hcju/isabelleps huggingface dataset:
rca0: shows "fps_ginv c a $ 0 = (0::'a)"
update_kno_dsn_greater_zero: fixes dsn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and hops :: "nat"   and ip :: "nat" assumes "1 \<le> dsn" shows "1 \<le> sqn (update rt dip (dsn, kno, val, hops, ip)) dip"
basis_expansion: fixes x :: "('c, 'd) vec" shows "(\<Sum>i\<in>UNIV. x $ i *s axis i (1::'c)) = x"
wf_below_mono1: fixes r :: "('a \<times> 'a) set"   and r' :: "('a \<times> 'a) set"   and A :: "'a set" assumes "r \<subseteq> r'"   and "wf_below r' A" shows "wf_below r A"
le_minus_self_iff: fixes a :: "'a" shows "(a \<le> - a) = (a \<le> (0::'a))"
set_spmf_return_pmf: fixes x :: "'c option" shows "set_spmf (return_pmf x) = set_option x"
m_5: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
assms_1: shows "\<U> (assert_bound_loop as (init t))"
norm_rep_ARR: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f" shows "free_monoidal_category.norm C (free_monoidal_category.rep C f) = free_monoidal_category.rep C f"
IH: fixes I :: "'a \<Rightarrow> 'b option" shows "length (trace_parallel_plan_sas_plus I \<psi>) \<le> length \<psi> + 1"
Schur_Ker: fixes smult' :: "('b, 'a) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and T :: "'c \<Rightarrow> 'e" assumes "GRepHom smult' T"   and "T ` V \<noteq> 0" shows "inj_on T V"
spmf_conv_measure_spmf: fixes p :: "'c spmf"   and x :: "'c" shows "spmf p x = Sigma_Algebra.measure (measure_spmf p) {x}"
INF_eqI: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and x :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq x (f i)"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq y (f i)) \<Longrightarrow> less_eq y x" shows "Inf (f ` A) = x"
mk_ode_ops_def: shows "mk_ode_ops \<equiv> map_fun id (map_fun id (map_option Abs_ode_ops)) (\<lambda>ode_e safe_form. if open_form safe_form \<and> max_Var_floatariths ode_e \<le> length ode_e \<and> max_Var_form safe_form \<le> length ode_e then Some (ode_e, safe_form) else None)"
G'_gb_fields: shows "G'.F = {f ` A |A. A \<in> F}"
Rep_Abs_STOP: shows "Rep_process (Abs_process ({(s, X). s = []}, {})) = ({(s, X). s = []}, {})"
negligible_differentiable_image_negligible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "DIM('a) \<le> DIM('b)"   and "negligible S"   and "f differentiable_on S" shows "negligible (f ` S)"
diff_numeral_special_7: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus one) (uminus (numeral.numeral one plus n)) = neg_numeral.sub minus plus one n num.One"
in_hhomE: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>B b\<guillemotright>" obtains "B.arr \<mu>" and "B.obj a" and "B.obj b" and "src\<^sub>B \<mu> = a" and "trg\<^sub>B \<mu> = b"
l1_inv2D: fixes Rb :: "fid_t"   and B :: "agent"   and A :: "agent"   and x :: "l1_state"   and KE :: "msg" assumes "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "progress x Rb = Some {xpkE, xsk}"   and "guessed_frame Rb xpkE = Some KE"   and "can_signal x A B"   and "x \<in> l1_inv2" shows "0 < l1_state.signals x (Running A B \<langle>KE, NonceF (Rb $ sk)\<rangle>)"
outsideOutside: fixes P :: "('a \<times> 'b) set"   and X :: "'a set"   and Y :: "'a set" shows "Outside' Y (Outside' X P) = Outside' (X \<union> Y) P"
lists_distinct_union_by_interleavings: fixes X :: "'a set"   and Y :: "'a set"   and n :: "nat" assumes "X \<inter> Y = {}" shows "{zs. length zs = n \<and> distinct zs \<and> set zs \<subseteq> X \<union> Y} = {0..n} \<bind> (\<lambda>k. {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> X} \<bind> (\<lambda>xs. {ys. length ys = n - k \<and> distinct ys \<and> set ys \<subseteq> Y} \<bind> (\<lambda>ys. {zs. interleavings xs ys zs})))"
inext_le_mono_infin_rev: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "inext a I \<le> inext b I"   and "a \<in> I"   and "b \<in> I"   and "infinite I" shows "a \<le> b"
