hcju/isabelleps huggingface dataset:
simply_connected_retraction_gen: fixes S :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and k :: "'c \<Rightarrow> 'b" assumes "simply_connected S"   and "continuous_on S h"   and "h ` S = T"   and "continuous_on T k"   and "k ` T \<subseteq> S"   and "\<And>y. y \<in> T \<Longrightarrow> h (k y) = y" shows "simply_connected T"
vertical_chop_width_mon: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and v3 :: "view"   and v4 :: "view"   and x :: "nat" shows "v=v1--v2 \<and> v2=v3--v4 \<and> |lan v3| = x \<longrightarrow> x \<le> |lan v|"
subtype_x_Tuple: fixes \<tau> :: "'a OCL_Types.type"   and \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" assumes "\<tau> \<sqsubset> Tuple \<pi>" obtains \<pi>' :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" where "\<tau> = Tuple \<pi>'" and "subtuple (\<lambda>\<tau> \<sigma>. \<tau> \<sqsubset> \<sigma> \<or> \<tau> = \<sigma>) \<pi>' \<pi>" and "\<pi>' \<noteq> \<pi>"
length_entries_rbtreeify_g: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> Suc (length kvs)" shows "length (entries (fst (rbtreeify_g n kvs))) = n - 1"
full_checker_l_def: fixes spec :: "(char list list \<times> int) list"   and A :: "(nat, (char list list \<times> int) list) fmap"   and st :: "((char list list \<times> int) list, char list, nat) pac_step list" shows "full_checker_l spec A st = full_normalize_poly spec \<bind> (\<lambda>spec'. remap_polys_l spec' {} A \<bind> (\<lambda>(b, \<V>, A). if is_cfailed b then RETURN (b, \<V>, A) else let \<V> = \<V> \<union> vars_llist spec in PAC_checker_l spec' (\<V>, A) b st))"
pred_intros_logic_2: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "Measurable.pred M P" shows "Measurable.pred M (\<lambda>x. \<not> P x)"
BisT_matchC_TMC_rev: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>T d" shows "matchC_TMC BisT d c"
assms_4: shows "p' \<in># image_mset set_mset P"
nonzero_mult_divide_mult_cancel_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "b \<noteq> \<zero>"   and "c \<noteq> \<zero>" shows "c \<otimes> a \<oslash> (c \<otimes> b) = a \<oslash> b"
Ag_ind_bijec: fixes A :: "('c, 'd) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and D :: "'e set" assumes "aGroup A"   and "f \<in> carrier A \<rightarrow> D"   and "bij_to f (carrier A) D" shows "bijec\<^bsub>A,Ag_ind A f\<^esub> Agii A f"
dis: shows "\<forall>j1\<in>K. \<forall>j2\<in>K. j1 \<noteq> j2 \<longrightarrow> R j1 \<inter> R j2 = {}"
correctCompositionDiffLevelsS15: shows "correctCompositionDiffLevels sS15"
hseqE: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" obtains "B.arr \<mu>" and "B.arr \<nu>" and "src \<nu> = trg \<mu>"
vbl_casesE: fixes x :: "vbl" obtains "x = zeroX"   | y :: "vbl" where "x = nextX y"
ScopeCommAux: fixes x :: "name"   and y :: "name"   and P :: "ccs" assumes "x \<noteq> y" shows "\<lparr>\<nu>x\<rparr>\<lparr>\<nu>y\<rparr>P \<equiv>\<^sub>s \<lparr>\<nu>y\<rparr>\<lparr>\<nu>x\<rparr>P"
fls_integral_X_inv_power: fixes n :: "nat" assumes "2 \<le> n" shows "fls_integral (fls_X_inv ^ n) = fls_const (inverse (of_int (1 - int n))) * fls_X_inv ^ (n - 1)"
ground_vars\<^sub>t: fixes t :: "fterm" assumes "ground\<^sub>t t" shows "vars\<^sub>t t = {}"
hyps_2: fixes \<rho> :: "Vars.var \<Rightarrow> 'Value" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho> f|` fv e\<^esub>"
sources_sinks_aux: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'d list" shows "sources_aux I D U xs = sinks_aux (I\<inverse>) D U (rev xs)"
while_inv_def: fixes b :: "'a upred"   and p :: "'a upred"   and S :: "'a hrel" shows "while b invr p do S od = while\<^sup>\<top> b do S od"
