hcju/isabelleps huggingface dataset:
face_cycle_set_def: fixes a :: "'b" shows "H.face_cycle_set a = orbit H.face_cycle_succ a"
sub_imp_same_verts: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and sub :: "('d \<times> nat) \<times> 'd \<times> nat" assumes "wf_sub_rel subs"   and "sub \<in> subs" shows "fst (subsumee sub) = fst (subsumer sub)"
Diff_Diff_Int: fixes A :: "'a set"   and B :: "'a set" shows "A - (A - B) = A \<inter> B"
Null_notin_Path: fixes p :: "ref"   and f :: "ref \<Rightarrow> ref"   and q :: "ref"   and ps :: "ref list" assumes "Path p f q ps" shows "Null \<notin> set ps"
inducts: fixes x :: "'b llist"   and A :: "'b set"   and P :: "'b llist \<Rightarrow> bool" assumes "x \<in> A\<^sup>\<star>"   and "P LNil"   and "\<And>l a. \<lbrakk>l \<in> A\<^sup>\<star>; P l; a \<in> A\<rbrakk> \<Longrightarrow> P (a ## l)" shows "P x"
arcs_ends_symmetric: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "symmetric G"   and "u \<rightarrow>\<^bsub>G\<^esub> v" shows "v \<rightarrow>\<^bsub>G\<^esub> u"
count_inject: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(count x = count y) = (x = y)"
minusinf_ex: fixes p :: "fm"   and a :: "int"   and bs :: "real list" assumes "iszlfm p (real_of_int a # bs)"   and "\<exists>x. Ifm (real_of_int x # bs) (minusinf p)" shows "\<exists>x. Ifm (real_of_int x # bs) p"
frp_\<alpha>_def: fixes S :: "('d \<times> 'e) set \<times> ('d \<times> 'e) list \<times> ('d \<times> 'e, 'f) ta_rule set" shows "frp_\<alpha> S \<equiv> let (Q, W, \<delta>) = S in (Q, W)"
ltl_const_equiv_implies_prop_equiv_1: fixes \<phi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>C true\<^sub>n = \<phi> \<sim>\<^sub>P true\<^sub>n"
h: fixes u :: "'b"   and v :: "'b" shows "\<exists>w z. h ` cbox u v = cbox w z"
gen_cond_def: fixes s :: "('a, 'c) state_scheme" shows "a.a.gen_dfs.gen_cond s \<equiv> (a.a.V0 \<subseteq> a.a.gen_dfs.gen_discovered s \<longrightarrow> \<not> gds_is_empty_stack a.a.gen_dfs.gds s) \<and> \<not> gds_is_break a.a.gen_dfs.gds s"
indexed_padd_def: fixes P :: "'d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a"   and Q :: "'d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a" shows "L.indexed_padd P Q = (\<lambda>m. P m \<oplus>\<^bsub>L\<^esub> Q m)"
has_integral_const: fixes c :: "'b"   and a :: "'a"   and b :: "'a" shows "((\<lambda>x. c) has_integral content (cbox a b) *\<^sub>R c) (cbox a b)"
alphaInp_trans: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option"   and qinp' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option"   and qinp'' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" assumes "qGoodInp qinp"   and "(qinp, qinp') \<in> alphaInp"   and "(qinp', qinp'') \<in> alphaInp" shows "(qinp, qinp'') \<in> alphaInp"
assms_1: shows "s1 \<subseteq>\<^sub>f s2"
smult_as_map_poly: fixes a :: "'b" shows "smult a = map_poly ((*) a)"
extremum_uniqueI: fixes a :: "'a::order_top" assumes "\<top> \<le> (a::'a::order_top)" shows "(a::'a::order_top) = \<top>"
min_union_empty_1: fixes A :: "'b fset set" shows "A \<union>\<^sub>m {} = min_set A"
fcompare_double_refl: fixes x :: "double" assumes "\<not> Code_Double.is_nan x" shows "fcompare_double x x = ccode.Eq"
split0_nb0': fixes t :: "tm" shows "tmbound0 (snd (split0 t))"
locally_path_connected_space_eq_open_path_component_of: fixes X :: "'c topology" shows "locally_path_connected_space X = (\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> openin X (path_component_of_set (subtopology X U) x))"