span_image: fixes S :: "'b set" shows "vs2.span (g ` S) = g ` vs1.span S"
simple_labels_def: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'd, 'b, 'e) seqp" shows "simple_labels \<Gamma> \<equiv> \<forall>pn. \<forall>p\<in>subterms (\<Gamma> pn). \<exists>!l. labels \<Gamma> p = {l}"
Min_singleton: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.linorder less_eq less" shows "linorder.Min less_eq {x} = x"
character_inv_character_iff: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" shows "character G (inv_character \<chi>) = character G \<chi>"
axiom4_F: fixes f1 :: "'l \<Rightarrow> 'o"   and f2 :: "'m \<Rightarrow> 'p"   and f3 :: "'n \<Rightarrow> 'q" shows "F'set3 \<circ> F'map f2 f3 f1 = (`) f1 \<circ> F'set3"
conic_sgn_abs: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0" shows "conic_sgn (proj2_abs v) = sgn (v \<bullet> (M *v v))"
sq_norm_vec_as_cscalar_prod: fixes v :: "'b vec" shows "\<parallel>v\<parallel>\<^sup>2 = v \<bullet>c v"
while_option_def: fixes b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'c"   and s :: "'c" shows "while_option b c s = (if \<exists>k. \<not> b ((c ^^ k) s) then Some ((c ^^ (LEAST k. \<not> b ((c ^^ k) s))) s) else None)"
list_of_vec_rw_map: fixes m :: "'c vec" shows "list_of_vec m = map (($) m) [0..<dim_vec m]"
vars\<^sub>s\<^sub>tI: fixes x :: "'c"   and S :: "('d, 'c) strand_step list" assumes "x \<in> fv\<^sub>s\<^sub>t S" shows "x \<in> vars\<^sub>s\<^sub>t S"
vdisjnt_nin_right: fixes A :: "V"   and B :: "V"   and a :: "V" assumes "vdisjnt A B"   and "a \<in>\<^sub>\<circ> A" shows "a \<notin>\<^sub>\<circ> B"
ChamberSubcomplex_def: fixes Y :: "'a set set" shows "ChamberSubcomplex Y \<equiv> order.greater_eq X Y \<and> ChamberComplex Y \<and> (\<forall>C. SimplicialComplex.maxsimp Y C \<longrightarrow> chamber C)"
osubst_eq_ODE_vars: fixes I :: "('sf, 'sc, 'sz) interp"   and ODE :: "('sf, 'sz) ODE"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "ODE_vars I (Osubst ODE \<sigma>) = ODE_vars (local.adjoint I \<sigma> \<nu>) ODE"
post_split_max: fixes t :: "('b \<times> nat) tree"   and t' :: "('b \<times> nat) tree"   and x :: "'b" assumes "invar t"   and "(t', x) = split_max t"   and "t \<noteq> \<langle>\<rangle>" shows "post_del t t'"
assms_4: fixes n :: "nat" shows "cruel n \<noteq> []"
is_isoI: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" assumes "h \<in> Group.iso G H" shows "G \<cong> H"
piecewise_continuous_on_induct: fixes a :: "'a"   and b :: "'a"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" assumes "piecewise_continuous_on a b I f"   and "\<And>a b f. continuous_on {a..b} f \<Longrightarrow> P a b {} f"   and "\<And>a i b I f1 f2 f. \<lbrakk>a \<le> i; i \<le> b; i \<notin> I; P a i I f1; P i b I f2; piecewise_continuous_on a i I f1; piecewise_continuous_on i b I f2; \<And>x. \<lbrakk>a \<le> x; x < i\<rbrakk> \<Longrightarrow> f1 x = f x; \<And>x. \<lbrakk>i < x; x \<le> b\<rbrakk> \<Longrightarrow> f2 x = f x; a < i \<Longrightarrow> (f \<longlongrightarrow> f1 i) (at_left i); i < b \<Longrightarrow> (f \<longlongrightarrow> f2 i) (at_right i)\<rbrakk> \<Longrightarrow> P a b (insert i I) f"   and "\<And>a b i I f. \<lbrakk>P a b I f; finite I; i \<notin> I\<rbrakk> \<Longrightarrow> P a b (insert i I) f" shows "P a b I f"
4: shows "coreflexive ((f \<sqinter> q)\<^sup>T * (f \<sqinter> - q)\<^sup>T)"
hyps_alongE: fixes f :: "'form"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "f |\<in>| hyps_along pth" obtains v :: "'vertex" and p :: "('form, 'var) antecedent" and h :: "('form, 'var) out_port" where "(vb__, p) \<in> snd ` set pth" and "f = labelAtOut vb__ h" and "h |\<in>| hyps_for (nodeOf vb__) p"   | v :: "'vertex" and pf :: "'form" where "v |\<in>| vertices" and "nodeOf v = Assumption pf" and "f = labelAtOut v (Reg pf)"