psubstAbs_substAbs_compose: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "goodEnv rho" shows "A $[Y / y]_ys $[rho] = A $[rho [y \<leftarrow> Y #[rho]]_ys]"
dependent_eq_real: shows "module_on.dependent (*\<^sub>R) = dependent"
InvariantEquivalentZLAfterInitialization: fixes F0 :: "Literal list list" shows "let state' = initialize F0 initialState; F0' = filter (\<lambda>c. \<not> clauseTautology c) F0 in getSATFlag state' = UNDEF \<and> InvariantEquivalentZL (getF state') (getM state') F0' \<or> getSATFlag state' = FALSE \<and> \<not> satisfiable F0'"
param_FAIL: fixes R :: "('b \<times> 'c) set" shows "(FAIL, FAIL) \<in> \<langle>R\<rangle>nres_rel"
of_complex_power_int: fixes x :: "complex"   and n :: "int" shows "of_complex (x powi n) = of_complex x powi n"
edge_kind_cases: fixes et :: "('a, 'b, 'c, 'd) edge_kind" obtains "intra_kind et"   | Q :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool" and r :: "'c" and p :: "'d" and fs :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list" where "et = Qa__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs"   | Q :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool" and p :: "'d" and f :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" where "et = Q\<hookleftarrow>\<^bsub>p\<^esub>f"
vec_vec_Tensor_right_id: fixes v :: "'a list" shows "vec_vec_Tensor v [id] = v"
r_one: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "x \<oplus>\<^bsub>K [X]\<^esub> \<zero>\<^bsub>K [X]\<^esub> = x"
a_2: shows "{- a--a} \<subseteq> existence_ivl0 x"
induct: fixes x :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>"   and P :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha> \<Rightarrow> bool" assumes "Formula.hereditarily_fs x"   and "\<And>tset\<^sub>\<alpha>. \<lbrakk>finite (supp tset\<^sub>\<alpha>); \<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> Formula.hereditarily_fs t\<^sub>\<alpha>; \<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Formula.Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"   and "\<And>t\<^sub>\<alpha>. \<lbrakk>Formula.hereditarily_fs t\<^sub>\<alpha>; P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Formula.Not\<^sub>\<alpha> t\<^sub>\<alpha>)"   and "\<And>\<phi>. P (Formula.Pred\<^sub>\<alpha> \<phi>)"   and "\<And>t\<^sub>\<alpha> \<alpha>. \<lbrakk>Formula.hereditarily_fs t\<^sub>\<alpha>; P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Formula.Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>)" shows "P x"
S_transform_tau_transition_iff: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.tau_transition (nominal_ts.S_transition satisfies transition) (S_action.Act \<tau>) P P' = weak_nominal_ts.tau_transition transition \<tau> P P'"
\<gamma>\<^sub>g: shows "\<guillemotleft>\<gamma>\<^sub>g : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma>\<^sub>g \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g)"
sol: shows "(sol solves_ode (\<lambda>a. ODE_sem I (OUrename x y ODE))) {0..t} {xa. mk_v I (OUrename x y ODE) (sol 0, b) xa \<in> fml_sem I (FUrename x y \<phi>)}"
atom_step2: fixes s :: "'a"   and q :: "'a"   and b :: "'a"   and r :: "'a"   and l :: "'a"   and a :: "'a" assumes "s = s \<cdot> q"   and "q \<cdot> b = (0::'a)"   and "r \<cdot> q \<le> q \<cdot> r"   and "q \<cdot> l \<le> l \<cdot> q"   and "r\<^sup>\<infinity> = r\<^sup>\<star>"   and "test q" shows "s \<cdot> l\<^sup>\<infinity> \<cdot> b\<^sup>\<infinity> \<cdot> r\<^sup>\<infinity> \<cdot> q \<cdot> (a \<cdot> b\<^sup>\<infinity> \<cdot> r\<^sup>\<infinity> \<cdot> q)\<^sup>\<infinity> \<le> s \<cdot> l\<^sup>\<infinity> \<cdot> r\<^sup>\<infinity> \<cdot> (a \<cdot> b\<^sup>\<infinity> \<cdot> q \<cdot> r\<^sup>\<infinity>)\<^sup>\<infinity>"
