hcju/isabelleps huggingface dataset:
flattening_lemma_i: fixes \<pi> :: "'a strips_operator list list"   and \<Pi> :: "'a strips_problem" assumes "\<forall>ops\<in>set \<pi>. \<forall>op\<in>set ops. op \<in> set (\<Pi>\<^sub>\<O>)" shows "\<forall>op\<in>set (concat \<pi>). op \<in> set (\<Pi>\<^sub>\<O>)"
prems_1: fixes P' :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "(P', Q') \<in> Rel" shows "thesis"
distinct_1: fixes nat :: "nat" shows "enat nat \<noteq> \<infinity>"
store_id_valid: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i1 :: "'a"   and e :: "'b"   and mo :: "'a"   and i2 :: "'a" assumes "xs prefix of j"   and "Deliver (i1, Store e mo i2) \<in> set xs" shows "i1 = i2"
elims_1: fixes x :: "'a extN"   and xa :: "'a extN"   and y :: "bool" assumes "(x < xa) = y" obtains x :: "'a extN" and y :: "'a extN" where "xa__ = x" and "xa = y" and "ya__ = (x \<le> y \<and> \<not> y \<le> x)"
orthogonal_orthogonal_transformation: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "orthogonal_transformation f" shows "orthogonal (f x) (f y) = orthogonal x y"
eq_game_rel_def: shows "eq_game_rel \<equiv> {(p, q). eq_game p q}"
Contracts_axioms: shows "Contracts Xd Xh BPd BCh"
distinct_5: shows "TernaryFalse \<noteq> TernaryUnknown"
cosh_dist_perp_special_case: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> < 1"   and "\<bar>y\<bar> < 1" shows "cosh_dist (proj2_abs (vector [x, 0, 1])) (proj2_abs (vector [0, y, 1])) = cosh_dist K2_centre (proj2_abs (vector [x, 0, 1])) * cosh_dist K2_centre (proj2_abs (vector [0, y, 1]))"
r_zero: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "x \<oplus>\<^bsub>K [X]\<^esub> \<zero>\<^bsub>K [X]\<^esub> = x"
sq_norm_factor_bound: fixes h :: "int poly"   and f :: "int poly" assumes "h dvd f"   and "f \<noteq> 0" shows "\<parallel>h\<parallel>\<^sup>2 \<le> 2 ^ (2 * degree h) * \<parallel>f\<parallel>\<^sup>2"
upt_conv_Cons_Cons: fixes m :: "nat"   and n :: "nat"   and ns :: "nat list"   and q :: "nat" shows "(m # n # ns = [m..<q]) = (n # ns = [Suc m..<q])"
init_fin_bisim_wait_simps: fixes status1 :: "status"   and x1 :: "'x1"   and status2 :: "status"   and x2 :: "'x2" shows "(status1, x1) \<approx>iw (status2, x2) = (status1 = Running \<and> status2 = Running \<and> x1 \<approx>w x2)"
exhaust: fixes y :: "Player" obtains "y = Even"   | "y = Odd"
degree_polyneg: fixes p :: "'b poly"   and n :: "nat" assumes "isnpolyh p n" shows "Polynomial_Expression.degree (~\<^sub>p p) = Polynomial_Expression.degree p"
mult_left_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times c a) (times c b)"
Abs_eq_res_set: fixes bs :: "atom set"   and x :: "'a"   and cs :: "atom set"   and y :: "'a" shows "([bs]res. x = [cs]res. y) = ([bs \<inter> supp x]set. x = [cs \<inter> supp y]set. y)"
takeWhile_exp1: fixes n :: "nat"   and xs :: "cell list" shows "takeWhile (\<lambda>a. a = Oc) (Oc \<up> n @ xs) = Oc \<up> n @ takeWhile (\<lambda>a. a = Oc) xs"
changes_gt_der_def: fixes a :: "real"   and p :: "real poly" shows "changes_gt_der a p = changes_poly_at (pders p) a"
create_gba_from_nodes__ipath: fixes r :: "nat \<Rightarrow> 'a node" shows "ipath gba.E r = (\<forall>i. r i \<in> qs \<and> name (r i) \<in> incoming (r (Suc i)))"
induct: fixes x1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and x2 :: "'Var \<Rightarrow> 'BExp set option"   and x3 :: "'Var set \<times> 'Var set"   and x4 :: "'BExp set"   and x5 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and P :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> ('Var \<Rightarrow> 'BExp set option) \<Rightarrow> 'Var set \<times> 'Var set \<Rightarrow> 'BExp set \<Rightarrow> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "x1 \<R>\<^sup>3\<^bsub>x2,x3,x4\<^esub> x5"   and "\<And>c\<^sub>1 mds mem\<^sub>1 \<Gamma> \<S> Pa c\<^sub>2 mem\<^sub>2 c \<Gamma>' \<S>' P'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,Pa\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; \<turnstile> \<Gamma>,\<S>,Pa {c} \<Gamma>',\<S>',P'\<rbrakk> \<Longrightarrow> P \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<Gamma>' \<S>' P' \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"   and "\<And>c\<^sub>1 mds mem\<^sub>1 \<Gamma> \<S> Pa c\<^sub>2 mem\<^sub>2 c \<Gamma>' \<S>' P'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,Pa\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; P \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<Gamma> \<S> Pa \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; \<turnstile> \<Gamma>,\<S>,Pa {c} \<Gamma>',\<S>',P'\<rbrakk> \<Longrightarrow> P \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<Gamma>' \<S>' P' \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>" shows "P x1 x2 x3 x4 x5"
r_cancel_one: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and a :: "'e" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(a \<otimes>\<^bsub>G\<^esub> x = x) = (a = \<one>\<^bsub>G\<^esub>)"
noStuckn_startD': fixes t :: "('d, 'e) xstate"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate"   and n :: "nat" assumes "t \<noteq> Stuck"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "s \<noteq> Stuck"
prime_factorization_primorial': fixes k :: "nat" shows "prime_factorization (primorial' k) = mset_set (nth_prime ` {..<k})"
abelian_groupE_3: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"
InternalSets_diff: fixes X :: "'a star set"   and Y :: "'a star set" assumes "X \<in> InternalSets"   and "Y \<in> InternalSets" shows "X - Y \<in> InternalSets"
separated_tmps_nth_update: fixes i :: "nat"   and t :: "nat"   and s :: "stmt"   and sb :: "(stmt \<times> nat) memref list"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "'a"   and \<O> :: "'b" assumes "i < length ts"   and "valid_sops_stmt t s"   and "\<forall>(s', t')\<in>prog_configs sb. valid_sops_stmt t' s'"   and "\<forall>i\<in>load_tmps is. i < t"   and "\<forall>i\<in>read_tmps sb. i < t"   and "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"   and "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"   and "dom \<theta> \<union> load_tmps is = {i. i < t}" shows "separated_tmps (ts[i := ((s, t), is, \<theta>, sb, \<D>, \<O>)])"
gcd_mult_right_left_cancel: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a c" shows "gcd a (times c b) = gcd a b"
nofail_def: fixes S :: "'b nres" shows "nofail S \<equiv> S \<noteq> FAIL"
measurable_rec_tree: fixes t :: "'d \<Rightarrow> 'e tree"   and B :: "'d measure"   and M :: "'e measure"   and l :: "'d \<Rightarrow> 'f"   and A :: "'f measure"   and n :: "'d \<Rightarrow> 'e tree \<Rightarrow> 'e \<Rightarrow> 'e tree \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f" assumes "t \<in> B \<rightarrow>\<^sub>M tree_sigma M"   and "l \<in> B \<rightarrow>\<^sub>M A"   and "(\<lambda>(x, l, v, r, al, ar). n x l v r al ar) \<in> B \<Otimes>\<^sub>M tree_sigma M \<Otimes>\<^sub>M M \<Otimes>\<^sub>M tree_sigma M \<Otimes>\<^sub>M A \<Otimes>\<^sub>M A \<rightarrow>\<^sub>M A" shows "(\<lambda>x. rec_tree (l x) (n x) (t x)) \<in> B \<rightarrow>\<^sub>M A"
measure_st: fixes m :: "'b \<Rightarrow> nat" assumes "(strict {(x, y). x \<sqsubseteq> y})\<inverse> \<subseteq> measure m"   and "\<forall>x y. x \<sqsubseteq> y \<and> y \<sqsubseteq> x \<longrightarrow> m x = m y" shows "(strict {(S, S'). S \<sqsubseteq> S'})\<inverse> \<subseteq> measure (\<lambda>fd. \<Sum>x | x \<in> set (Abs_State.dom fd) \<and> \<not> \<top> \<sqsubseteq> fun fd x. m (fun fd x) + 1)"
weak_join_assoc_lemma: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion> (y \<squnion> z) .= \<Squnion>{x, y, z}"
\<iota>_in_hom: shows "\<guillemotleft>\<iota> : b \<star> b \<Rightarrow> b\<guillemotright>"
map_add_le_mapE: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option"   and h :: "'a \<Rightarrow> 'b option" assumes "g \<Oplus> f \<subseteq>\<^sub>m h" shows "g \<subseteq>\<^sub>m h"
