hcju/isabelleps huggingface dataset:
ZFfunCompAppZFfun: fixes f :: "ZF"   and g :: "ZF" assumes "isZFfun f"   and "isZFfun g"   and "|dom|g = |cod|f" shows "isZFfun (f |o| g)"
at0_def: shows "at0 = Cn 1 r_nth [recf.Id 1 0, Z]"
smap2_szip: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and s1 :: "'d stream"   and s2 :: "'e stream" shows "smap2 f s1 s2 = smap (\<lambda>(x, y). f x y) (szip s1 s2)"
vpra_Cons_Return: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and cs :: "'edge list"   and as :: "'edge list" assumes "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "valid_path_rev_aux cs as" shows "valid_path_rev_aux cs (a # as)"
Un_set_offending_flows_bound_minus_subseteq: fixes m :: "'b SecurityInvariant_configured"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and X :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "\<Union> (c_offending_flows m \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> X" shows "\<Union> (c_offending_flows m \<lparr>nodes = V, edges = E - E'\<rparr>) \<subseteq> X - E'"
conjugate_square_eq_0_vec: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(inner_prod v v = (0::'c)) = (v = 0\<^sub>v n)"
image_zmset_union: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b zmultiset"   and N :: "'b zmultiset" shows "Auxiliary.image_zmset f (M + N) = Auxiliary.image_zmset f M + Auxiliary.image_zmset f N"
inject: fixes x5 :: "char list"   and y5 :: "char list" shows "(Class x5 = Class y5) = (x5 = y5)"
CD_on_Xd: fixes x :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "x \<in> CD_on ds A" shows "Xd x \<in> ds"
memo_commute: fixes f :: "'v \<Rightarrow> 'm"   and x :: "'v"   and y :: "'v"   and g :: "int \<Rightarrow> int \<Rightarrow> 'm" assumes "\<And>m x g. bind m (\<lambda>a. bind (f x) (g a)) = bind (f x) (\<lambda>b. bind m (\<lambda>a. g a b))"   and "\<And>x g. get (\<lambda>s. bind (f x) (g s)) = bind (f x) (\<lambda>a. get (\<lambda>s. g s a))" shows "bind (memo f x) (\<lambda>a. bind (memo f y) (g a)) = bind (memo f y) (\<lambda>b. bind (memo f x) (\<lambda>a. g a b))"
matches_tuple: fixes \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d" shows "Matching_Ternary.matches (\<beta>, \<alpha>) m a p = ternary_to_bool_unknown_match_tac \<alpha> a p (ternary_ternary_eval (map_match_tac \<beta> p m))"
sqrt_prime_irrational: fixes p :: "nat" assumes "prime p" shows "sqrt (real p) \<notin> \<rat>"
minus_lucas_lehmer_ring: fixes x :: "int \<times> int" shows "\<ominus>\<^bsub>lucas_lehmer_ring\<^esub> x = (case x of (a, b) \<Rightarrow> (- a, - b))"
alwaysD: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'g"   and i :: "nat" assumes "(\<box>P) \<sigma>" shows "P (\<sigma> |\<^sub>s i)"
dist_bound: fixes f :: "'c \<Rightarrow>\<^sub>C 'd"   and g :: "'c \<Rightarrow>\<^sub>C 'd"   and b :: "real" assumes "\<And>x. dist (apply_bcontfun f x) (apply_bcontfun g x) \<le> b" shows "dist f g \<le> b"
null_sets_density_iff: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and A :: "'c set" assumes "f \<in> borel_measurable M" shows "(A \<in> null_sets (density M f)) = (A \<in> sets M \<and> (AE x\<in>A in M. f x = 0))"
subst_uop: fixes \<sigma> :: "'f \<Rightarrow> 'g"   and f :: "'h \<Rightarrow> 'e"   and v :: "('h, 'g) uexpr" shows "\<sigma> \<dagger> uop f v = uop f (\<sigma> \<dagger> v)"
assms_2: shows "vcard \<SS> = 3\<^sub>\<nat>"
clock_simAction_jAction: fixes t :: "'s Trace"   and aec :: "'s odlist"   and a :: "'a"   and cec :: "'s odlist" assumes "t \<in> Clock.jkbpC"   and "toSet aec = agent_abs a t"   and "toSet cec = common_abs t" shows "set (local.clock_simAction a (cec, aec)) = set (jAction (clock_repMC (toSet cec)) (tLast t) a)"