complete_iff_full: fixes t :: "'b tree23" shows "complete t = (\<exists>n. full n t)"
eval_expr_eq: fixes G :: "prog"   and s :: "state"   and t :: "expr"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>s \<midarrow>In1l t\<succ>\<rightarrow> (w, s') = (\<exists>v. w = In1 v \<and> G\<turnstile>s \<midarrow>t-\<succ>v\<rightarrow> s')"
bij_betw_nth: fixes xs :: "'c list"   and A :: "nat set"   and B :: "'c set" assumes "distinct xs"   and "A = {..<length xs}"   and "B = set xs" shows "bij_betw ((!) xs) A B"
prob_space_distr: fixes f :: "'b \<Rightarrow> 'c"   and M' :: "'c measure" assumes "X.random_variable M' f" shows "prob_space (distr (distr M S X) M' f)"
exp_dvd_iff_exp_udvd: fixes n :: "nat"   and w :: "'b word" shows "(2 ^ n dvd w) = (2 ^ n udvd w)"
f: shows "f = (\<lambda>x. if getRel S_Idt G `` {x} = {} then x else SOME y. P x y)"
monom_one_Suc2: fixes n :: "nat" shows "monom P \<one> (Suc n) = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
generatorE: fixes G :: "('d, 'e) cyclic_group_scheme"   and x :: "'d" assumes "Cyclic_Group.cyclic_group G"   and "x \<in> carrier G" obtains n :: "nat" where "x = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> n"
verit_sko_forall': fixes P :: "'a \<Rightarrow> bool"   and A :: "bool" assumes "P (SOME x. \<not> P x) = A" shows "(\<forall>x. P x) = A"
l_inv_ex: fixes x :: "'c" assumes "x \<in> carrier V" shows "\<exists>y\<in>carrier V. y \<oplus>\<^bsub>V\<^esub> x = \<zero>\<^bsub>V\<^esub>"
self_append_conv2: fixes ys :: "'d list"   and xs :: "'d list" shows "(ys = xs @ ys) = (xs = [])"
noS: shows "\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S"
finer_def: fixes P :: "'c set set"   and Q :: "'c set set" shows "finer P Q \<equiv> (\<forall>J\<in>Q. J = \<Union> {I \<in> P. I \<subseteq> J}) \<and> (P \<noteq> {} \<longrightarrow> Q \<noteq> {})"
cloT1S: shows "closedin (top_of_set ({0::'c..1::'c} \<times> T)) ({0::'c..1::'c} \<times> S)"
prems_2: fixes i :: "nat" assumes "i < length zs" shows "(x # xs) ! i \<le> zs ! i"
IH: shows "vars (sum f S) \<subseteq> (\<Union>m\<in>S. vars (f m))"
hyps_4: shows "\<forall>e f g. e,f,g,a[App n []/0] # G \<Turnstile> b"
preserves_arr: fixes f :: "'a" assumes "Adj.C.arr f" shows "Adj.D.arr (G' f)"
Infinitesimal_inverse_HInfinite: fixes x :: "'a star" assumes "x \<in> Infinitesimal"   and "x \<noteq> 0" shows "inverse x \<in> HInfinite"
wt_s: shows "wtc c pc s \<noteq> \<top>"
af_G_ltl_continuation: fixes \<phi> :: "'b ltl"   and w' :: "nat \<Rightarrow> 'b set"   and w :: "'b set list" assumes "\<forall>\<psi>\<in>\<^bold>G \<phi>. w' \<Turnstile> \<psi> = (w \<frown> w') \<Turnstile> \<psi>" shows "(w \<frown> w') \<Turnstile> \<phi> = w' \<Turnstile> af\<^sub>G \<phi> w"
R_def: shows "R = {(V, V'). length V = length V' \<and> (\<forall>i<length V. [V ! i] \<approx>\<^bsub>d\<^esub> [V' ! i])}"
scaleR_mtx_vec_assoc: fixes c :: "real"   and A :: "'b sq_mtx"   and x :: "(real, 'b) vec" shows "(c *\<^sub>R A) *\<^sub>V x = c *\<^sub>R A *\<^sub>V x"
residual_l_refl: fixes x :: "'a" shows "(1::'a) \<le> x \<leftarrow> x"
simps_8: fixes x31 :: "nat"   and x32 :: "nat" shows "recf.Id x31 x32 \<noteq> Z"
open_empty: fixes S :: "'b set"   and is_open :: "'b set \<Rightarrow> bool" assumes "topological_space S is_open" shows "is_open {}"
multpwp_multpw_eq: fixes ns :: "('a \<times> 'a) set" shows "multpwp (\<lambda>x xa. (x, xa) \<in> ns) = (\<lambda>x xa. (x, xa) \<in> multpw ns)"
l2_trans_def: shows "l2_trans \<equiv> \<Union>m M X Rb Ra A B K. (l2_step1 Ra A B \<union> l2_step2 Rb A B X \<union> l2_step3 Ra A B X \<union> l2_step4 Rb A B X \<union> l2_dy_fake_chan M \<union> l2_dy_fake_msg m \<union> l2_lkr_others A \<union> l2_lkr_after A \<union> l2_skr Ra K)\<^sup>="
