hcju/isabelleps huggingface dataset:
wf_bounded_set: fixes r :: "('c \<times> 'c) set"   and ub :: "'c \<Rightarrow> 'd set"   and f :: "'c \<Rightarrow> 'd set" assumes "\<And>a b. (b, a) \<in> r \<Longrightarrow> finite (ub a) \<and> ub b \<subseteq> ub a \<and> f b \<subseteq> ub a \<and> f a \<subset> f b" shows "wf r"
floor_exists1: fixes x :: "'a" shows "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)"
the_def: fixes option :: "'c option" shows "the option = (case option of Some x2 \<Rightarrow> x2)"
dvd_mult_unit_iff': fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b dvd (1::'a)" shows "(a dvd b * c) = (a dvd c)"
gr_zeroI: fixes n :: "'k multiset" assumes "n = {#} \<Longrightarrow> False" shows "{#} \<subset># n"
simps: fixes i :: "name"   and x :: "tm"   and y :: "tm"   and z :: "tm" assumes "atom i \<sharp> (x, y, z)" shows "UnionP x y z = SyntaxN.All i (Var i IN z IFF Var i IN x OR Var i IN y)"
p'_stop_zero: shows "ego2.p' ego2.t_stop = (if a\<^sub>e = 0 then v\<^sub>e else 0)"
sup_ge1: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y \<le> x"
Step_def: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b"   and r :: "'a" shows "Partial_Cost_Model.Step A s r = (let (a, is') = snd A s r in (step (fst s) r a, is'))"
1: shows "ref = None \<or> ref = Some r \<and> r \<in> set (a # list)"
semilattice_neutr_order_axioms: shows "semilattice_neutr_order max 0 (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
a_inv_H: shows "inv a \<in> H \<and> inv a \<noteq> \<one>"
uniM: shows "\<exists>!M. matrix_seq.lowner_is_lub g M"
simps_1: fixes xs :: "'a list"   and ys :: "'a list"   and zss :: "'a list list" shows "local.c_merge_pairs (xs # ys # zss) = local.c_merge xs ys + local.c_merge_pairs zss"
inorder_delete: fixes x :: "'a"   and t :: "'a tree"   and dl :: "nat"   and t' :: "'a tree"   and dl' :: "nat" assumes "local.delete x (t, dl) = (t', dl')"   and "Sorted_Less.sorted (inorder t)" shows "inorder t' = del_list x (inorder t)"
split_strict: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c" shows "(case \<bottom> of (x, xa) \<Rightarrow> f x xa) = f \<bottom> \<bottom>"
remove_shadow_root_def: shows "remove_shadow_root = (\<lambda>element_ptr. Heap_Error_Monad.bind (get_shadow_root element_ptr) (case_option (error HierarchyRequestError) (\<lambda>shadow_root_ptr. Heap_Error_Monad.bind (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 shadow_root_ptr)) (\<lambda>children. Heap_Error_Monad.bind (get_disconnected_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>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)) (\<lambda>disconnected_nodes. if children = [] \<and> disconnected_nodes = [] then Heap_Error_Monad.bind (set_shadow_root element_ptr None) (\<lambda>_. delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr) else error HierarchyRequestError)))))"
y0: shows "real_of_1 (p2, l2, r2) < 0 \<or> 0 < real_of_1 (p2, l2, r2)"
multiple_t_wise_balance: fixes n :: "nat" assumes "0 < n" shows "t_wise_balance \<V> (multiple_blocks n) \<t> (\<Lambda>\<^sub>t * int n)"
multLessListDiff: fixes a :: "Literal list"   and b :: "Literal list"   and M :: "(Literal \<times> bool) list"   and x :: "Literal list" assumes "(a, b) \<in> multLess M" shows "(list_diff a x, b) \<in> multLess M"
less_oInvD: fixes x :: "ordinal"   and a :: "ordinal" assumes "x < oInv F a" shows "F (oSuc x) \<le> a"
type_definition_ccs: shows "type_definition Rep_ccs Abs_ccs (Collect ccs_Rep_set)"
s_cond_def_2: fixes t :: "real" assumes "0 \<le> t"   and "t \<le> ego.t_stop" shows "ego.s t = ego.p t"
