hcju/isabelleps huggingface dataset:
r_into_rtrancl: fixes p :: "'a \<times> 'a"   and r :: "('a \<times> 'a) set" assumes "p \<in> r" shows "p \<in> r\<^sup>*"
m_lcomm: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier H"   and "y \<in> carrier H"   and "z \<in> carrier H" shows "x \<oplus>\<^bsub>H\<^esub> (y \<oplus>\<^bsub>H\<^esub> z) = y \<oplus>\<^bsub>H\<^esub> (x \<oplus>\<^bsub>H\<^esub> z)"
r_length_prim: shows "prim_recfn 1 r_length"
presheaf_of_rings_axioms: shows "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
assms_3: shows "\<forall>x. x \<le> (g \<circ> f) x"
\<zeta>_def: shows "\<zeta> \<equiv> Re (zeta (complex_of_real (\<alpha> + 1)))"
checkI_OKI: fixes inp :: "'c \<Rightarrow> 'd option" shows "checkI (OKI inp) = inp"
mk_monpol_list_correct: fixes l :: "'a list"   and pps :: "(polex \<times> polex) list" assumes "in_carrier l"   and "Ipolex_polex_list l pps" shows "Imon_pol_list l (mk_monpol_list pps)"
merge_def: fixes qs :: "('c \<times> 'd) list"   and ps :: "('c \<times> 'd) list" shows "merge qs ps = foldr (\<lambda>(k, v). update k v) ps qs"
affine_dependent_imp_dependent: fixes s :: "'a set" assumes "affine_dependent s" shows "dependent s"
runit_def: fixes a :: "'c" shows "\<r>\<^sub>C[a] \<equiv> THE f. \<guillemotleft>f : a \<otimes>\<^sub>C \<I>\<^sub>C \<rightarrow>\<^sub>C a\<guillemotright> \<and> f \<otimes>\<^sub>C \<I>\<^sub>C = (a \<otimes>\<^sub>C \<iota>\<^sub>C) \<cdot>\<^sub>C \<a>\<^sub>C[a, \<I>\<^sub>C, \<I>\<^sub>C]"
weak_map_of_SomeI: fixes k :: "'c"   and x :: "'d"   and l :: "('c \<times> 'd) list" assumes "(k, x) \<in> set l" shows "\<exists>x. map_of l k = Some x"
nonneg_dirichlet_series_axioms: shows "nonneg_dirichlet_series f"
REP_in_hhom_2: fixes F :: "('a, 'a set) Maps.arr"   and A :: "('a, 'a set) Maps.arr"   and B :: "('a, 'a set) Maps.arr" assumes "Maps.in_hom F A B" shows "is_left_adjoint (Maps.REP F)"
\<alpha>_def: fixes f :: "'c local.term set \<times> 'c local.term set \<times> 'c local.term set" shows "\<F>\<^sub>SC.\<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C f \<equiv> if \<F>\<^sub>SC.CCC.arr f then \<F>\<^sub>SC.comp (fst f \<otimes>\<^sub>S fst (snd f) \<otimes>\<^sub>S snd (snd f)) \<a>\<^sub>S[\<F>\<^sub>SC.dom (fst f), \<F>\<^sub>SC.dom (fst (snd f)), \<F>\<^sub>SC.dom (snd (snd f))] else \<F>\<^sub>SC.null"
vector_rectangle: fixes x :: "'a" assumes "is_vector x" shows "x ; 1 ; x = x"
2: shows "Sel (\<T> k u) (TokensAt k (\<J> k u)) \<subseteq> TokensAt k (\<J> k u)"
inter_ABC_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points" assumes "line A B \<noteq> line C A" shows "Pascal_Property.inter (line A B) (line C A) = A"
multrow_mat_carrier: fixes n :: "nat"   and k :: "nat"   and a :: "'b" shows "multrow_mat n k a \<in> carrier_mat n n"
vec_elements: fixes v :: "'b Matrix.vec" shows "vec_elements v = {v $v i |i. i < dim_vec v}"
f_aggregate_append_mod: fixes xs :: "'d list"   and k :: "nat"   and ys :: "'d list"   and ag :: "'d list \<Rightarrow> 'd" assumes "length xs mod k = 0" shows "f_aggregate (xs @ ys) k ag = f_aggregate xs k ag @ f_aggregate ys k ag"
Sups_upper: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X" shows "\<Sqinter> X \<le> x"
recAbs_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) Recursion.model" shows "recAbs MOD \<equiv> fromIMorAbs (iterAbs (fromMOD MOD))"
assms_4: shows "gen_dfs_outer E U V0 it V False"
A: fixes self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(\<lambda>_. self \<tau>.boss \<doteq> null) \<tau> = (self.boss \<doteq> null) \<tau>"