UNION_eq: fixes B :: "'e \<Rightarrow> 'd set"   and A :: "'e set" shows "\<Union> (B ` A) = {y. \<exists>x\<in>A. y \<in> B x}"
cos_geD: fixes a :: "real"   and x :: "real" assumes "cos a \<le> cos x"   and "0 \<le> a"   and "a \<le> pi"   and "- pi \<le> x"   and "x \<le> pi" shows "x \<in> {- a..a}"
pnsubst_def: fixes P1 :: "pol"   and M :: "mon"   and P2 :: "pol"   and n :: "nat" shows "pnsubst P1 M P2 n = (case ponesubst P1 M P2 of None \<Rightarrow> None | Some P3 \<Rightarrow> Some (pnsubst1 P3 M P2 n))"
valid_Act_strong: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and X :: "'d"   and P :: "'a"   and \<alpha> :: "'c"   and x :: "('e, 'b, 'c) formula" assumes "nominal_ts satisfies transition"   and "finite (supp X)" shows "nominal_ts.valid satisfies transition P (Act \<alpha> x) = (\<exists>\<alpha>'' x' P'. Act \<alpha> x = Act \<alpha>'' x' \<and> transition P \<langle>\<alpha>'',P'\<rangle> \<and> nominal_ts.valid satisfies transition P' x' \<and> bn \<alpha>'' \<sharp>* X)"
eventuallyI: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter" assumes "\<And>x. P x" shows "eventually P F"
omega_\<omega>1: shows "\<omega> \<in> elts \<omega>1"
lprefix_lSup_revert_2: shows "lprefix = (\<le>)"
lfp_while_lattice: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f"   and "finite UNIV" shows "lfp f = while (\<lambda>A. f A \<noteq> A) f bot"
linord_of_listP_empty: fixes xs :: "'a list" shows "(\<forall>x y. \<not> linord_of_listP x y xs) = (xs = [])"
rel_stacktraces_rewrite_fundef: fixes F2 :: "'fenv_ubx"   and xs :: "('fun, 'dyn) frame list"   and ys :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list"   and opt :: "'fun option"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and instr :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr" assumes "rel_stacktraces (Fubx_get F2) xs ys opt"   and "Fubx_get F2 f = Some fd"   and "pc < length (body fd)"   and "\<forall>\<Sigma>. Subx.sp_instr (Fubx_get F2) (body fd ! pc) \<Sigma> = Subx.sp_instr (Fubx_get F2) instr \<Sigma>"   and "\<not> is_fun_call (body fd ! pc)" shows "rel_stacktraces (Fubx_get (Fubx_add F2 f (rewrite_fundef_body fd pc instr))) xs ys opt"
add_poly_l_def: shows "add_poly_l = REC\<^sub>T (\<lambda>add_poly_l (p, q). case (p, q) of ([], []) \<Rightarrow> RETURN [] | ([], a # list) \<Rightarrow> RETURN (a # list) | ((aa, ba) # list, []) \<Rightarrow> RETURN ((aa, ba) # list) | ((aa, ba) # list, (ys, m) # qa) \<Rightarrow> if aa = ys then if ba + m = 0 then add_poly_l (list, qa) else add_poly_l (list, qa) \<bind> (\<lambda>pq. RETURN ((aa, ba + m) # pq)) else if (aa, ys) \<in> term_order_rel then add_poly_l (list, (ys, m) # qa) \<bind> (\<lambda>pq. RETURN ((aa, ba) # pq)) else add_poly_l ((aa, ba) # list, qa) \<bind> (\<lambda>pq. RETURN ((ys, m) # pq)))"
tarski_space_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_space C B \<equiv> tarski_absolute_space C B \<and> tarski_space_axioms B"
nodes_connected_sym: fixes G :: "('a, 'b) graph"   and a :: "'a"   and b :: "'a" assumes "valid_graph G" shows "nodes_connected G a b = nodes_connected G b a"
simps_2: fixes n :: "nat"   and k :: "nat"   and B :: "binding"   and \<Gamma> :: "binding list" shows "\<up>\<^sub>e n k (B \<Colon> \<Gamma>) = mapB (\<up>\<^sub>\<tau> n (k + \<parallel>\<Gamma>\<parallel>)) B \<Colon> \<up>\<^sub>e n k \<Gamma>"
card_Inv_sym: fixes xs :: "'b list"   and ys :: "'b list" shows "card (Inv xs ys) = card (Inv ys xs)"