ranges: fixes i :: "char list \<Rightarrow> tv"   and U :: "nat set"   and p :: "fm" assumes "range i \<subseteq> domain U" shows "eval i p \<in> domain U"
chain_extend: fixes C :: "'a set"   and z :: "'a" assumes "chain_on P C {x \<in> A. \<not> Q x}"   and "z \<in> {x \<in> A. \<not> Q x}"   and "\<forall>x\<in>C. x \<sqsubseteq> z" shows "chain_on P ({z} \<union> C) {x \<in> A. \<not> Q x}"
dickson_less_p_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "dickson_less_p d m p q = ({p, q} \<subseteq> dgrad_p_set d m \<and> p \<prec>\<^sub>p q)"
strong_reduction_simulations_impl_bisimulation: fixes Rel :: "('a \<times> 'a) set"   and Cal :: "'a processCalculus" assumes "strong_reduction_simulation Rel Cal"   and "strong_reduction_simulation (Rel\<inverse>) Cal" shows "strong_reduction_simulation Rel Cal \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>Cal Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Cal P' \<and> (P', Q') \<in> Rel))"
norm_lt: fixes x :: "'b"   and y :: "'c" shows "(norm x < norm y) = (x \<bullet> x < y \<bullet> y)"
measurable_ennreal: shows "ennreal \<in> borel_measurable borel"
ord_option_trans: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c option"   and y :: "'c option"   and z :: "'c option" assumes "ord_option ord x y"   and "ord_option ord y z"   and "\<And>a b c. \<lbrakk>a \<in> set_option x; b \<in> set_option y; c \<in> set_option z; ord a b; ord b c\<rbrakk> \<Longrightarrow> ord a c" shows "ord_option ord x z"
trm_rep_aux_def: fixes t :: "'a trm"   and S :: "'a eclause set" assumes "\<forall>y. (y, t) \<in> trm_ord \<longrightarrow> y \<noteq> trm_rep y S \<longrightarrow> (trm_rep y S, y) \<in> trm_ord" shows "trm_rep t S = (if subterm_reduction_applicable S t then subterm_reduction S t else get_min t (set_of_candidate_values S t))"
A: shows "A \<in> smallest_ccdi_sets \<Omega> M"
assms: shows "(TP, TQ) \<in> (TRel \<union> _Collect (T1, T2) (\<exists>S. T1 = \<lbrakk>S\<rbrakk> \<and> T2 = \<lbrakk>S\<rbrakk>))\<^sup>+"
1: shows "(LEAST x. x < upper \<and> P x) < Suc upper \<and> P (LEAST x. x < upper \<and> P x)"
case1: shows "queue_invar (Node e2 a2 r ts2 # ts1)"
lower_times: fixes A :: "'b interval"   and B :: "'b interval" shows "lower (A * B) = Min {lower A * lower B, lower A * upper B, upper A * lower B, upper A * upper B}"
IH: fixes i :: "int"   and s :: "(char list \<Rightarrow> int) \<times> int list"   and j :: "int" assumes "P @ c @ P' \<turnstile> (size P + i, s) \<rightarrow>^n (j, s')"   and "0 \<le> i"   and "i < size c"   and "j \<notin> {size P..<size P + size c}" shows "\<exists>s'' i' k m. c \<turnstile> (i, s) \<rightarrow>^k (i', s'') \<and> i' \<in> exits c \<and> P @ c @ P' \<turnstile> (size P + i', s'') \<rightarrow>^m (j, s') \<and> n = k + m"
zero_sheaf_spec_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.zero_sheaf_spec R addition multiplication zero unit U \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.zero_rel (R\<setminus>\<pp>) R addition multiplication zero unit"
bind_SE_def: fixes f :: "'c \<Rightarrow> ('e \<times> 'c) option"   and g :: "'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'c) option" shows "bind\<^sub>S\<^sub>E f g = (\<lambda>\<sigma>. case f \<sigma> of None \<Rightarrow> None | Some (out, \<sigma>') \<Rightarrow> g out \<sigma>')"