OclIf_true: fixes B\<^sub>1 :: "'a state \<times> 'a state \<Rightarrow> 'b"   and B\<^sub>2 :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(if true then B\<^sub>1 else B\<^sub>2 endif) = B\<^sub>1"
inorder_insert: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (AVL_Bal_Set.insert x t) = ins_list x (Tree2.inorder t)"
disjI2_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and q :: "'b"   and A :: "'b set"   and p :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "q \<in> Cn A" shows "disj_PL p q \<in> Cn A"
vdw_lemma_def: fixes k :: "nat"   and m :: "nat"   and l :: "nat"   and n :: "nat" shows "vdw_lemma k m l n = (\<forall>a b col. a + int n \<le> b + 1 \<and> col \<in> {a..b} \<rightarrow> {..<k} \<longrightarrow> (\<exists>start steps. (\<forall>i<m. 0 < steps i) \<and> is_multi_arith_prog_on (l + 1) m start steps a b \<and> (let f = multi_arith_prog m start steps in \<forall>c\<in>{0..<m} \<rightarrow> {0..l}. \<forall>s<m. (\<forall>j\<le>s. c j < l) \<longrightarrow> col (f c) = col (f (\<lambda>i. if i \<le> s then 0 else c i)))))"
finite_lin_spanTr1: fixes A :: "'b set"   and z :: "'a"   and h :: "nat \<Rightarrow> 'a"   and n :: "nat"   and t :: "nat \<Rightarrow> 'b" assumes "ideal R A"   and "z \<in> carrier M" shows "h \<in> {j. j \<le> n} \<rightarrow> {z} \<and> t \<in> {j. j \<le> n} \<rightarrow> A \<longrightarrow> (\<exists>s\<in>{0} \<rightarrow> A. \<Sigma>\<^sub>e M (\<lambda>j. t j \<cdot>\<^sub>s h j) n = s 0 \<cdot>\<^sub>s z)"
homeomorphismI: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set"   and g :: "'d \<Rightarrow> 'c" assumes "continuous_on S f"   and "continuous_on T g"   and "f ` S \<subseteq> T"   and "g ` T \<subseteq> S"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"   and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y" shows "homeomorphism S T f g"
one_sheaf_spec_def: fixes U :: "('a \<times> 'a) set set set" shows "local.local.one_sheaf_spec U \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.one_rel (local.carrier_quotient_ring\<setminus>\<pp>) local.carrier_quotient_ring local.add_rel local.mult_rel local.zero_rel local.one_rel"
l_distr: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "z \<in> carrier Zp" shows "(x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
eq: shows "(\<lambda>x. (X x, Y x)) ` msgs = (\<lambda>(x, y). (y, x)) ` (\<lambda>x. (Y x, X x)) ` msgs"
D_def: shows "D \<equiv> Im (Ln (f x)) + Im (Ln (prod f A))"
avl_insert_2: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "avl t" shows "height (AVL_Set_Code.insert x t) \<in> {height t, height t + 1}"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "C.arr f"   and "C.dom f = a" shows "f \<cdot> a = f"
eql: shows "row (two_block_non_interfering A A\<^sup>T) (db + i) = 0\<^sub>v (dim_col A) @\<^sub>v row A\<^sup>T i"
proots_count_of_root_set: fixes P :: "'a poly"   and R :: "'a set"   and S :: "'a set" assumes "P \<noteq> 0"   and "R \<subseteq> S"   and "\<And>x. x \<in> R \<Longrightarrow> poly P x = (0::'a)" shows "card R \<le> proots_count P S"
J_def: shows "J \<equiv> relcycle_group (p - Suc 0) (subtopology X S) {}"
borel_eq_halfspace_less: shows "borel = sigma UNIV ((\<lambda>(a, i). {x. x \<bullet> i < a}) ` (UNIV \<times> Basis))"
welltyped_Apps: fixes ts :: "expr list"   and Ts :: "Unary_PCF.type list"   and f :: "expr"   and U :: "Unary_PCF.type" assumes "list_all2 (:::) ts Ts"   and "f ::: Ts \<rightarrow>\<rightarrow> U" shows "f \<bullet> ts ::: U"