r1: shows "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = P"
x_leq_triple_x: fixes x :: "'a" shows "x \<le> x ; x\<^sup>T ; x"
invc2_baldL: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invc2 l"   and "invc r" shows "invc2 (baldL l a r)"
allUses_finite: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "finite (allUses g n)"
nhd_iff: shows "(\<exists>v\<in>nhds y. x \<notin> v) = (\<exists>u\<in>nhds x. y \<notin> u)"
Longest_common_prefix_prefix: fixes xs :: "'c list"   and L :: "'c list set" assumes "xs \<in> L" shows "prefix (Longest_common_prefix L) xs"
ereal_minus_8: fixes x :: "ereal" shows "0 - x = - x"
alphaAbs_ex_qAFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_ex_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
P1_def: shows "P1 = {#p \<in># P. Polynomial.degree p = 0#}"
fps_shift_mult: fixes n :: "nat"   and g :: "'a fps"   and h :: "'a fps" assumes "n \<le> subdegree g" shows "fps_shift n (h * g) = h * fps_shift n g"
defval_conf: fixes P :: "'d prog"   and T :: "ty" shows "P \<turnstile> default_val T :\<le> T"
fun_upd_idem: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a" assumes "f x = y" shows "f(x := y) = f"
density_context_axioms: shows "density_context V V' \<Gamma> \<delta>"
effect_heapI: fixes h' :: "heap"   and f :: "heap \<Rightarrow> 'b \<times> heap"   and h :: "heap"   and r :: "'b" assumes "h' = snd (f h)"   and "r = fst (f h)" shows "effect (Heap_Monad.heap f) h h' r"
IH_1: fixes x :: "'a \<times> 'b"   and xa :: "('a \<times> 'b) set"   and xb :: "('a \<times> 'b) set" assumes "\<not> (infinite A \<or> A = {})"   and "x = arg_min_on snd A"   and "xa = {p \<in> A. fst p < fst x}"   and "xb = {p \<in> A. fst x < fst p}" shows "set_tree (treap_of xa) \<subseteq> xa"
at1_def: shows "at1 = Cn 1 r_nth [recf.Id 1 0, r_const 1]"
simps_29: shows "cospan_shape.comp cospan_shape.AA cospan_shape.BB = cospan_shape.Null"
augment_mono_aux: fixes A :: "nat \<Rightarrow> bool option"   and B :: "nat \<Rightarrow> bool option"   and x :: "nat set"   and C :: "nat set" assumes "dom A \<subseteq> dom B" shows "dom (A \<oplus>\<^bsub>x\<^esub> C) \<subseteq> dom (B \<oplus>\<^bsub>x\<^esub> C)"
mult_scalar_tail_rec_left: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "p \<odot> q = monom_mult (punit.lc p) (punit.lt p) q + punit.tail p \<odot> q"
power2_le_imp_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (power.power one times x 2) (power.power one times y 2)"   and "less_eq zero y" shows "less_eq x y"
hcis_eq: fixes a :: "real star" shows "hcis a = hcomplex_of_hypreal ((*f* cos) a) + iii * hcomplex_of_hypreal ((*f* sin) a)"
Nonce_Agent: fixes X :: "nonce_t"   and X' :: "agent" shows "Nonce X \<noteq> Messages.Agent X'"
I: shows "impl_lasso_loc Re node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G"
scast_2_power: fixes x :: "nat" shows "scast (2 ^ x) = 2 ^ x"
simps: fixes s :: "atom_sort"   and n :: "nat" shows "sort_of (Atom s n) = s"
simplex_unsat0: fixes cs :: "('i \<times> constraint) list"   and I :: "'i list" assumes "solve_exec cs = Inl I" shows "minimal_unsat_core (set I) cs"
red_ch_cd_def: fixes G :: "('e, 'f) Group_scheme"   and f :: "nat \<Rightarrow> 'e set"   and m :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "red_ch_cd G f m g = (tW_cmpser G (card (f ` {i. i \<le> m}) - 1) g \<and> g ` {i. i \<le> card (f ` {i. i \<le> m}) - 1} = f ` {i. i \<le> m})"
inject: fixes x2 :: "form"   and x1 :: "form_list"   and y2 :: "form"   and y1 :: "form_list" shows "(FCons x2 x1 = FCons y2 y1) = (x2 = y2 \<and> x1 = y1)"