only_one_SOME_edge: fixes a :: "'edge"   and mex :: "'node"   and x :: "nat" assumes "valid_edge a"   and "intra_kind (kind a)"   and "distance (targetnode a) mex x" shows "\<exists>!a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') mex x \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = (SOME n'. \<exists>a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') mex x \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = n')"
ordered_map_image: fixes S :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite S"   and "is_map S" shows "map (\<lambda>(a, b). (a, f a b)) (ordered_map S) = ordered_map ((\<lambda>(a, b). (a, f a b)) ` S)"
dist_eq_sqrt_dist_code: fixes p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "int \<times> int" shows "dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) = sqrt (real_of_int (dist_code p\<^sub>0 p\<^sub>1))"
c3_1: shows "conv_mirror \<gamma>3 \<in> conv ars"
pathCard_Cons: fixes x :: "Vars.var"   and xs :: "Vars.var list" shows "pathCard (x # xs) x = two_add\<cdot>once\<cdot>(pathCard xs x)"
simps_10: fixes var :: "nat"   and a :: "atom fm" shows "freeIn var (ExQ a) = freeIn (var + 1) a"
ind_hyp: assumes "0 < n * 10 + char_to_digit c \<or> is_strong_nat_string s" shows "nat_to_string (string_to_nat_aux (n * 10 + char_to_digit c) s) = (if n * 10 + char_to_digit c = 0 then [] else nat_to_string (n * 10 + char_to_digit c)) @ s"
zip_commute: fixes xs :: "'d list"   and ys :: "'e list" shows "zip xs ys = Map2.map2 (\<lambda>x y. (y, x)) ys xs"
strictly_generalizes_atm_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'b"   and B :: "'b" shows "substitution_ops.strictly_generalizes_atm subst_atm A B = (substitution_ops.generalizes_atm subst_atm A B \<and> \<not> substitution_ops.generalizes_atm subst_atm B A)"
Y: shows "Complete_Partial_Order.chain (\<sqsubseteq>) Y"
GatewaySystem_L1: fixes i :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and vc :: "nat \<Rightarrow> vcType list"   and k :: "nat"   and t :: "nat" assumes "ServiceCenter i a"   and "GatewayReq req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"   and "i (t + 3 + k + d) \<noteq> []"   and "\<forall>x\<le>d + d. ack (t + 3 + k + x) = [sending_data]" shows "vc (2 * d + (t + (4 + k))) = [vc_com]"
f_shrink_last_message: fixes xs :: "'b message_af list"   and k :: "nat" assumes "length xs mod k = 0" shows "last_message (xs \<div> k) = last_message xs"
elims: fixes x :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and xa :: "ipt_l4_ports"   and y :: "'b common_primitive match_expr" assumes "l4_ports_negate_one x xa = y" obtains C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive" and proto :: "8 word" and pts :: "(16 word \<times> 16 word) list" where "x = C" and "xa = L4Ports proto pts" and "y = MatchOr (MatchNot (Match (Prot (Proto proto)))) (Match (C (L4Ports proto (raw_ports_invert pts))))"
dom_subid: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.domain_near_semiring_one plus times one domain_op less_eq less" shows "less_eq (domain_op x) one"
int_eq: shows "LINT x|MX. f x * log b (f x) = LINT x|distr M MX X. log b (f x)"
stone_minus_def: fixes x :: "'a"   and y :: "'a" shows "x - y = x \<sqinter> - y"
intro: fixes get_scdom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_scdom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_scdom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "get_scdom_component = l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_scdom_component get_owner_document get_disconnected_nodes to_tree_order"   and "is_strongly_scdom_component_safe = l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order"   and "is_weakly_scdom_component_safe = l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order" shows "l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_scdom_component is_strongly_scdom_component_safe is_weakly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order"
vpeq_reflexive: shows "\<forall>a u. vpeq u a a"