dghm_def: shows "\<FF> = [\<FF>\<lparr>ObjMap\<rparr>, \<FF>\<lparr>ArrMap\<rparr>, \<FF>\<lparr>HomDom\<rparr>, \<FF>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
fst_map_prod: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'e"   and x :: "'f \<times> 'g" shows "fst (map_prod f g x) = f (fst x)"
less_linear: fixes y :: "'d"   and x :: "'d" shows "y < x \<or> x = y \<or> x < y"
cases: fixes trsys :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and \<tau>move :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a" assumes "\<tau>trsys.silent_move trsys \<tau>move a1 a2" obtains s :: "'a" and s' :: "'a" and tl :: "'b" where "a1 = s" and "a2 = s'" and "trsys s tl s'" and "\<tau>move s tl s'"
Cons: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "x \<rightarrow> y"   and "steps (y # xs)" shows "steps (x # y # xs)"
fun_sum_commute: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "f (0::'c) = (0::'b)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (sum g A) = (\<Sum>a\<in>A. f (g a))"
bound_on_path: fixes Q :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and b :: "'a" assumes "Q \<in> \<P>"   and "[f[f 0 ..]X]"   and "X \<subseteq> Q"   and "is_bound_f b X f" shows "b \<in> Q"
leg0_composite: shows "HH\<nu>\<pi>\<rho>.leg0 = \<rho>.leg0 \<cdot> \<nu>\<pi>_\<rho>.prj\<^sub>0"
S: shows "S = {y \<in> K. f y = (0::'a)}"
span_sum: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and S :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> vs2.span S" shows "sum f A \<in> vs2.span S"
nonneg_cexpr_def: fixes V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" shows "nonneg_cexpr V \<Gamma> e \<equiv> \<forall>\<sigma>\<in>space (state_measure V \<Gamma>). 0 \<le> extract_real (cexpr_sem \<sigma> e)"
wellformed_items_Scan: fixes I :: "item set"   and T :: "(symbol \<times> character list) set"   and k :: "nat" assumes "wellformed_items I"   and "T \<subseteq> \<X> k" shows "wellformed_items (Scan T k I)"
OV_inv3I: fixes x :: "'a obsv_state_scheme" assumes "\<And>r p v. obsv_state.obs x r p = Some v \<Longrightarrow> obs_safe r x v" shows "x \<in> OV_inv3"
opnet_node_trans: fixes s :: "(nat \<Rightarrow> 'd) \<times> 'e net_state"   and a :: "'f node_action"   and s' :: "(nat \<Rightarrow> 'd) \<times> 'e net_state"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and i :: "nat"   and R :: "nat set" assumes "(s, a, s') \<in> automaton.trans (opnet onp \<langle>i; R\<rangle>)" shows "(s, a, s') \<in> onode_sos (automaton.trans (onp i))"
list_order_memb2: fixes ops :: "('a \<times> 'a option) list"   and x :: "'a"   and y :: "'a" assumes "list_order ops x y" shows "y \<in> set (interp_ins ops)"
select_convs_1: fixes init :: "'a set"   and trans :: "('a \<times> 'a) set"   and obs :: "'a \<Rightarrow> 'b"   and more :: "'c" shows "obs \<lparr>init = init, trans = trans, obs = obs, \<dots> = more\<rparr> = obs"
exec_to_exec_mark_guards_Fault: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and f :: "'c"   and x :: "'c" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f" shows "\<exists>f'. \<Gamma>\<turnstile> \<langle>mark_guards x c,s\<rangle> \<Rightarrow> Fault f'"
multi_member_this: fixes x :: "'k"   and XS :: "'k multiset" shows "x \<in># {#x#} + XS"
interval_integral_discrete_difference: fixes X :: "real set"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and M :: "real measure" assumes "countable X"   and "\<And>x. \<lbrakk>a \<le> b; a < ereal x; ereal x < b; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x"   and "\<And>x. \<lbrakk>b \<le> a; b < ereal x; ereal x < a; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" shows "interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