eventually_at_leftI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.order_topology less_eq less open"   and "\<And>x. x \<in> ord.greaterThanLessThan less a b \<Longrightarrow> P x"   and "less a b" shows "eventually P (topological_space.at_within open b (ord.lessThan less b))"
exhaust: fixes y :: "atom" obtains x :: "nat" where "y = atom x"
takeWhile_cong: fixes l :: "'a list"   and k :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> set l \<Longrightarrow> P x = Q x" shows "takeWhile P l = takeWhile Q k"
that: fixes as :: "nat \<Rightarrow> 'a" assumes "A.is_lrun cl as"   and "\<forall>i. R (as i) (cs i)" shows "thesis"
rel_intros_2: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x21 :: "'d"   and y21 :: "'e"   and x22 :: "'d list"   and y22 :: "'e list" assumes "R x21 y21"   and "list_all2 R x22 y22" shows "list_all2 R (x21 # x22) (y21 # y22)"
trm_in_hom: fixes f :: "'a" assumes "arr f" shows "\<guillemotleft>\<t>[f] : local.dom f \<rightarrow> \<one>\<guillemotright>"
strictly_subsumes_neq: fixes D' :: "'a literal multiset"   and D :: "'a literal multiset"   and \<sigma> :: "'s" assumes "strictly_subsumes D' D" shows "D' \<noteq> D \<cdot> \<sigma>"
disjunctiveD: fixes x :: "'b"   and sup_b :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup_c :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and times_abc :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and y :: "'c"   and z :: "'c" assumes "x \<in> disjunctive.disjunctive sup_b sup_c times_abc" shows "times_abc x (sup_b y z) = sup_c (times_abc x y) (times_abc x z)"
jointfun_hom: fixes f :: "nat \<Rightarrow> 'h"   and n :: "nat"   and A :: "'h set"   and g :: "nat \<Rightarrow> 'h"   and m :: "nat"   and B :: "'h set" assumes "f \<in> {i. i \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> B" shows "jointfun n f m g \<in> {j. j \<le> Suc (n + m)} \<rightarrow> A \<union> B"
eval_monom_collapse: fixes c :: "real"   and c' :: "real"   and es :: "real list"   and basis :: "(real \<Rightarrow> real) list"   and x :: "real" shows "c * eval_monom (c', es) basis x = eval_monom (c * c', es) basis x"
cone_mkCone: fixes a :: "'c"   and F :: "'j \<Rightarrow> 'c" assumes "C.ide a"   and "\<And>j. J.arr j \<Longrightarrow> \<guillemotleft>F j : a \<rightarrow> D j\<guillemotright>" shows "D.cone a (D.mkCone F)"
changes_smods_congr: fixes a :: "real"   and a' :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a \<noteq> a'"   and "poly p a \<noteq> 0"   and "\<forall>p\<in>set (smods p q). \<forall>x. a < x \<and> x \<le> a' \<or> a' \<le> x \<and> x < a \<longrightarrow> poly p x \<noteq> 0" shows "changes_poly_at (smods p q) a = changes_poly_at (smods p q) a'"
inv_end_in_segment: fixes b :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "b \<in> orbit f a"   and "f a \<noteq> b"   and "bij f" shows "inv f b \<in> segment f a b"
lebesgue_real_affine: fixes c :: "real"   and t :: "real" assumes "c \<noteq> 0" shows "lebesgue = density (distr lebesgue lebesgue (\<lambda>x. t + c * x)) (\<lambda>_. ennreal \<bar>c\<bar>)"
lNxt_active: fixes active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "singleton active" shows "dynamic_component.nxtAct active (singleton.the_singleton active) t n = n"
bernoulli_stream_preimage: fixes N :: "bool stream measure"   and p :: "real"   and f :: "bool stream \<Rightarrow> 'b"   and A :: "'b set" assumes "N = bernoulli_stream p" shows "f -` A \<inter> space N = f -` A"
2: fixes x :: "'a" assumes "x \<in> S" shows "f x \<in> T \<and> (x, f x) \<in> A"
simps_7: fixes I :: "interp"   and \<alpha> :: "game" shows "game_sem I \<alpha>^d = (\<lambda>X. - game_sem I \<alpha> (- X))"
abs_minus_le_zero: fixes abs :: "'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 a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (uminus (abs a)) zero"