pw_bind_le_iff: fixes M :: "'b nres"   and f :: "'b \<Rightarrow> 'a nres"   and S :: "'a nres" shows "(M \<bind> f \<le> S) = ((nofail S \<longrightarrow> nofail M) \<and> (\<forall>x. nofail M \<and> inres M x \<longrightarrow> f x \<le> S))"
suffixes_eq_snoc: fixes ys :: "'c list"   and xs :: "'c list list"   and x :: "'c list" shows "(suffixes ys = xs @ [x]) = ((ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = z # zs \<and> xs = suffixes zs)) \<and> x = ys)"
non_empty: fixes i :: "nat" shows "space (M i) \<noteq> {}"
matchC_C_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" shows "matchC_C theta c d \<equiv> \<forall>s t. s \<approx> t \<longrightarrow> (\<exists>P F. mC_C theta c d s t P F)"
simps_18: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and f4 :: "('c, 'd) Combinators \<Rightarrow> ('c, 'd) Combinators \<Rightarrow> 'b"   and x31 :: "'c"   and x32 :: "'c"   and x33 :: "'d" shows "(case AllowPortFromTo x31 x32 x33 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f3 x31 x32 x33"
dagger_plus_one: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "t_one \<union> t_star x = t_star x"
fst_netgmap_none: fixes i :: "nat"   and s :: "'d net_state"   and sr :: "'d \<Rightarrow> 'e \<times> 'f" assumes "i \<notin> net_ips s" shows "fst (netgmap sr s) i = None"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [t, t]"
sigma_algebra_axioms: shows "sigma_algebra \<Omega> (sigma_sets \<Omega> A)"
mult_zero_right: fixes times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and a :: "'k" assumes "class.mult_zero times zero" shows "times a zero = zero"
deduct_if_closure_synth: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "decomp_closure M M'"   and "M' \<turnstile>\<^sub>c t" shows "M \<turnstile> t"
add_mono_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "a + c \<le> b + c"
psi_elem: fixes x :: "'a sort_key extended \<times> 'a sort_key extended"   and i :: "'a" assumes "is_interval x" shows "elem \<lbrakk>\<Psi> x i\<rbrakk> x"
univ_is_not_pr: shows "univ_for_pr \<notin> PrimRec1"
scalar_prod_left_zero: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "0\<^sub>v n \<bullet> v = (0::'a)"
zerofun_is_aezfun: shows "0 \<in> aezfun_set"
t2: shows " (x, close \<Gamma> T\<^sub>1) # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2 "
\<Delta>\<Phi>_merge: fixes h1 :: "'b tree"   and hs1 :: "'b tree"   and x1 :: "'b"   and h2 :: "'b tree"   and hs2 :: "'b tree"   and x2 :: "'b" assumes "h1 = \<langle>hs1, x1, \<langle>\<rangle>\<rangle>"   and "h2 = \<langle>hs2, x2, \<langle>\<rangle>\<rangle>" shows "\<Phi> (merge h1 h2) - \<Phi> h1 - \<Phi> h2 \<le> log 2 (real (size h1 + size h2)) + 1"
borel_adapt_stoch_proc_borel_measurable: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b" assumes "borel_adapt_stoch_proc G X" shows "\<forall>n. random_variable borel (X n)"
atoms_list_disjE: fixes a :: "'c"   and fs :: "'c fm list" assumes "a \<in> atoms (list_disj fs)" shows "a \<in> \<Union> (atoms ` set fs)"
inline1_bind_gpv: fixes callee :: "'e \<Rightarrow> 'i \<Rightarrow> ('h \<times> 'e, 'f, 'g) gpv"   and gpv :: "('j, 'i, 'h) gpv"   and f :: "'j \<Rightarrow> ('d, 'i, 'h) gpv"   and s :: "'e" shows "inline1 callee (gpv \<bind> f) s = inline1 callee gpv s \<bind> (\<lambda>res. case res of Inl (x, s') \<Rightarrow> inline1 callee (f x) s' | Inr (out, rpv', rpv) \<Rightarrow> return_spmf (Inr (out, rpv', rpv \<bind> f)))"
Ri_def: fixes x :: "'c"   and y :: "'d"   and z :: "'d" shows "Ri x y z = \<Sqinter> {f z |f. x \<le> f y \<and> mono f}"
small_elts_of_set: fixes x :: "V set" assumes "small x" shows "elts (ZFC_in_HOL.set x) = x"
Inf_utrans_def: fixes S :: "(('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real) set" shows "Inf_utrans S = (if S = {} then \<lambda>P s. 1 else Inf_trans S)"
assms: shows "h \<turnstile> ok create_element document_ptr tag"
assms_2: shows "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = A"
fset_of_list_simps_2: fixes x21 :: "'c"   and x22 :: "'c list" shows "fset_of_list (x21 # x22) = finsert x21 (fset_of_list x22)"