unfold2': fixes E :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> 'res) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'\<AA> state \<times> '\<AA> state"   and self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2)"   and POST' :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0) \<Rightarrow> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and BODY :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0) \<Rightarrow> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> 'res" assumes "cp E"   and "\<tau> \<Turnstile> \<delta> self \<and> (case a1 of (a1, a2) \<Rightarrow> \<lambda>\<tau>. \<tau> \<Turnstile> \<upsilon> a1 \<and> \<tau> \<Turnstile> \<upsilon> a2) \<tau>"   and "\<tau> \<Turnstile> case a1 of (a, b) \<Rightarrow> PRE self a b"   and "\<tau> \<Turnstile> POST' self a1"   and "\<And>res. (case a1 of (a, b) \<Rightarrow> POST self a b) res = (POST' self a1 and (res \<triangleq> BODY self a1))" shows "(\<tau> \<Turnstile> E (case a1 of (a, b) \<Rightarrow> f self a b)) = (\<tau> \<Turnstile> E (BODY self a1))"
vals_throw_no_step: fixes vs :: "val list"   and a :: "nat"   and es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "((map Val vs @ Throw a # es, s, b), es', s', b') \<notin> reds P"
Disj: fixes A :: "fm"   and B :: "fm"   and i :: "name"   and x :: "tm" shows "(A OR B)(i::=x) = A(i::=x) OR B(i::=x)"
cD: shows "char_poly D = (\<Prod>a\<leftarrow>ds. [:- a, 1::'a:])"
fold_mset_single: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and s :: "'l"   and x :: "'k" assumes "comp_fun_commute f" shows "fold_mset f s {#x#} = f x s"
fundef_ex1_eqvt: fixes f :: "'a \<Rightarrow> 'b"   and d :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and p :: "perm" assumes "f \<equiv> \<lambda>x. THE_default (d x) (G x)"   and "eqvt G"   and "\<exists>!y. G x y" shows "p \<bullet> f x = f (p \<bullet> x)"
simps: fixes c :: "nat" shows "rght c = lo c (Pi 2)"
hb'_def: shows "hb' = max 1 (Max {hb h |h. h \<in> set hs})"
subst_ext_def: fixes \<sigma> :: "'d \<Rightarrow> 'd"   and x :: "'d \<Longrightarrow> 'c" shows "\<sigma> \<oplus>\<^sub>s x = (\<lambda>s. put\<^bsub>x\<^esub> s (\<sigma> (get\<^bsub>x\<^esub> s)))"
a_subset: fixes M :: "('a, 'b) message set"   and N :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended" assumes "M \<subseteq> N"   and "a_conditions (insert_messages N) a" shows "a_conditions (insert_messages M) a"
univ: shows "UNIV = Abs_bit0 ` {0..<2 * int CARD('a)}"
continuous_infdist: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd"   and A :: "'d set" assumes "continuous F f" shows "continuous F (\<lambda>x. infdist (f x) A)"
elims_2: fixes x :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs x" obtains xs :: "'varSort" and xa :: "'var" and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" where "x = qAbs xs xa X" and "qGood X"
derivations1_def: shows "derivations1 = {(u, v) |u v. derives1 u v}"
subgame_def: fixes V' :: "'a set" shows "subgame V' \<equiv> G\<lparr>verts := V \<inter> V', arcs := Restr E V', player0 := V0 \<inter> V'\<rparr>"
let_to_bind_conv: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b nres" shows "Let x f = RETURN x \<bind> f"
distinct_concat_map_E_2: fixes f :: "'b \<Rightarrow> 'a list"   and xs :: "'b list"   and x :: "'b" assumes "distinct (concat (map f xs))"   and "x \<in> set xs" shows "distinct (f x)"
inject_2: fixes x21 :: "'e"   and x22 :: "'f"   and y21 :: "'e"   and y22 :: "'f" shows "(IO x21 x22 = IO y21 y22) = (x21 = y21 \<and> x22 = y22)"
ldrop_all: fixes xs :: "'a llist"   and m :: "enat" assumes "llength xs \<le> m" shows "ldrop m xs = []\<^sub>l"
not_S_least: fixes f :: "'f"   and t :: "('f, 'b) Term.term" assumes "least f" shows "\<not> S (Fun f []) t"