coplanar_perm_10: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet B D A C"
set_enqueue_new_thread: fixes queue :: "'a list"   and nta :: "('a, 'b, 'c) new_thread_action" shows "set (Round_Robin.enqueue_new_thread queue nta) = set queue \<union> {t. \<exists>x m. nta = NewThread t x m}"
hn_refine_cons: fixes P :: "assn"   and P' :: "assn"   and c :: "'a Heap"   and Q :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres"   and Q' :: "assn"   and R' :: "'b \<Rightarrow> 'a \<Rightarrow> assn" assumes "P \<Longrightarrow>\<^sub>t P'"   and "hn_refine P' c Q R m"   and "Q \<Longrightarrow>\<^sub>t Q'"   and "\<And>x y. R x y \<Longrightarrow>\<^sub>t R' x y" shows "hn_refine P c Q' R' m"
column_Gram_Schmidt_column_k: fixes k :: "'c"   and A :: "(('a, 'c) vec, 'b) vec" shows "column k (Gram_Schmidt_column_k A (mod_type_class.to_nat k)) = column k A - (\<Sum>x\<in>{column i A |i. i < k}. (x \<bullet> column k A / (x \<bullet> x)) *\<^sub>R x)"
neg_char: fixes x :: "'a" assumes "test x" shows "are_complementary x (! x)"
Quotient_dlist_remdups: shows "Quotient dlist_eq abs_dlist list_of_dlist qcr_dlist"
monocontlub2cont: fixes f :: "'a \<Rightarrow> 'b" assumes "monofun f"   and "contlub f" shows "cont f"
assms_3: shows "t \<le> v ; v\<^sup>T"
size_list_simp_2: fixes xs :: "'a list" shows "length xs = (if xs = [] then 0 else Suc (length (tl xs)))"
monad_envT': assumes "monad Monad_Overloading.return Monad_Overloading.bind" shows "monad Monad_Overloading.return Monad_Overloading.bind"
\<eta>_def: shows "Adj.\<eta> \<equiv> transformation_by_components.map (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) Adj.D.map Adj.\<eta>o"
assms: shows "u \<in> unaffected_domains I D {D y} zs"
simps_3: fixes e1 :: "expr"   and e2 :: "expr" shows "free_vars (LET e1 IN e2) = free_vars e1 \<union> Suc -` free_vars e2"
not_in_env_bigger: fixes x :: "char list"   and e :: "'b environment"   and a :: "char list"   and X :: "'b" assumes "x \<notin> env_dom e"   and "x \<noteq> a" shows "x \<notin> env_dom (e\<lparr>a:X\<rparr>)"
convergent_eq_Cauchy: fixes S :: "nat \<Rightarrow> 'd" shows "(\<exists>l. S \<longlonglongrightarrow> l) = Cauchy S"
tm_dghm_HomDom_is_tiny_digraph: shows "tiny_digraph \<alpha> \<AA>"
lformula_Restrict: fixes k :: "'k"   and i :: "nat" shows "lformula (Restrict k i)"
swap_simp: fixes x :: "'c"   and y :: "'b" shows "swap (x, y) = (y, x)"
Max_in_lits: fixes C :: "'a multiset" assumes "C \<noteq> {#}" shows "Max_mset C \<in># C"
prv_eql_refl: fixes x :: "'var" assumes "x \<in> var" shows "prv (eql (Var x) (Var x))"
commute: fixes a :: "'r"   and b :: "'r" shows "a + b = b + a"
arrI: fixes f :: "'a" assumes "left a f" shows "L.arr f"
simps_2: fixes a :: "'c"   and t1 :: "'c bt"   and t2 :: "'c bt" shows "n_leaves (Br a t1 t2) = n_leaves t1 + n_leaves t2"
stable_Un: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> stable A"   and "F \<in> stable A'" shows "F \<in> stable (A \<union> A')"
Units_def: fixes G :: "('b, 'c) monoid_scheme" shows "Units G = {y \<in> carrier G. \<exists>x\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>}"
step_eq_traces: fixes A :: "('e, 'b, 'f) ioa_scheme"   and e_A' :: "'c \<times> ('b \<times> 'd) list"   and e_B' :: "'g \<times> ('b \<times> 'h) list"   and a :: "'b"   and e :: "'i \<times> ('b \<times> 'd) list"   and t :: "'h" assumes "trace (ioa.asig A) e_A' = trace (ioa.asig A) e_B'"   and "if a \<in> externals (ioa.asig A) then trace (ioa.asig A) e = [a] else trace (ioa.asig A) e = []" shows "trace (ioa.asig A) (append_exec e_A' e) = trace (ioa.asig A) (cons_exec e_B' (a, t))"