plus_lens_distr: fixes Z :: "'a \<Longrightarrow> 'b"   and X :: "'c \<Longrightarrow> 'a"   and Y :: "'d \<Longrightarrow> 'a" assumes "mwb_lens Z" shows "(X +\<^sub>L Y) ;\<^sub>L Z = X ;\<^sub>L Z +\<^sub>L Y ;\<^sub>L Z"
sum_list_closed_prod: fixes R :: "'a set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xys :: "('b \<times> 'c) list" assumes "Ring1 R"   and "set (map (\<lambda>(x, y). f x y) xys) \<subseteq> R" shows "(\<Sum>(x, y)\<leftarrow>xys. f x y) \<in> R"
unity_def: shows "M\<^sub>\<omega>.unity \<equiv> S.cod \<omega>"
zero_in_Field: assumes "Field r \<noteq> {}" shows "zero \<in> Field r"
rep_eq: fixes x :: "'d \<Rightarrow> bool"   and xa :: "('d, 'e) fmap" shows "fmlookup (fmfilter x xa) = map_filter x (fmlookup xa)"
C_eq_if_mr_eq2: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators list"   and r :: "((int \<times> int) set set, int) Combinators"   and b :: "((int \<times> int) set set, int) Combinators list" assumes "applied_rule_rev C x a = \<lfloor>r\<rfloor>"   and "applied_rule_rev C x b = \<lfloor>r\<rfloor>"   and "a \<noteq> []"   and "b \<noteq> []" shows "C (list2FWpolicy a) x = C (list2FWpolicy b) x"
fresh_bij: fixes pi :: "(name \<times> name) list"   and a :: "name"   and x :: "'d" shows "pi \<bullet> a \<sharp> pi \<bullet> x = a \<sharp> x"
ultra_filter: fixes F :: "'a set" assumes "proper_filter F" shows "\<exists>G. ultra_filter G \<and> F \<subseteq> G"
matrix_to_iarray_nrows: fixes A :: "(('d, 'e) vec, 'f) vec" shows "nrows A = nrows_iarray (matrix_to_iarray A)"
lunit_char_3: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and a :: "'d" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "\<exists>!f. partial_magma.in_hom C f (T (monoidal_category.unity C \<iota>, a)) a \<and> T (monoidal_category.unity C \<iota>, f) = C (T (\<iota>, a)) (category.inv C (\<alpha> (monoidal_category.unity C \<iota>, monoidal_category.unity C \<iota>, a)))"
simps_1: fixes T :: "typ"   and t :: "Core.term" shows "eta_norm (Abs T t) = (case eta_norm t of Ct literal typ \<Rightarrow> Abs T (Ct literal typ) | Fv variable typ \<Rightarrow> Abs T (Fv variable typ) | Bv nat \<Rightarrow> Abs T (Bv nat) | Abs typ term \<Rightarrow> Abs T (Abs typ term) | f $ Ct literal typ \<Rightarrow> Abs T (f $ Ct literal typ) | f $ Fv variable typ \<Rightarrow> Abs T (f $ Fv variable typ) | f $ Bv 0 \<Rightarrow> if is_dependent f then Abs T (f $ Bv 0) else decr 0 (eta_norm f) | f $ Bv (Suc nata) \<Rightarrow> Abs T (f $ Bv (Suc nata)) | f $ Abs typ term \<Rightarrow> Abs T (f $ Abs typ term) | f $ (term1b $ term2a) \<Rightarrow> Abs T (f $ (term1b $ term2a)))"
proj_stoch_proj_invariant: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and n :: "nat"   and w :: "bool stream" assumes "borel_adapt_stoch_proc F X" shows "proj_stoch_proc X n w = proj_stoch_proc X n (pseudo_proj_True n w)"
extend_def: fixes forms :: "sequent"   and seq :: "sequent" shows "extend forms seq \<equiv>  antec forms + antec seq \<Rightarrow>* succ forms + succ seq"
ereal_power: fixes x :: "real"   and n :: "nat" shows "ereal x ^ n = ereal (x ^ n)"
distinct_9: shows "Unlock \<noteq> ReleaseAcquire"
fmlookup_of_list: fixes m :: "('e \<times> 'f) list" shows "fmlookup (fmap_of_list m) = map_of m"
dests_vD_inc_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:9, PRreq-:17, PRrep-:8} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> the (dests \<xi> ip) = inc (sqn (rt \<xi>) ip))) \<and> (l = PRerr-:1 \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> sqn (rt \<xi>) ip < the (dests \<xi> ip))))"
interpret_floatarith_sin: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) vs = sin (interpret_floatarith a vs)"
Carmichael_twopow: fixes k :: "nat" shows "Carmichael (2 ^ k) = (if k \<le> 2 then 2 ^ (k - 1) else 2 ^ (k - 2))"
emeasure_countable_singleton: fixes X :: "'a set"   and M :: "'a measure" assumes "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "countable X" shows "emeasure M X = \<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space X"