linorder_cases: fixes x :: "('x, nat) pp"   and y :: "('x, nat) pp" obtains "ord_strict x y"   | "x = y"   | "ord_strict y x"
multiple_t_wise_balance: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int"   and n :: "nat" assumes "t_wise_balance point_set block_collection grouping index"   and "0 < n" shows "t_wise_balance point_set (repeat_mset n block_collection) grouping (index * int n)"
cBex_triv_one_point1: fixes A :: "'f cset"   and a :: "'f" shows "cBex A (\<lambda>x. x = a) = cin a A"
properfactor_unitE: fixes u :: "'a"   and a :: "'a"   and P :: "bool" assumes "u \<in> Units G"   and "properfactor G a u"   and "a \<in> carrier G" shows "P"
inner_simps_4: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x - y) \<bullet> z = x \<bullet> z - y \<bullet> z"
mset_tree_empty_iff: fixes t :: "'b tree" shows "(mset_tree t = {#}) = (t = \<langle>\<rangle>)"
wp_assume: fixes p :: "'b Assertion"   and q :: "'b Assertion" shows "wp [\<cdot> p ] q = - p \<squnion> q"
eq: shows "(\<lambda>(x, y). (if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0) * log b ((if (x, y) \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy (x, y) else 0) / Py y)) = (\<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Pxy x * log b (Pxy x / Py (snd x)) else 0)"
measurable_compose_n: fixes T :: "'a \<Rightarrow> 'a"   and M :: "'a measure"   and n :: "nat" assumes "T \<in> M \<rightarrow>\<^sub>M M" shows "T ^^ n \<in> M \<rightarrow>\<^sub>M M"
simps_2: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and L :: "lock_action"   and Ls :: "lock_action list" shows "lock_actions_ok' l t (L # Ls) = (L = Lock \<and> \<not> may_lock l t \<or> lock_action_ok l t L \<and> lock_actions_ok' (upd_lock l t L) t Ls)"
B: fixes a :: "'a" shows "(1::'a) l\<rightarrow> a \<le> (1::'a)"
lfp_loop_fp: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "sound P" shows "\<lambda>s. \<guillemotleft> G \<guillemotright> s * wp body (\<lambda>s. bound_of P) s + \<guillemotleft> \<N> G \<guillemotright> s * P s \<tturnstile> \<lambda>s. bound_of P"
hyps_5: assumes "t \<notin> Fault ` F" shows "strip F \<Gamma>\<turnstile> \<langle>While c s,s'\<rangle> =n\<Rightarrow> t"
delete_prefix_trie: fixes xs :: "bool list"   and b :: "bool"   and l :: "trie"   and r :: "trie" shows "delete xs (prefix_trie xs (Nd b (l, r))) = (if (l, r) = (Lf, Lf) then Lf else prefix_trie xs (Nd False (l, r)))"
prime_power_minus_oneD: fixes a :: "nat"   and n :: "nat" assumes "prime (a ^ n - 1)" shows "n = 1 \<or> a = 2"
RedFAcc: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and Ds :: "char list list"   and Cs' :: "char list list"   and Cs :: "char list list"   and fs :: "char list \<Rightarrow> val option"   and F :: "char list"   and v :: "val"   and E :: "char list \<Rightarrow> ty option"   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" assumes "hp s a = \<lfloor>(D, S)\<rfloor>"   and "Ds = Cs' @\<^sub>p Cs"   and "(Ds, fs) \<in> S"   and "fs F = \<lfloor>v\<rfloor>" shows "P,E \<turnstile> \<langle>ref (a, Cs')\<bullet>F{Cs},s\<rangle> \<rightarrow> \<langle>Val v,s\<rangle>"