Var\<^sub>0_def: fixes n :: "'b" shows "Var\<^sub>0 n \<equiv> Poly_Mapping.single (Poly_Mapping.single n 1) (1::'c)"
gen_gbg_impl_rel_sv: fixes Rm :: "('a \<times> 'b) set"   and Rf :: "('c \<times> 'd set set) set" assumes "single_valued Rm"   and "single_valued Rf" shows "single_valued (\<langle>Rm, Rf\<rangle>gen_gbg_impl_rel_eext)"
invertible_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "invertible A = (\<exists>A'. A ** A' = mat (1::'d) \<and> A' ** A = mat (1::'d))"
rep_eq: fixes x :: "nat"   and xa :: "uint16" shows "Rep_uint16 (drop_bit x xa) = drop_bit x (Rep_uint16 xa)"
set_add_atoms: fixes r :: "'a rexp"   and as :: "'a list" shows "set (add_atoms r as) = atoms r \<union> set as"
uedge_img_disj_iff: fixes g\<^sub>1 :: "'b ugraph"   and g\<^sub>2 :: "'b ugraph" shows "(uedge ` edges g\<^sub>1 \<inter> uedge ` edges g\<^sub>2 = {}) = (edges g\<^sub>1 \<inter> edges g\<^sub>2 = {})"
elims: fixes x :: "sterm"   and y :: "name option" assumes "unfree x = y" obtains name :: "name" where "x = Svar name" and "y = Some name"   | v :: "name" where "x = Sconst v__" and "y = None"   | v :: "(Term_Class.term \<times> sterm) list" where "x = Sabs va__" and "y = None"   | v :: "sterm" and va :: "sterm" where "x = v $\<^sub>s va" and "y = None"
adj: shows "\<exists>n. Rep_nat_int (res ts c) \<union> Rep_nat_int (clm ts c) = {n, n + 1}"
strand_fv_bvars_disjointD_1: fixes S :: "('b, 'a) strand_step list"   and X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S = {}"   and "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S" shows "set X \<subseteq> bvars\<^sub>s\<^sub>t S"
vec_lambda_eta: fixes g :: "('b, 'c) vec" shows "(\<chi>i. g $ i) = g"
1: shows "bprv (SS \<langle>\<chi>\<rangle> \<langle>\<psi>\<rangle>)"
Units_minus_one_closed: shows "\<ominus>\<^bsub>S\<^esub> \<one>\<^bsub>S\<^esub> \<in> Units S"
remainderexp: fixes a :: "nat"   and n :: "nat"   and i :: "nat" shows "(a mod n) ^ i mod n = a ^ i mod n"
insert_def: fixes e :: "'a"   and a :: "'b"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.insert e a bq = BinomialHeapStruc.ins (BinomialHeapStruc.Node e a 0 []) bq"
prog_sifum_secure_def: fixes cmds :: "'Com list" shows "prog_sifum_secure cmds = (no_assumptions_on_termination cmds \<and> (\<forall>mem\<^sub>1 mem\<^sub>2. mem\<^sub>1 =\<^sup>l mem\<^sub>2 \<longrightarrow> (\<forall>k cms\<^sub>1' mem\<^sub>1'. (add_initial_modes cmds, mem\<^sub>1) \<rightarrow>\<^bsub>k\<^esub> (cms\<^sub>1', mem\<^sub>1') \<longrightarrow> (\<exists>cms\<^sub>2' mem\<^sub>2'. (add_initial_modes cmds, mem\<^sub>2) \<rightarrow>\<^bsub>k\<^esub> (cms\<^sub>2', mem\<^sub>2') \<and> map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> length cms\<^sub>2' = length cms\<^sub>1' \<and> (\<forall>x. dma x = Low \<and> (\<forall>i<length cms\<^sub>1'. x \<notin> snd (cms\<^sub>1' ! i) AsmNoRead) \<longrightarrow> mem\<^sub>1' x = mem\<^sub>2' x)))))"
mod_diff_self1: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "(m - n) mod m = m - n"
wf_update_matchF_base_invar: fixes r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and mrs :: "mregex list"   and n :: "nat"   and R :: "event_data list set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and j :: "nat"   and rels :: "event_data option list set list"   and I :: "\<I>" assumes "safe_regex Futu Strict r"   and "to_mregex r = (mr, \<phi>s)"   and "mrs = sorted_list_of_set (LPDs mr)"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) j \<phi>)) \<phi>s rels" shows "wf_matchF_invar \<sigma> V n R I r (update_matchF_base n I mr mrs rels (\<tau> \<sigma> j)) j"