fMx: shows "finite {y \<in> S. y \<le> x}"
simps_1: fixes l :: "'a set list" shows "all_in_list [] l = True"
tbisim_cases: fixes nw :: "bool"   and t :: "'t"   and ts1 :: "('x1 \<times> 'l \<Rightarrow>f nat) option"   and m1 :: "'m1"   and ts2 :: "('x2 \<times> 'l \<Rightarrow>f nat) option"   and m2 :: "'m2" assumes "tbisim nw t ts1 m1 ts2 m2" obtains "ts1 = None" and "ts2 = None"   | x :: "'x1" and ln :: "'l \<Rightarrow>f nat" and x' :: "'x2" where "ts1 = \<lfloor>(x, ln)\<rfloor>" and "ts2 = \<lfloor>(x', ln)\<rfloor>" and "t \<turnstile> (x, m1) \<approx> (x', m2)" and "nw \<or> x \<approx>w x'"
Labels_CondFalse: fixes c\<^sub>2 :: "cmd"   and l :: "nat"   and c :: "cmd"   and b :: "expr"   and c\<^sub>1 :: "cmd" assumes "labels c\<^sub>2 l c" shows "labels (if (b) c\<^sub>1 else c\<^sub>2) (l + #:c\<^sub>1 + 1) c"
covering_space_def: fixes c :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set" shows "covering_space c p S \<equiv> continuous_on c p \<and> p ` c = S \<and> (\<forall>x\<in>S. \<exists>T. x \<in> T \<and> openin (top_of_set S) T \<and> (\<exists>v. \<Union> v = c \<inter> p -` T \<and> (\<forall>u\<in>v. openin (top_of_set c) u) \<and> disjoint v \<and> (\<forall>u\<in>v. \<exists>q. homeomorphism u T p q)))"
null_sets_completion_subset: fixes B :: "'c set"   and A :: "'c set"   and M :: "'c measure" assumes "B \<subseteq> A"   and "A \<in> null_sets (completion M)" shows "B \<in> null_sets (completion M)"
esize_cont: fixes C :: "'a set" assumes "Complete_Partial_Order.chain (\<le>) C"   and "C \<noteq> {}" shows "esize (\<Squnion> C) = \<Squnion> (esize ` C)"
rep_eq: fixes x :: "('b, 'c, 'd) formula" shows "Rep_formula (Formula.Not x) = Not\<^sub>\<alpha> (Rep_formula x)"
M2: fixes s :: "'a"   and t :: "'a" assumes "s \<in> M"   and "t \<in> S \<union> N" shows "\<Delta> t \<le> \<Delta> s"
insertll_append: fixes xs :: "'a list"   and vs :: "'a list list"   and k :: "nat"   and x :: "'a"   and v :: "'a list" assumes "length xs = length vs" shows "insertll k (xs @ [x]) (vs @ [v]) = insertll k xs vs @ [insertl k x v]"
derives_implies_leftderives_cons: fixes a :: "symbol list"   and u :: "symbol list"   and X :: "symbol"   and b :: "symbol list" assumes "is_word a"   and "derives u (a @ X # b)" shows "\<exists>c. leftderives u (a @ X # c)"
zero_order_2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
deriv_c: shows "\<Gamma>,\<Theta>'\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
compactin_def: fixes X :: "'b topology"   and S :: "'b set" shows "compactin X S = (S \<subseteq> topspace X \<and> (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> S \<subseteq> \<Union> \<U> \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> S \<subseteq> \<Union> \<F>)))"
z_zero_def: shows "z_zero \<equiv> proj2_line_abs (vector [0, 0, 1])"
leq_zero_imp: fixes x :: "'b \<Rightarrow> 'a" assumes "(x, o.zero) \<in> oexp" shows "x = o.zero"
a: shows "(\<Sum>b2\<in>#remove1_mset b1 \<B>. int (nat (b1 |\<inter>| b2) choose 2)) = (\<Sum>bl\<in>#remove1_mset b1 \<B>. b1 |\<inter>| bl * (b1 |\<inter>| bl - 1) div 2)"
FunctorMapExt: shows "(##) F \<in> extensional (mor\<^bsub>CatDom F\<^esub>)"
arr_RelD_8: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
eint_ord_Suc: fixes x :: "eint"   and y :: "eint" assumes "x < y" shows "x + 1 < y + 1"
x: shows "(\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z))"