af_simps_3: fixes \<phi> :: "'b ltln"   and x :: "'b set"   and xs :: "'b set list" shows "af (X\<^sub>n \<phi>) (x # xs) = af \<phi> xs"
has_derivative_transform_within: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e"   and x :: "'d"   and s :: "'d set"   and d :: "real"   and g :: "'d \<Rightarrow> 'e" assumes "(f has_derivative f') (at x within s)"   and "0 < d"   and "x \<in> s"   and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "(g has_derivative f') (at x within s)"
post_meas1_bell01_snd: shows "post_meas1 2 |\<beta>\<^sub>0\<^sub>1\<rangle> 1 = |unit_vec 4 1\<rangle>"
support_on_def: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "support_on S f = {x \<in> S. f x \<noteq> (0::'d)}"
rel_funE: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and x :: "'a"   and y :: "'b" assumes "(A ===> B) f g"   and "A x y" obtains "B (f x) (g y)"
simps_1: fixes uu :: "typ list"   and uv :: "String.literal"   and T :: "typ" shows "typ_of1 uu (Ct uv T) = Some T"
ex: fixes i :: "nat" obtains a :: "nat" where "a \<in> A" and "card (A \<inter> {..<a}) = i"
assms_5: shows "c = insertion (nth_default 0 (xs' @ y # xs)) C"
braun_Node': fixes l :: "'a tree"   and x :: "'a"   and r :: "'a tree" shows "braun \<langle>l, x, r\<rangle> = (size r \<le> size l \<and> size l \<le> size r + 1 \<and> braun l \<and> braun r)"
monic_of_int_poly: fixes D :: "int poly" assumes "monic D" shows "monic (of_int_poly (Mp D))"
is_Inr_def: shows "is_Inr \<equiv> \<lambda>uu. case uu of Inl uv_ \<Rightarrow> False | Inr uu_ \<Rightarrow> True"
le_INF_iff: fixes u :: "'b \<times> 'c"   and f :: "'d \<Rightarrow> 'b \<times> 'c"   and A :: "'d set" shows "less_eq_prod u (Inf_prod (f ` A)) = (\<forall>i\<in>A. less_eq_prod u (f i))"
simps_2: fixes name :: "name" shows "consts (Sconst name) = {|name|}"
sub_block_comp: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and block :: "('e, 'h) fm list \<times> 'h" shows "sub_block f (sub_block g block) = sub_block (f \<circ> g) block"
eventuallyI: fixes P :: "'d \<Rightarrow> bool"   and F :: "'d filter" assumes "\<And>x. P x" shows "eventually P F"
lt_of_key_order_alt: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "lt_of_key_order ko x y = (key_compare ko x y = Lt)"
aff_dim_eq_hyperplane: fixes S :: "'a set" shows "(aff_dim S = int (DIM('a) - 1)) = (\<exists>a b. a \<noteq> (0::'a) \<and> affine hull S = {x. a \<bullet> x = b})"
open_blinfun_def: fixes S :: "('a \<Rightarrow>\<^sub>L 'b) set" shows "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
weak_arrow_of_homs_axioms: shows "weak_arrow_of_homs (\<cdot>\<^sub>D) src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) src\<^sub>C trg\<^sub>C F"
ScopeExtPar: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<parallel> Q) \<equiv>\<^sub>s P \<parallel> <\<nu>x>Q"
inverse_scaleR_distrib: fixes a :: "real"   and x :: "'b" shows "inverse (a *\<^sub>R x) = inverse x /\<^sub>R a"
inverse_unique: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "group f z inverse"   and "f a b = z" shows "inverse a = b"
DD_subset_helper: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma>' :: "'a list"   and \<sigma> :: "'a list"   and S :: "'a set" assumes "trans r"   and "(r|\<tau> @ \<sigma>'|, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r"   and "set_mset (r|\<tau>| + r|\<sigma>|) \<subseteq> r \<down>s S" shows "set_mset r|\<sigma>'| \<subseteq> r \<down>s S"
assms_2: shows "g \<in> \<Theta>[F](h)"
undual_equality: fixes x' :: "'c dual"   and y' :: "'c dual" shows "(undual x' = undual y') = (x' = y')"