max2_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "((r.max2 a b, c) \<in> r) = ((a, c) \<in> r \<and> (b, c) \<in> r)"
maxim_mono: fixes X :: "'b set"   and Y :: "'b set" assumes "X \<subseteq> Y"   and "finite Y"   and "X \<noteq> {}"   and "Y \<subseteq> Field s" shows "(s.maxim X, s.maxim Y) \<in> s"
integral_snd: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" shows "LINT y|M2. LINT x|M1. f x y = LINT (x, y)|(M1 \<Otimes>\<^sub>M M2). f x y"
bind_map_spmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf"   and g :: "'c \<Rightarrow> 'b spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
eval_ternary_simps_simple_4: fixes x :: "ternaryvalue" shows "eval_ternary_And x TernaryFalse = TernaryFalse"
cod_cod: fixes f :: "'j" assumes "J.arr f" shows "J.cod (J.cod f) = J.cod f"
ssufD1: fixes a :: "'b list"   and b :: "'b list" assumes "a <s b" shows "a \<le>s b"
list_relE4: fixes l :: "'d list"   and x' :: "'e"   and xs' :: "'e list"   and R :: "('d \<times> 'e) set" assumes "(l, x' # xs') \<in> \<langle>R\<rangle>list_rel" obtains x :: "'d" and xs :: "'d list" where "l = x # xs" and "(x, x') \<in> R" and "(xs, xs') \<in> \<langle>R\<rangle>list_rel"
RETURN_refine: fixes x :: "'a"   and x' :: "'b"   and R :: "('a \<times> 'b) set" assumes "(x, x') \<in> R" shows "RETURN x \<le> \<Down> R (RETURN x')"
scaleC: fixes f :: "'b \<Rightarrow> 'c"   and r :: "complex"   and b :: "'b" assumes "clinear f" shows "f (r *\<^sub>C b) = r *\<^sub>C f b"
power2_diff_nat: fixes y :: "nat"   and x :: "nat" assumes "y \<le> x" shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
Q_in: shows "Q \<in> carrier_mat (dim_row A) (dim_row A)"
plus_monotone_left: fixes s :: "'a"   and t :: "'a"   and u :: "'a" assumes "s \<preceq> t" shows "u + s \<preceq> u + t"
InvGlobPointstampsEq_def: fixes c :: "('b, 't, 'loc) Combined.configuration" shows "InvGlobPointstampsEq c = (\<forall>p loc t. zcount (c_pts (prop_config c p) loc) t = zcount (c_glob (exchange_config c) p) (loc, t))"
A: shows "X \<subseteq> X' \<union> Y' \<union> Z'"
dim_col_mat_2: fixes nr :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'a vec" shows "dim_col (mat\<^sub>r nr nc g) = nc"
ivl_disj_un_one_4: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{..<l} \<union> {l..u} = {..u}"
induct: fixes P :: "'b option list \<times> 'b option list \<Rightarrow> bool"   and a0 :: "'b option list \<times> 'b option list" assumes "P ([], [])"   and "\<And>xs ys. P (xs, ys) \<Longrightarrow> P (None # xs, None # ys)"   and "\<And>x xs ys. P (xs, ys) \<Longrightarrow> P (Some x # xs, None # ys)"   and "\<And>xs y ys. P (xs, ys) \<Longrightarrow> P (None # xs, Some y # ys)"   and "\<And>x xs y ys. (x = y \<Longrightarrow> P (xs, ys)) \<Longrightarrow> P (Some x # xs, Some y # ys)"   and "\<And>v vc. P (Some v # vc, [])"   and "\<And>vb vc. P (vb # vc, [])"   and "\<And>vb vc. P ([], vb # vc)"   and "\<And>va vc. P ([], Some va # vc)" shows "P a0"
SK_returnS_returnK: fixes x :: "nat discr" shows "SK\<cdot>(returnS\<cdot>x) = returnK\<cdot>x"
prove_HRP: fixes A :: "fm" shows "{} \<turnstile> HRP \<guillemotleft>A\<guillemotright> \<guillemotleft>\<guillemotleft>A\<guillemotright>\<guillemotright>"
Least_mono: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "mono f"   and "\<exists>x\<in>S. \<forall>y\<in>S. x \<le> y" shows "(LEAST y. y \<in> f ` S) = f (LEAST x. x \<in> S)"
integrable_real_density: fixes f :: "'d \<Rightarrow> real"   and M :: "'d measure"   and g :: "'d \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "g \<in> borel_measurable M" shows "integrable (density M (\<lambda>x. ennreal (f x))) g = integrable M (\<lambda>x. f x * g x)"