m1_inv1r_cacheD: fixes x :: "'a m1_state_scheme"   and Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat"   and nl :: "atom list" assumes "runs x Rb = Some (Resp, [A, B], aKey Kab # aNum Ts # aNum Ta # nl)"   and "clk x < Ta + La"   and "x \<in> m1_inv1r_cache" shows "(B, Kab, Ta) \<in> cache x"
prod_mset_image: fixes X :: "'a multiset" shows "\<Prod>\<^sub># (image_mset hom X) = hom (\<Prod>\<^sub># X)"
dom\<sigma>: shows "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i"
pow_generator_mod: fixes G :: "('a, 'b) cyclic_group_scheme"   and k :: "nat" assumes "Cyclic_Group.cyclic_group G" shows "\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> (k mod order G) = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> k"
cefi: assumes "f\<inverse> \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "e O f\<inverse> = f\<inverse>"
image_inv_into_cancel: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and A' :: "'c set"   and B' :: "'c set" assumes "f ` A = A'"   and "B' \<subseteq> A'" shows "f ` inv_into A f ` B' = B'"
finite_ne_induct: fixes F :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite F"   and "F \<noteq> {}"   and "\<And>x. P {x}"   and "\<And>x F. \<lbrakk>finite F; F \<noteq> {}; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
meet_pres_def: fixes X :: "('a, 'b) gorder_scheme"   and Y :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "meet_pres X Y f \<equiv> lattice X \<and> lattice Y \<and> (\<forall>x\<in>carrier X. \<forall>y\<in>carrier X. f (x \<sqinter>\<^bsub>X\<^esub> y) = f x \<sqinter>\<^bsub>Y\<^esub> f y)"
EmptyChain: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q = resChain [] (P \<parallel> Q)"
nsupp_zero: fixes x :: "'a"   and f :: "'a \<Rightarrow> real" assumes "x \<notin> supp f" shows "f x = 0"
sp_spec_subj_subj_def: fixes a :: "'a"   and b :: "'a" shows "sp_spec_subj_subj a b \<equiv> \<exists>f m1 m2. sp_spec_subj_obj a (FILEP f) m1 \<and> sp_spec_subj_obj b (FILEP f) m2"
mapping_of_inverse: fixes x :: "('a, 'b) pp" shows "PP (pp.mapping_of x) = x"
path2_hd_in_\<alpha>n: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "n \<in> set (\<alpha>n g)"
run_fail_nondet: shows "run_nondet fail_nondet = Monad_Overloading.return {#}"
split_x: fixes y :: "'a" obtains "y = x" and "y \<notin> S"   | "y \<in> S" and "y \<noteq> x"   | "y \<notin> S" and "y \<noteq> x"
perm_empty_imp: fixes ys :: "'a list" assumes "[] <~~> ys" shows "ys = []"
thetaParZO_ZObis: shows "thetaParZO \<subseteq> ZObis"
sem\<^sub>e\<^sub>s\<^sub>t_c_decomp_rm\<^sub>e\<^sub>s\<^sub>t_deduct: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> A"   and "ik\<^sub>e\<^sub>s\<^sub>t A \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t" shows "ik\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t"
2: shows "g \<turnstile> Entry g-ns\<^sub>v @ [defNode g v]\<rightarrow>defNode g v"
A2: shows "CLF = (SIGMA cl:CL. {f. f \<in> pset cl})"
subtype_implies_closed: fixes \<Gamma> :: "binding list"   and S :: "ty"   and T :: "ty" assumes "\<Gamma>\<turnstile>S<:T" shows "S closed_in \<Gamma> \<and> T closed_in \<Gamma>"
step_cfunc_adv: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and ia :: "'iadv_core" assumes "S p q"   and "ia \<in> IA" shows "p \<bind> (\<lambda>s1. map_spmf fst (cfunc_adv core1 s1 ia)) = q \<bind> (\<lambda>s2. map_spmf fst (cfunc_adv core2 s2 ia))"
length_Tensor: fixes M1 :: "'a list list"   and M2 :: "'a list list" shows "length (M1 \<otimes> M2) = length M1 * length M2"
prec_F_grounding: fixes q :: "'q"   and C2 :: "'f"   and C1 :: "'f" assumes "q \<in> Q"   and "C2 \<prec>\<cdot> C1" shows "\<G>_F_q q C1 \<subseteq> \<G>_F_q q C2"
add_eq_0_iff_both_eq_0: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