Q: shows "((\<mu> U\<^sub>r \<eta> \<in> old (fst (n, ns)) \<longrightarrow> \<eta> \<in> old (fst (n, ns)) \<union> new (fst (n, ns)) \<or> \<mu> \<in> old (fst (n, ns)) \<union> new (fst (n, ns)) \<and> \<mu> U\<^sub>r \<eta> \<in> next (fst (n, ns))) \<and> (\<forall>nd\<in>snd (n, ns). \<mu> U\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst (n, ns)) \<longrightarrow> \<eta> \<in> old nd \<or> \<mu> \<in> old nd \<and> \<mu> U\<^sub>r \<eta> \<in> old (fst (n, ns)) \<union> new (fst (n, ns)))) \<and> (\<forall>na\<in>snd (n, ns). \<forall>nd\<in>snd (n, ns). \<mu> U\<^sub>r \<eta> \<in> old na \<and> name na \<in> incoming nd \<longrightarrow> \<eta> \<in> old na \<or> \<mu> \<in> old na \<and> \<mu> U\<^sub>r \<eta> \<in> old nd) \<and> ((\<forall>nm\<in>incoming (fst (n, ns)). nm < name (fst (n, ns))) \<and> 0 < name (fst (n, ns)) \<and> (\<forall>q\<in>snd (n, ns). name q < name (fst (n, ns)) \<and> (\<forall>nm\<in>incoming q. nm < name (fst (n, ns))))) \<and> expand_name_ident (snd (n, ns))"
heap_is_wellformed_one_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and ptr' :: "(_) object_ptr"   and children' :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children'"   and "set children \<inter> set children' \<noteq> {}" shows "ptr = ptr'"
Ana_subst'_or_Ana_keys_subterm: shows "(\<forall>f T \<delta> K R. Ana (Fun f T) = (K, R) \<longrightarrow> Ana (Fun f T \<cdot> \<delta>) = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>, R \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)) \<or> (\<forall>t K R k. Ana t = (K, R) \<longrightarrow> k \<in> set K \<longrightarrow> k \<sqsubset> t)"
repc_consts: fixes I :: "interp"   and f :: "char"   and d :: "real"   and c :: "char" shows "Consts (repc I f d) c = (if c = f then d else Consts I c)"
rev_singleton_conv: fixes xs :: "'b list"   and x :: "'b" shows "(rev xs = [x]) = (xs = [x])"
VectorSpace_VectorSpaceHomSet: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and W :: "'c set" assumes "VectorSpace smult V"   and "VectorSpace smult' W" shows "VectorSpace (\<lambda>a T v. smult' a (T v)) (VectorSpaceHomSet smult V smult' W)"
prv_imp_eqvER: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "bprv (eqv \<phi>1 \<phi>2)" shows "bprv (imp \<phi>2 \<phi>1)"
B: shows "B = \<phi> o\<^sub>C\<^sub>L \<phi>*"
simps_2: fixes i :: "nat"   and j :: "nat"   and uu :: "'b option"   and t :: "'b tree"   and u :: "'b tree" shows "discharge (Node i j uu t u) = Node i j None t u"
new_shadow_root_no_child_nodes: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr) \<rightarrow>\<^sub>r []"
cinsertI2: fixes a :: "'f"   and B :: "'f cset"   and b :: "'f" assumes "cin a B" shows "cin a (cinsert b B)"
exhaust_sel: fixes writerT :: "('d, 'e, 'f) writerT" obtains "writerT = WriterT (run_writer writerT)"
bounded_iff: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" shows "(a \<subseteq># b \<inter># c) = (a \<subseteq># b \<and> a \<subseteq># c)"
inner_aforms'_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) list"   and Y :: "(real \<times> real pdevs) list" shows "inner_aforms' p X Y = (let fas = [inner_floatariths (map floatarith.Var [0..<length X]) (map floatarith.Var [length X..<length X + length Y])] in approx_slp_outer p (length fas) fas (X @ Y))"
simps_12: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'d \<Rightarrow> 'a"   and x3 :: "'d" shows "(case In3 x3 of In1 x \<Rightarrow> f1 x | In2 x \<Rightarrow> f2 x | In3 x \<Rightarrow> f3 x) = f3 x3"
canceling_def: fixes a :: "bool \<times> 'c"   and b :: "bool \<times> 'c" shows "canceling a b = (snd a = snd b \<and> fst a \<noteq> fst b)"
scale: fixes S1 :: "'c set"   and S2 :: "'d set"   and s1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and b :: "'c"   and r :: "'e" assumes "module_hom_on S1 S2 s1 s2 f"   and "b \<in> S1" shows "f (s1 r b) = s2 r (f b)"