convex_hull_translation: fixes a :: "'c"   and S :: "'c set" shows "convex hull (+) a ` S = (+) a ` (convex hull S)"
path_Entry_target_2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as Entry" shows "as = []"
ib_block: shows "(i_begin, i_end) \<in> set (identify_blocks A k)"
prob_space_stream_space: shows "prob_space (stream_space (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x)))))"
sel_1: fixes x1 :: "'a"   and x2 :: "'a" shows "low (Interval x1 x2) = x1"
oghoare_sound: fixes p :: "'a set"   and c :: "'a com"   and q :: "'a set" assumes "\<parallel>- p c q" shows "\<parallel>= p  c q"
abs_less_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (abs a) b = (less a b \<and> less (uminus a) b)"
fact3: fixes i :: "'c" assumes "i \<in> - ODE_vars I ODE" shows "(\<lambda>t. fst (mk_v I ODE (sol 0, b) (sol t)) $ i) = (\<lambda>t. sol 0 $ i)"
det_code: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and A :: "'a mat" assumes "det_selection_fun sel_fun"   and "mute_fun mf" shows "det_code sel_fun mf A = det A"
ounicastT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>i\<^sub>p :: "'e \<Rightarrow> nat"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, {l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q), unicast (s\<^sub>i\<^sub>p (\<sigma> i)) (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i)), \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
map_poly_pderiv: fixes p :: "'a poly" shows "map_poly hom (pderiv p) = pderiv (map_poly hom p)"
assms_4: fixes z :: "'a" assumes "z \<in> A"   and "\<forall>y\<in>S. (y, z) \<in> r" shows "(L, z) \<in> r"
LeastI2_wellorder_ex: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x"   and "\<And>a. \<lbrakk>P a; \<forall>b. P b \<longrightarrow> less_eq a b\<rbrakk> \<Longrightarrow> Q a" shows "Q (ord.Least less_eq P)"
is_constant_seqE: fixes R :: "('a, 'b) partial_object_scheme"   and s :: "nat \<Rightarrow> 'a"   and k :: "nat"   and a :: "'a"   and n :: "nat" assumes "is_constant_seq R s"   and "s k = a" shows "s n = a"
trivial_hom: fixes H :: "('e, 'f) monoid_scheme"   and G :: "('g, 'h) monoid_scheme" assumes "Group.group H" shows "(\<lambda>x. \<one>\<^bsub>H\<^esub>) \<in> hom G H"
scope_valid_inport: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and v' :: "'b \<times> nat list"   and v :: "'b \<times> nat list"   and ip :: "('b, 'a) antecedent" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "v' |\<in>| Solved_Task.vertices freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "Solved_Task.scope' freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions v ip v' = (\<exists>i. fst v = fst v' \<and> prefix (snd v @ [i]) (snd v') \<and> ip = Solved_Task.in_port_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions v i)"
Inf_UNIV_nat: shows "\<Sqinter> UNIV = 0"
PRes_imp_LID_used: fixes M :: "agent"   and N :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Crypt (priSK M) (Hash \<lbrace>N, X\<rbrace>) \<in> parts (knows Spy evs)"   and "M \<notin> bad"   and "evs \<in> set_pur" shows "N \<in> used evs"
rev: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and xs :: "'c list" assumes "comm_monoid_list f z" shows "monoid_list.F f z (rev xs) = monoid_list.F f z xs"
p_dist_inf: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) = - x \<squnion> - y"
assms_1: fixes a :: "'a"   and b1 :: "'a"   and b2 :: "'a" assumes "a \<rightarrow>\<^sup>* b1"   and "a \<rightarrow>\<^sup>* b2"   and "is_final b1"   and "is_final b2" shows "b1 = b2"
de_Morgan_disj: fixes x :: "'c"   and y :: "'c" shows "NOT (x OR y) = NOT x AND NOT y"
infsuff_inf: fixes x :: "'b llist"   and A :: "'b set"   and s :: "'b llist" assumes "x \<in> infsuff A s" shows "x \<in> A\<^sup>\<omega>"