parts_singleton: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> parts H" shows "\<exists>Y\<in>H. X \<in> parts {Y}"
CSP2_join: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "x is CSP2 healthy"   and "y is CSP2 healthy" shows "x \<or> y is CSP2 healthy"
closed_halfspace_component_ge: fixes a :: "real"   and i :: "'c" shows "closed {x. a \<le> x \<bullet> i}"
block_closed: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> orbit.Partition" shows "a \<in> S"
well_def_wp_cts: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "well_def prog" shows "bd_cts (wp prog)"
valid_is_valid_in_all: fixes \<phi> :: "fml" shows "valid \<phi> = (\<forall>I. valid_in I \<phi>)"
subst_unrest: fixes x :: "'e \<Longrightarrow> 'f"   and P :: "('g, 'f) uexpr"   and \<sigma> :: "'h \<Rightarrow> 'f"   and v :: "('e, 'h) uexpr" assumes "x \<sharp> P" shows "\<sigma>(x \<mapsto>\<^sub>s v) \<dagger> P = \<sigma> \<dagger> P"
inverse_ennreal_unique: fixes a :: "ennreal"   and b :: "ennreal" assumes "a * b = 1" shows "b = inverse a"
HC_intros_12: fixes F :: "'b formula" shows "AX10 \<turnstile>\<^sub>H \<^bold>\<not> F \<^bold>\<rightarrow> F \<^bold>\<rightarrow> \<bottom>"
obtain_subset_with_card_n: fixes n :: "nat"   and S :: "'g set" assumes "n \<le> card S" obtains T :: "'g set" where "T \<subseteq> S" and "card T = n" and "finite T"
intros_1: fixes S :: "'d \<Rightarrow> bool" shows "treesp S \<langle>\<rangle>"
is_Pointwise_Output_Shrink_list_imp_is_Pointwise_Output_Shrink: fixes pointwise_shrink :: "'d list \<Rightarrow> 'd"   and fs :: "('d \<Rightarrow> 'e message_af) list"   and output_fun :: "'d \<Rightarrow> 'e message_af" assumes "is_Pointwise_Output_Shrink_list pointwise_shrink fs"   and "output_fun \<in> set fs" shows "is_Pointwise_Output_Shrink pointwise_shrink output_fun"
rep_eq: fixes x :: "real_alg_3" shows "info_3 x = info_2 (rep_real_alg_3 x)"
le_inf_iff: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "(x \<subseteq># y \<inter># z) = (x \<subseteq># y \<and> x \<subseteq># z)"
refine_trans: fixes a :: "'a::order"   and b :: "'a::order"   and c :: "'a::order" assumes "(a::'a::order) \<le> (b::'a::order)"   and "(b::'a::order) \<le> (c::'a::order)" shows "(a::'a::order) \<le> (c::'a::order)"
hyps_3: fixes f :: "'a \<Rightarrow>\<^sub>0 'b" assumes "K = keys f" shows "P f"
h2b: assumes "Av = - Bv * sqrt Cv" shows "Av\<^sup>2 = (- Bv * sqrt Cv)\<^sup>2"
msgrel_def: shows "msgrel \<equiv> {(xa, x). msgrelp xa x}"
CP_def: shows "CP = (\<lambda>g. (*\<cdot>) (- g) \<circ> P \<circ> (*\<cdot>) g)"
diff_ge_0_iff_ge: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (minus a b) = less_eq b a"
env_restr_def: fixes S :: "'a set"   and m :: "'a \<Rightarrow> 'b" shows "m f|` S = (\<lambda>x. if x \<in> S then m x else \<bottom>)"
Re_sgn: fixes z :: "complex" shows "Re (sgn z) = Re z / cmod z"
DirProds_iso2: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'d \<Rightarrow> ('e, 'f) monoid_scheme" assumes "inj_on f A"   and "Group.group (DirProds g (f ` A))" shows "DirProds (g \<circ> f) A \<cong> DirProds g (f ` A)"
setdist_le_dist: fixes x :: "'c"   and s :: "'c set"   and y :: "'c"   and t :: "'c set" assumes "x \<in> s"   and "y \<in> t" shows "setdist s t \<le> dist x y"
complete_latticeI: fixes L :: "('a, 'b) gorder_scheme" assumes "partial_order L"   and "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>s. is_lub L s A"   and "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>i. is_glb L i A" shows "complete_lattice L"
expand: fixes complex :: "complex"   and complex' :: "complex" assumes "Re complex = Re complex' \<and> Im complex = Im complex'" shows "complex = complex'"