False_imp_not_eventually: fixes P :: "'c \<Rightarrow> bool"   and net :: "'c filter" assumes "\<forall>x. \<not> P x"   and "net \<noteq> bot" shows "\<not> eventually P net"
simps_11: shows "iszlfm T = (\<lambda>bs. isatom T \<and> bound0 T)"
add_le_imp_le_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less_eq (plus c a) (plus c b)" shows "less_eq a b"
O_assoc: fixes R :: "('c \<times> 'f) set"   and S :: "('f \<times> 'e) set"   and T :: "('e \<times> 'd) set" shows "(R O S) O T = R O S O T"
basis_iff_basis_in: fixes B :: "'a set" shows "basis B = basis_in carrier B"
alist_delete_min: fixes xs :: "('d, 'c) bintree option list"   and a :: "'d"   and v :: "'c"   and ts :: "('d, 'c) bintree list" assumes "distinct (vals xs)"   and "distinct (prios xs)"   and "find_min xs = Some (Node a v ts)" shows "set (elements (delete_min xs)) = set (elements xs) - {(v, a)}"
pe: shows "fs ! i - gso i \<in> local.span (gso ` {0..<i})"
path_component_eq_connected_component_of: fixes X :: "'c topology"   and x :: "'c" assumes "locally_path_connected_space X" shows "path_component_of_set X x = connected_component_of_set X x"
assms_1: shows "prov_triple (p1, c1, q1)"
matrix_scalar_mult_def: fixes k :: "'d"   and A :: "(('d, 'e) vec, 'f) vec" shows "k *k A \<equiv> \<chi>i j. k * A $ i $ j"
can_step_def: fixes l :: "nat"   and s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" shows "B.can_step l s \<equiv> \<exists>s'. blstep l s s'"
assms: shows "[\<^bold>\<forall>x. \<^bold>\<box>(\<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>G,x\<^sup>P\<rparr>) in v]"
comp_ide_self: fixes a :: "'a" assumes "\<phi>.A.ide a" shows "a \<cdot>\<^sub>A a = a"
asubg_subset1: fixes H :: "'a set"   and x :: "'a" assumes "M +> H"   and "x \<in> H" shows "x \<in> carrier M"
subgame_V': fixes V' :: "'a set" assumes "V' \<subseteq> V" shows "V\<^bsub>subgame V'\<^esub> = V'"
arev_arev: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "bidirected_digraph G arev" shows "arev (arev a) = a"
Def_bool3: fixes x :: "bool" shows "(Def x = TT) = x"
nths_out_of_bounds: fixes I :: "nat set"   and xs :: "'b list" assumes "\<forall>i\<in>I. length xs \<le> i" shows "nths xs I = []"
UnI2: fixes c :: "'a"   and B :: "'a set"   and A :: "'a set" assumes "c \<in> B" shows "c \<in> A \<union> B"
float_cases_finite: fixes a :: "('a, 'b) IEEE.float" shows "is_nan a \<or> is_infinity a \<or> is_finite a"
waux3: fixes x :: "('b, 'c) Combinators"   and a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "x \<noteq> a"   and "x \<notin> set p" shows "x \<notin> set (FWNormalisationCore.insort a p l)"
1: shows "super_orientation (x \<sqinter> - (1::'a)) y"
seq_comp_list_weakly_sequential: fixes P :: "'c option process"   and PS :: "'c option process list" assumes "\<And>x. x \<in> set (P # PS) \<Longrightarrow> weakly_sequential x" shows "weakly_sequential (foldl (;) P PS)"
rel_fundefs_def: fixes f :: "'e \<Rightarrow> ('f, 'g, 'h, 'op) Std.instr fundef option"   and g :: "'e \<Rightarrow> ('f, 'g, 'h, 'op, 'opinl) Inca.instr fundef option" shows "rel_fundefs f g = (\<forall>x. rel_option (rel_fundef (\<lambda>x y. x = norm_instr y)) (f x) (g x))"
Abs_permutation_inject: fixes x :: "'a \<Rightarrow> 'a"   and y :: "'a \<Rightarrow> 'a" assumes "x \<in> {f. bij f}"   and "y \<in> {f. bij f}" shows "(Abs_permutation x = Abs_permutation y) = (x = y)"
stream_all_iff: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b stream" shows "stream_all P s = Ball (sset s) P"
prob_space: shows "N.prob (space (density M (\<lambda>x. ennreal (D x)))) = 1"