region_cover_spec: fixes u :: "'c \<Rightarrow> real" assumes "\<forall>x\<in>\<X>. 0 \<le> u x" shows "\<exists>R. R \<in> \<R> \<and> u \<in> R"
ordered_semiring_record_simps_11: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'b" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
singleDSourceEmpty4isNotSource: fixes i :: "AbstrLevelsID"   and C :: "CSet"   and S :: "CSet"   and Z :: "CSet" assumes "DAcc i C = {S}"   and "isNotDSource i S"   and "Z \<noteq> S" shows "C \<notin> Sources i Z"
minim_in: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "local.minim B \<in> B"
card_S_chamber: fixes C :: "'a set" assumes "chamber C" shows "card C = card (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
completeTA_is_ta: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option" assumes "det_tree_automaton TA A" shows "complete_tree_automaton (det_tree_automaton.completeTA TA A) A"
mk_regular1_def: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "'b set" shows "mk_regular1 P F I = I \<union> {F q x |q x. x \<in> I \<and> P q x}"
mapl_G_comp: fixes l1 :: "'w \<Rightarrow> 'u"   and l2 :: "'x \<Rightarrow> 'v"   and l1' :: "'n \<Rightarrow> 'w"   and l2' :: "'o \<Rightarrow> 'x" shows "mapl_G l1 l2 \<circ> mapl_G l1' l2' = mapl_G (l1 \<circ> l1') (l2 \<circ> l2')"
partial_function_lift: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lb :: "'a set \<Rightarrow> 'a" assumes "partial_function_definitions ord lb" shows "partial_function_definitions (fun_ord ord) (fun_lub lb)"
p_def: shows "p = (\<Prod>a\<in>UNIV. [:- a, 1::'a:]) + [:- (1::'a):]"
insert: fixes x :: "'a"   and xs :: "'a list" assumes "dxs = fromList (x # xs)"   and "distinct (x # xs)"   and "sorted (x # xs)" shows "P"
RedInitBlock: fixes V :: "char list"   and T :: "ty"   and v :: "val"   and u :: "val"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; Val u},s,b\<rangle> \<rightarrow> \<langle>Val u,s,b\<rangle>"
v0_v1_Digraph_axioms: shows "v0_v1_Digraph H_x v0 v1"
embed_bool_def: fixes P :: "'b \<Rightarrow> bool" shows "\<guillemotleft> P \<guillemotright> \<equiv> \<lambda>s. if P s then 1 else 0"
progress_binop_relop_f: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and cs :: "e list"   and t :: "t"   and e :: "b_e"   and fop :: "binop_f"   and frop :: "relop_f"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [t, t]"   and "is_float_t t"   and "const_list cs"   and "e = Binop_f t fop \<or> e = Relop_f t frop" shows "\<exists>a s' vs' es'. \<lparr>s;vs;cs @ [$e]\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
min_add_distrib_left: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "subset_mset.min x y + z = subset_mset.min (x + z) (y + z)"
hyps_2: fixes b :: "trm" assumes "\<Gamma> \<turnstile> q \<leftrightarrow> b : T\<^sub>1\<rightarrow>T\<^sub>2" shows "\<Gamma> \<turnstile> p \<leftrightarrow> b : T\<^sub>1\<rightarrow>T\<^sub>2"
\<phi>_simps_1: fixes \<mu> :: "'a" assumes "\<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>" shows "arr (d\<^sub>0e\<^sub>0.\<phi> \<mu>)"
sum_SDG_summary_edge_matched: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and p :: "'f"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n None (Some (p, True)) True n'" obtains ns :: "'b SDG_node list" where "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" and "n \<in> set ns" and "get_proc (parent_node (last ns)) = p"
cases: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "a \<in> orbit f x" obtains "a = f x"   | y :: "'b" where "a = f y" and "y \<in> orbit f x"
fnmember_intro: fixes x :: "'b"   and xs :: "'b fset" assumes "x \<notin> \<langle>xs\<rangle>\<^sub>f" shows "x |\<notin>| xs"
all_after_unl_6: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'a" shows "w \<Turnstile> (\<forall>x. P x)$ = (\<forall>x. P x$)"