is_closed_eq: fixes A :: "'a set"   and x :: "'a"   and x' :: "'a" assumes "is_closed A"   and "x \<in> A"   and "x .= x'"   and "x' \<in> carrier L" shows "x' \<in> A"
ideal_vimage: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and I :: "'g set" assumes "ring_hom_ring R S h"   and "ideal I S" shows "ideal {r \<in> carrier R. h r \<in> I} R"
lcm_proj1_if_dvd_int: fixes x :: "int"   and y :: "int" assumes "x dvd y" shows "lcm y x = \<bar>y\<bar>"
index_row_2: fixes A :: "'b mat"   and i :: "nat" shows "dim_vec (Matrix.row A i) = dim_col A"
pure_pure: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog" assumes "h \<turnstile> ok f"   and "pure f h" shows "h \<turnstile> f \<rightarrow>\<^sub>h h"
bind_returns_heap_E2: fixes h :: "'m"   and f :: "('m, 'n, 'p) prog"   and g :: "'p \<Rightarrow> ('m, 'n, 'o) prog"   and h' :: "'m" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h'"   and "pure f h" obtains x :: "'p" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
inf_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "inf_prod x y = (fst x \<sqinter> fst y, snd x \<sqinter> snd y)"
eq: shows "new_syz_sigs ss bs p = filter_min_append (adds\<^sub>t) ss ss'"
q_def: shows "q = p \<circ>\<^sub>p r"
wt_step_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and \<tau>s :: "'b list" shows "wt_step r T step \<tau>s = (\<forall>p<length \<tau>s. \<tau>s ! p \<noteq> T \<and> stable r step \<tau>s p)"
op_inter_fst_ivl_scaleR2_def: fixes X :: "('c \<times> 'd) set"   and Y :: "'c set" shows "op_inter_fst_ivl_scaleR2 X Y = scaleR2_rep X \<bind> (\<lambda>((l, u), X). ivl_rep (op_inter_fst X Y) \<bind> (\<lambda>(i, s). let R = op_inter_fst (op_atLeastAtMost_ivl i s) Y in scaleRe_ivl_coll_spec l u (filter_empty_ivls (mk_coll R))))"
wd_DC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "well_def a"   and "well_def b" shows "well_def (a \<Sqinter> b)"
insertCI: fixes a :: "'b"   and B :: "'b set"   and b :: "'b" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> Set.insert b B"
actionFreshChain_2: fixes xvec :: "name list"   and \<alpha> :: "'d action" shows "xvec \<sharp>* \<alpha> = (xvec \<sharp>* subject \<alpha> \<and> xvec \<sharp>* bn \<alpha> \<and> xvec \<sharp>* object \<alpha>)"
intro: fixes jkbp :: "'a \<Rightarrow> ('a, 'b, 'c) GC list"   and jview :: "'a \<Rightarrow> 'd Trace \<Rightarrow> 'e"   and envObs :: "'a \<Rightarrow> 'd \<Rightarrow> 'f"   and jviewInit :: "'a \<Rightarrow> 'f \<Rightarrow> 'e"   and jviewIncr :: "'a \<Rightarrow> 'f \<Rightarrow> 'e \<Rightarrow> 'e"   and envInit :: "'d list"   and envAction :: "'d \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'd"   and envVal :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and simf :: "'d Trace \<Rightarrow> 'h"   and simRels :: "'a \<Rightarrow> ('h \<times> 'h) set"   and simVal :: "'h \<Rightarrow> 'b \<Rightarrow> bool" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"   and "SimIncrEnvironment_axioms jkbp envInit envAction envTrans envVal jview simf simRels simVal" shows "SimIncrEnvironment jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal"
T_ker: fixes v :: "'c" assumes "v \<in> kerT" shows "T v = \<zero>\<^bsub>W\<^esub>"
unique_connecting_path_properties_1: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "path (v \<leadsto> w)"
hd_in_awalk_verts_1: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" shows "u \<in> set (awalk_verts u p)"
