGTH('d))))"
assms: fixes k :: "'b"   and v :: "'c" assumes "(k, v) \<in> set xs" shows "f k v = g k v"
return_time_least: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b"   and s :: "real" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x"   and "closed P"   and "0 < s"   and "s < auto_ll_on_open.return_time f X P x" shows "auto_ll_on_open.flow0 f X x s \<notin> P"
fps_zero_def: shows "0 = Abs_fps (\<lambda>n. 0::'a)"
neq_dP_disp: fixes p :: "'b ref"   and q :: "'b ref"   and h :: "'b \<Rightarrow> 'b ref"   and Ps :: "'b list" assumes "p \<noteq> q"   and "distPath h p Ps q" shows "\<exists>a Qs. p = Ref a \<and> Ps = a # Qs \<and> a \<notin> set Qs"
focus_plus: fixes X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "focus X (p + q) = focus X p + focus X q"
minus_closed: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)" shows "x \<ominus>\<^bsub>poly_ring R\<^esub> y \<in> carrier (poly_ring R)"
map_ident: fixes t :: "'l option" shows "map_option (\<lambda>x. x) t = t"
mono_neutral_left: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = z" shows "abel_semigroup_set.F f z g S = abel_semigroup_set.F f z g T"
g_integrable: fixes x :: "real" assumes "x\<^sub>0 \<le> x" shows "integrable (\<lambda>u. g u / u powr (p + 1)) x\<^sub>0 x"
inf_Sup: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and a :: "'b::type"   and B :: "'b::type set" assumes "class.complete_distrib_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) ((Sup::'b::type set \<Rightarrow> 'b::type) (B::'b::type set)) = Sup (inf a ` B)"
induced_automorphism_fixespointwise_C0_int_D0: shows "fixespointwise \
 (C0 \<inter> D0)"
oarrivemsg_simps_3: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and ii :: "nat set"   and ni :: "nat set"   and m :: "'b" shows "oarrivemsg P \<sigma> (ii\<not>ni:arrive(m)) = P \<sigma> m"
perp_not_col: fixes A :: "'p"   and B :: "'p"   and P :: "'p" assumes "A B Perp P A" shows "\<not> Col A B P"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Fext of Cext \<Rightarrow> f1 | Text \<Rightarrow> f2 | Fext \<Rightarrow> f3) = f3"
distrib_left: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * (b + c) = a * b + a * c"
unity_def: shows "\<I>\<^sub>1 \<equiv> D.cod \<iota>\<^sub>1"
lt_monomial_plus: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'f"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and v :: "'c" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "c \<noteq> (0::'f)"   and "ord_term_strict (ordered_term.lt term_of_pair ord_term p) v" shows "ordered_term.lt term_of_pair ord_term (monomial c v + p) = v"
inj_hom_imp_solvable: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h"   and "inj_on h (carrier G)"   and "solvable H" shows "solvable G"
unit_cube_convex_hull: obtains S :: "'b set" where "finite S" and "cbox (0::'b) One = convex hull S"
shadow_root_delete_get_3: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr h = Some h'" shows "get node_ptr h' = get node_ptr h"
remove_vertex_V: fixes x :: "'a" shows "V\<^bsub>remove_vertex x\<^esub> = V - {x}"