nonempty_path_components_of: fixes C :: "'c set"   and X :: "'c topology" assumes "C \<in> path_components_of X" shows "C \<noteq> {}"
Inf_greatest: fixes A :: "('a \<times> 'b) set"   and z :: "'a \<times> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod z x" shows "less_eq_prod z (Inf_prod A)"
poly_cmult: fixes c :: "'a"   and p :: "'a list"   and x :: "'a" shows "poly (c %* p) x = c * poly p x"
RedLAss: fixes V :: "char list"   and v :: "val"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>V:=Val v,(h, l, sh),b\<rangle> \<rightarrow> \<langle>unit,(h, l(V \<mapsto> v), sh),b\<rangle>"
simps_2: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and accu :: "('b \<times> 'b) list"   and e :: "'b \<times> 'b"   and Es :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations_accu G M accu (e # Es) = (if TopoS_Composition_Theory.all_security_requirements_fulfilled (TopoS_Composition_Theory.get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set (e # accu)\<rparr>) then TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations_accu G M (e # accu) Es else TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations_accu G M accu Es)"
ket_plus_k_decode: fixes k :: "nat" assumes "k < n" shows "ket_plus_k k = Matrix.vec (2 ^ Suc k) (\<lambda>l. 1 / csqrt (2 ^ Suc k))"
cases: fixes a1 :: "pi"   and a2 :: "pi" assumes "a1 \<sim>\<^sub>e a2" obtains P :: "pi" and Q :: "pi" where "a1 = P" and "a2 = Q" and "P \<leadsto>\<^sub>e[Strong_Early_Bisim.bisim] Q" and "Q \<sim>\<^sub>e P"
append_dot_product_split: fixes u :: "'a vec"   and w :: "'a vec"   and x :: "'a vec" assumes "dim_vec (u @\<^sub>v w) = dim_vec x" shows "(u @\<^sub>v w) \<bullet> x = (\<Sum>i = 0..<dim_vec u. u $ i * x $ i) + (\<Sum>i = 0..<dim_vec w. w $ i * x $ (i + dim_vec u))"
lemma_2_10_33: fixes b :: "'a"   and a :: "'a" shows "(((b l\<rightarrow> a) r\<rightarrow> a) l\<rightarrow> b) l\<rightarrow> b l\<rightarrow> a = b l\<rightarrow> a"
cobounded2: fixes b :: "'k"   and a :: "'k" shows "b \<le> max a b"
chine_composite_2: shows "H\<mu>H\<nu>H\<pi>\<rho>.chine = \<mu>.leg0 \<down>\<down> \<nu>.leg1 \<cdot> \<nu>_\<pi>\<rho>.prj\<^sub>1"
comp_arr_inv': fixes f :: "'a" assumes "S.iso f" shows "f \<cdot>\<^sub>S S.inv f = S.cod f"
simps_1: fixes i :: "nat"   and p :: "'c"   and R\<^sub>i :: "nat set" shows "netmap (NodeS i p R\<^sub>i) = [i \<mapsto> p]"
prod_cont_iff: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" shows "cont f = ((\<forall>y. cont (\<lambda>x. f (x, y))) \<and> (\<forall>x. cont (\<lambda>y. f (x, y))))"
goedel_first_theHardHalf: assumes "consistent" shows "\<not> prv (neg \<phi>G)"
butlast_rev: fixes p :: "'b list" shows "butlast (rev p) = rev (tl p)"
solution_pell_power: fixes z :: "'a \<times> 'a"   and n :: "nat" assumes "solution z" shows "solution (pell_power z n)"
kingman_theorem_nonergodic_2: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u"   and "- \<infinity> < subcocycle_avg_ereal u" shows "integrable M (subcocycle_lim u)"
comp_fun_idem_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "comp_fun_idem f \<equiv> comp_fun_commute f \<and> comp_fun_idem_axioms f"
numeral_eq_neg_one_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(uminus (numeral.numeral one plus n) = uminus one) = (n = num.One)"
topological_basis_def: fixes B :: "'a set set" shows "topological_basis B = (Ball B open \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union> B' = x)))"
ide_self_inverse: fixes a :: "'b" assumes "B.ide a" shows "B.inverse_arrows a a"
set_list: fixes A :: "'a set" assumes "finite A" shows "set (list A) = A"