gallery_rev: fixes xs :: "'w set set list" assumes "gallery xs" shows "gallery (rev xs)"
rbt_lookup_rbt_insert: fixes t :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c" assumes "is_rbt t" shows "rbt_lookup (rbt_insert k v t) = rbt_lookup t(k \<mapsto> v)"
induct: fixes x :: "'b set"   and sp :: "'b set"   and A :: "'b set set"   and P :: "'b set \<Rightarrow> bool" assumes "x \<in> sigma_sets sp A"   and "\<And>a. a \<in> A \<Longrightarrow> P a"   and "P {}"   and "\<And>a. \<lbrakk>a \<in> sigma_sets sp A; P a\<rbrakk> \<Longrightarrow> P (sp - a)"   and "\<And>a. \<lbrakk>\<And>i. a i \<in> sigma_sets sp A; \<And>i. P (a i)\<rbrakk> \<Longrightarrow> P (\<Union> (range a))" shows "P x"
hyps_5: fixes b :: "real" assumes "b \<le> s" shows "finite_Psegments P a b"
elims: fixes key_gen :: "nat \<Rightarrow> ('b \<times> 'c) spmf"   and sign :: "nat \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a spmf"   and valid_message :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "nat"   and xa :: "('b \<times> 'c \<times> ('d \<times> 'a) list) option"   and xb :: "'d"   and y :: "('a \<times> ('b \<times> 'c \<times> ('d \<times> 'a) list) option) spmf" assumes "suf_cma.sign_oracle key_gen sign valid_message x xa xb = y" obtains \<eta> :: "nat" and m :: "'d" where "x = \<eta>__" and "xa = None" and "xb = m__" and "y = suf_cma.vkey_oracle key_gen \<eta>__ None () \<bind> (\<lambda>(uu_, \<sigma>). suf_cma.sign_oracle key_gen sign valid_message \<eta>__ \<sigma> m__)"   | \<eta> :: "nat" and vkey :: "'b" and skey :: "'c" and m :: "'d" and log :: "('d \<times> 'a) list" where "x = \<eta>" and "xa = Some (vkey, skey, log)" and "xb = m" and "y = (if valid_message \<eta> m then sign \<eta> skey m \<bind> (\<lambda>sig. return_spmf (sig, Some (vkey, skey, (m, sig) # log))) else return_pmf None)"
p_def: shows "p = (\<lambda>n. \<Prod>k = 1..n. 4 * (real k)\<^sup>2 / (4 * (real k)\<^sup>2 - 1))"
not_inject: fixes P :: "bool"   and Q :: "bool" assumes "(\<not> P) = (\<not> Q)" shows "P = Q"
Liminf_ereal_mult_right: fixes F :: "'a filter"   and c :: "real"   and f :: "'a \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "0 \<le> c" shows "Liminf F (\<lambda>n. f n * ereal c) = Liminf F f * ereal c"
inter_guards_execn_Normal_noFault: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "'d"   and n :: "nat"   and t :: "('d, 'f) xstate" assumes "(c1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t"   and "\<not> isFault t" shows "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> =n\<Rightarrow> t \<and> \<Gamma>\<turnstile> \<langle>c2,Normal s\<rangle> =n\<Rightarrow> t"
total_quasi_ordered_set_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "total_quasi_ordered_set A r = total_quasi_ordered_set A r'"
vimage_diff: fixes C :: "'b set" shows "carrier - f -` (carrier\<^bsub>T\<^esub> - C) = carrier \<inter> f -` C"
Rep_JF2: fixes x :: "'d JF2" shows "Rep_JF2 x \<in> car_final2"
empty_subset_add_mset: fixes x :: "'b"   and M :: "'b multiset" shows "{#} \<subset># add_mset x M"
num_2: shows "\<forall>f\<in>\<R>. \<exists>i. \<psi> i = f"
echelon_form_interchange: shows "echelon_form_upt_k (interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n)) k"
RG_proj_mult_leftdelta: fixes G :: "'a set"   and g :: "'a"   and r :: "'b"   and x :: "('b, 'a) aezfun" assumes "Group G"   and "g \<in> G" shows "aezfun_setspan_proj G (r \<delta>\<delta> g * x) = r \<delta>\<delta> g * aezfun_setspan_proj G x"