measurable_cong_simp: fixes M :: "'b measure"   and N :: "'b measure"   and M' :: "'c measure"   and N' :: "'c measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "M = N"   and "M' = N'"   and "\<And>w. w \<in> space M \<Longrightarrow> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> N \<rightarrow>\<^sub>M N')"
sum_mset_delta: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and y :: "'c"   and c :: "'b"   and A :: "'c multiset" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum_mset plus zero {#if x = y then c else zero. x \<in># A#} = times c (semiring_1.of_nat one plus zero (count A y))"
orbit_list_conv_impl: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "x \<in> orbit f x" shows "orbit_list f x = orbit_list_impl f x [] x"
rwy: shows "y = wp (p x) (\<lambda>s. a * P s + b * Q s \<ominus> c) s"
finsum_cong': fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'a" assumes "A = B"   and "g \<in> B \<rightarrow> carrier R"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum R f A = finsum R g B"
symm: fixes x :: "'b"   and y :: "'b" shows "norm_dist x y = norm_dist y x"
abs_xy: shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> in v]"
effect_while_induct: fixes a :: "'c array"   and key :: "'c"   and i :: "nat ref"   and f :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and u :: "unit"   and I :: "heap \<Rightarrow> bool" assumes "effect (while (shiftr_p a key i) f) h h' u"   and "I h"   and "\<And>h h' u. \<lbrakk>I h; success (shiftr_p a key i) h; cond (shiftr_p a key i) h; effect f h h' u\<rbrakk> \<Longrightarrow> I h'" shows "I h'"
quot_simps_6: fixes A :: "fm" shows "\<guillemotleft>Neg A\<guillemotright> = Q_Neg \<guillemotleft>A\<guillemotright>"
strips_equivalent_to_sas_plus_i_a: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and I :: "'c \<Rightarrow> 'd option"   and G :: "'c \<Rightarrow> 'd option"   and \<psi> :: "('c, 'd) sas_plus_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "dom I \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom I. the (I v) \<in> \<R>\<^sub>+ \<Psi> v"   and "dom G \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom G. the (G v) \<in> \<R>\<^sub>+ \<Psi> v"   and "\<forall>ops\<in>set \<psi>. \<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "G \<subseteq>\<^sub>m execute_parallel_plan_sas_plus I \<psi>" shows "\<phi>\<^sub>S \<Psi> G \<subseteq>\<^sub>m execute_parallel_plan (\<phi>\<^sub>S \<Psi> I) (\<phi>\<^sub>P \<Psi> \<psi>)"
bi_total_rel_gpv: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "bi_total A"   and "bi_total B" shows "bi_total (rel_gpv A B)"
simps_2: fixes uu :: "('c \<times> nat) tree"   and uv :: "'c \<times> nat" shows "sngl \<langle>uu, uv, \<langle>\<rangle>\<rangle> = True"
diagonal_least_nonzero: fixes associates :: "'a set"   and residues :: "'a \<Rightarrow> 'a set"   and H :: "(('a, 'b) vec, 'b) vec"   and i :: "'b" assumes "Hermite associates residues H"   and "invertible H"   and "upper_triangular H" shows "(LEAST n. H $ i $ n \<noteq> (0::'a)) = i"
option_to_is_option_of: fixes option :: "game"   and g :: "game" shows "((option, g) \<in> option_of) = ((Rep_game option, Rep_game g) \<in> is_option_of)"
vlrestriction_vsubset_rel: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>l\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> r"
distinct_1: fixes x1 :: "agent"   and x2 :: "fresh_t" shows "aAgt x1 \<noteq> aNon x2"
add_power_poly_mod_ring: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly" shows "(x + y) ^ CARD('a) = x ^ CARD('a) + y ^ CARD('a)"
sup1: fixes A :: "'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "A \<equiv> sup k a" shows "sup A b \<equiv> sup k (sup a b)"