atLeastLessThan_borel: fixes a :: "'c"   and b :: "'c" shows "{a..<b} \<in> sets borel"
b: shows "\<lbrace>p \<cdot> b\<rbrace>x\<lbrace>q\<rbrace>"
set_dom_\<Phi>: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "partial_magma.arr S f" shows "set_category_data.set S' (set_category.some_img S' \<AA>) (partial_magma.dom S' (two_set_categories_bij_betw_Univ.\<Phi> S S' \<AA> \<phi> f)) = \<phi> ` set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f)"
halt_lemma2'': fixes P :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "P (f n)"   and "\<not> P (f 0)" shows "\<exists>n. P (f n) \<and> (\<forall>i<n. \<not> P (f i))"
f1_abs_summable_on: shows "(\<lambda>i. f (i + 1)) abs_summable_on UNIV"
prems_2: shows "list_all (\<lambda>d. fst d = i) Di"
continuous_def: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'd" shows "continuous F f = (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
expansion_level_eq_Cons: fixes xs :: "'a list"   and x :: "'a" assumes "length xs = expansion_level TYPE('b)" shows "length (x # xs) = expansion_level TYPE('b ms)"
mem_Sup_iff: fixes x :: "V"   and X :: "V set" shows "(x \<in>\<^sub>\<circ> \<Squnion> X) = (x \<in> \<Union> (elts ` X) \<and> small X)"
simplicial_simplex_simplex_cone: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and v :: "nat \<Rightarrow> real"   and T :: "(nat \<Rightarrow> real) set" assumes "simplicial_simplex p S f"   and "\<And>x u. \<lbrakk>0 \<le> u; u \<le> 1; x \<in> S\<rbrakk> \<Longrightarrow> (\<lambda>i. (1 - u) * v i + u * x i) \<in> T" shows "simplicial_simplex (Suc p) T (simplex_cone p v f)"
length_filter_map: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "length (filter P (map f xs)) = length (filter (P \<circ> f) xs)"
ID_Some: fixes x :: "'b" shows "ID (Some x) = Some x"
simps: fixes a1 :: "char list"   and a2 :: "char list"   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" shows "P \<turnstile> a1 \<prec>\<^sub>S a2 = (\<exists>C Bs rest D. a1 = C \<and> a2 = D \<and> class P C = \<lfloor>(Bs, rest)\<rfloor> \<and> Shares D \<in> set Bs)"
vec_uniqueness: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and v :: "'c list"   and n :: "nat" assumes "Matrix_Tensor.mult id f"   and "vec m v"   and "vec n v" shows "m = n"
insert_compr: fixes a :: "'k"   and B :: "'k set" shows "insert a B = {x. x = a \<or> x \<in> B}"
double_compl: fixes x :: "'a" shows "NOT (NOT x) = x"
head_com_def: fixes c :: "('a, 'b, 'c) com" shows "head_com c = (case c of Seq c\<^sub>1 c\<^sub>2 \<Rightarrow> c\<^sub>1 | Catch c\<^sub>1 c\<^sub>2 \<Rightarrow> c\<^sub>1 | _ \<Rightarrow> c)"
outmarked_def: fixes x :: "'v" shows "outmarked x = (x \<in> B \<Gamma> \<or> Inl x \<in> S)"
obs_elem: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and S :: "'node set" assumes "n -as\<rightarrow>* n'"   and "\<forall>nx\<in>set (sourcenodes as). nx \<notin> S"   and "n' \<in> S" shows "n' \<in> obs n S"
wfp_on_imp_inductive_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on P A" shows "inductive_on P A"
axioms_1: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and default_node_properties :: "'d"   and receiver_violation :: "bool" assumes "SecurityInvariant sinvar default_node_properties receiver_violation" shows "SecurityInvariant_preliminaries sinvar"
sur_comp: fixes x :: "'a"   and y :: "'a" assumes "is_sur x"   and "is_sur y" shows "is_sur (x ; y)"
rbb_rbd_demorgan: fixes R :: "('c \<times> 'd) set" shows "\<partial> \<circ> bb\<^sub>\<R> R = bd\<^sub>\<R> R \<circ> \<partial>"
divide_self: fixes R :: "('b, 'c) ring_scheme"   and a :: "'b" assumes "field R"   and "a \<in> carrier R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>" shows "a \<oslash>\<^bsub>R\<^esub> a = \<one>\<^bsub>R\<^esub>"