OUTfromVCorrect1_data22: shows "OUTfromVCorrect1 data22"
no_nonce_NS1_NS2: fixes evs :: "event list"   and C :: "agent"   and NA' :: "msg"   and NA :: "nat"   and B :: "agent"   and A :: "agent" assumes "evs \<in> ns_public"   and "Crypt (pubK C) \<lbrace>NA', Nonce NA\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (knows Spy evs)" shows "Nonce NA \<in> analz (knows Spy evs)"
int: shows "integrable M (\<lambda>x. g x * log b (g x / f x))"
r_one: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "x \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = x"
loxodromic_attracting_fixed_point_2: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "Gromov_extension f (attracting_fixed_point f) = attracting_fixed_point f"
intro: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e" assumes "monad return bind"   and "monad_fail_axioms bind fail" shows "monad_fail return bind fail"
simps: fixes x :: "'b"   and v :: "'b \<Rightarrow> 'c"   and lb :: "'b \<Rightarrow> 'c option"   and ub :: "'b \<Rightarrow> 'c option" shows "in_bounds x v (lb, ub) = (v x \<ge>\<^sub>l\<^sub>b lb x \<and> v x \<le>\<^sub>u\<^sub>b ub x)"
vec_lambda_unique: fixes f :: "('c, 'b) vec"   and g :: "'b \<Rightarrow> 'c" shows "(\<forall>i. f $ i = g i) = (vec_lambda g = f)"
OV_inv3_def: shows "OV_inv3 = {s. \<forall>r p v. obsv_state.obs s r p = Some v \<longrightarrow> obs_safe r s v}"
cnj_def: fixes A :: "fmla"   and B :: "fmla" shows "A AND B \<equiv> neg (A IMP neg B)"
WT_callee: fixes s :: "'s2" assumes "I2 s" shows "\<I> \<turnstile>c callee2 s \<surd>"
section_map_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "section_map X Y f \<equiv> \<exists>g. retraction_maps Y X g f"
y_hy: shows "(y, h y) \<in> graph H' h'"
chart_basis_basis_defined: fixes a :: "'b"   and b :: "'b"   and xa :: "'b proj_space" assumes "a \<in> Basis"   and "b \<in> Basis"   and "a \<noteq> b"   and "chart_basis_domainP a xa"   and "chart_basis_domainP b xa" shows "chart_basis a xa \<bullet> b \<noteq> 0"
prime_card_int: assumes "class.prime_card TYPE('a)" shows "prime (int CARD('a))"
path_component_of_mono: fixes X :: "'a topology"   and S :: "'a set"   and x :: "'a"   and y :: "'a"   and T :: "'a set" assumes "path_component_of (subtopology X S) x y"   and "S \<subseteq> T" shows "path_component_of (subtopology X T) x y"
div_geq: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "\<not> m < n" shows "m div n = Suc ((m - n) div n)"
has_integral_component_le: fixes k :: "'c"   and f :: "'d \<Rightarrow> 'c"   and i :: "'c"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'c"   and j :: "'c" assumes "k \<in> Basis"   and "(f has_integral i) S"   and "(g has_integral j) S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<bullet> k \<le> g x \<bullet> k" shows "i \<bullet> k \<le> j \<bullet> k"
y: fixes x :: "'a pmf"   and y :: "'b pmf" assumes "(x, y) \<in> set_pmf pq" shows "map_pmf snd (PQ x y) = y"
map_disj_empty_left: fixes h :: "'c \<Rightarrow> 'd option" shows "Map.empty \<bottom> h"
exp_conj_comm: fixes a :: "'a \<Rightarrow> real"   and b :: "'a \<Rightarrow> real" shows "a && b = b && a"
exec_ts: fixes G :: "prog"   and s :: "state"   and c :: "stmt"   and s' :: "state"   and L :: "lenv"   and C :: "qtname"   and A :: "assigned" assumes "G\<turnstile>s \<midarrow>c\<rightarrow> s'"   and "wf_prog G"   and "s\<Colon>\<preceq>(G, L)"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s)) \<guillemotright>In1r c\<guillemotright> A" shows "s'\<Colon>\<preceq>(G, L) \<and> (error_free s \<longrightarrow> error_free s')"