inner_quad_ge: fixes a :: "'b"   and b :: "'b" shows "2 * (a \<bullet> b) \<le> a \<bullet> a + b \<bullet> b"
Append_to_free0: fixes i :: "nat"   and e :: "(nat \<times> nat) list" shows "length (Append_to_free (i, e)) = length e"
landau_trans_31: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "g \<in> O[F](h)" shows "f \<in> O[F](h)"
degrees_def: fixes X :: "'b pdevs list" shows "degrees X = Max (insert 0 (degree ` set X))"
eq_zero_or_degree_less: fixes p :: "'b poly"   and n :: "nat" assumes "degree p \<le> n"   and "coeff p n = (0::'b)" shows "p = 0 \<or> degree p < n"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'k list" assumes "i < length xs" shows "xs ! i # drop (Suc i) xs = drop i xs"
fmdom_fmap_of_list: fixes xs :: "('a \<times> 'b) list" shows "fmdom (fmap_of_list xs) = fset_of_list (map fst xs)"
prefix_length_code: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "prefix_length P xs = prefix_length_aux P 0 xs"
hyps_2: shows "subdivision_step I rev_I H rev_H (u, v, w) (uv, uw, vw)"
emeasure_lborel_countable: fixes A :: "'b set" assumes "countable A" shows "emeasure lborel A = 0"
eq_sqrt_squared: fixes x :: "real" assumes "0 \<le> x" shows "sqrt x * sqrt x = x"
assms: shows "((\<lambda>x. ln (f x) + F x) expands_to L) basis"
iMODb_iff: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" shows "(x \<in> [ r, mod m, c ]) = (x mod m = r mod m \<and> r \<le> x \<and> x \<le> r + m * c)"
obtain_atom: fixes X :: "atom set"   and s :: "atom_sort" assumes "finite X" obtains a :: "atom" where "a \<notin> X" and "sort_of a = s"
assms_6: shows "(_Entry_) -as'\<rightarrow>\<^sub>\<surd>* (_Exit_)"
inf_sup_aci_5: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
int_pow_neg: fixes x :: "'c"   and i :: "int" assumes "x \<in> carrier H" shows "x [^]\<^bsub>H\<^esub> - i = inv\<^bsub>H\<^esub> (x [^]\<^bsub>H\<^esub> i)"
typeII_twoCube_def: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" shows "typeII_twoCube twoC \<equiv> \<exists>a b g1 g2. a < b \<and> (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and> twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b)) \<and> g1 piecewise_C1_differentiable_on {a..b} \<and> g2 piecewise_C1_differentiable_on {a..b}"
canI_unitor_0_2: fixes f :: "'a" assumes "B.ide f" shows "B.runit f = B.can \<^bold>\<langle>f\<^bold>\<rangle> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>src\<^sub>B f\<^bold>\<rangle>\<^sub>0)"
IrrE_2: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "pirreducible K (Irr K x)"
exec_elim_cases_13: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Throw,s\<rangle> \<Rightarrow> t" obtains f :: "'f" where "s = Fault f" and "t = Fault f"   | "s = Stuck" and "t = Stuck"   | sa :: "'e" where "s = Normal sa__" and "t = Abrupt sa__"   | sa :: "'e" where "s = Abrupt sa" and "t = Abrupt sa"
uniform_alt_def_all: shows "\<forall>bl\<in>#\<B>'. int (card bl) = \<k>"
pt_perm_compose: fixes pi2 :: "('c \<times> 'c) list"   and pi1 :: "('c \<times> 'c) list"   and x :: "'b" assumes "pt TYPE('b) TYPE('c)"   and "at TYPE('c)" shows "pi2 \<bullet> pi1 \<bullet> x = (pi2 \<bullet> pi1) \<bullet> pi2 \<bullet> x"
hyps_1: shows "class P C = \<lfloor>(D, fs, ms)\<rfloor>"
sim_f_def: fixes M :: "('d, 'e, 'f) KripkeStructure"   and M' :: "('d, 'e, 'g) KripkeStructure"   and f :: "'f \<Rightarrow> 'g" shows "sim_f M M' f \<equiv> \<forall>a u v. (u, v) \<in> relations M a \<longrightarrow> (f u, f v) \<in> relations M' a"
final_iff_Nil: fixes s :: "'s" shows "final s = ([] \<in> L s)"