hcju/isabelleps huggingface dataset:
valid_SBE_def: fixes \<sigma> :: "'b"   and m :: "'b \<Rightarrow> ('c \<times> 'b) set option" shows "\<sigma> \<Turnstile>\<^sub>S\<^sub>B\<^sub>E m \<equiv> m \<sigma> \<noteq> None"
1: shows "rat_poly_inv (rat_poly_times B (rat_poly_times A B)) = rat_poly_inv B"
hcomp_in_vhom: fixes \<mu> :: "'b"   and f :: "'b"   and g :: "'b"   and \<nu> :: "'b"   and h :: "'b"   and k :: "'b" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow> k\<guillemotright>"   and "src h = trg f" shows "\<guillemotleft>\<nu> \<star> \<mu> : h \<star> f \<Rightarrow> k \<star> g\<guillemotright>"
antisym_conv2: fixes x :: "'a word"   and y :: "'a word" assumes "x \<le>s y" shows "(\<not> x <s y) = (x = y)"
assms_3: shows "all_fulfill (\<lambda>x. redundant x S2) S1"
Complexs_cases: fixes q :: "'b" assumes "q \<in> \<complex>" obtains c :: "complex" where "q = of_complex c"
simps_12: fixes m :: "nat"   and p :: "fm" shows "bound m (E p) = bound (Suc m) p"
nodes_of_component: fixes g :: "'a ugraph"   and r :: "'a" shows "nodes (component_of g r) = reachable_nodes g r"
Smith_normal_form_upt_k_condition1: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and a :: "'d"   and b :: "'c" assumes "Smith_normal_form_upt_k A k"   and "mod_type_class.to_nat a = mod_type_class.to_nat b"   and "mod_type_class.to_nat a + 1 < k"   and "mod_type_class.to_nat b + 1 < k" shows "A $ a $ b dvd A $ (a + (1::'d)) $ (b + (1::'c))"
is_join_and_compatible: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> z"   and "y \<sqsubseteq> z"   and "\<And>a. \<lbrakk>x \<sqsubseteq> a; y \<sqsubseteq> a\<rbrakk> \<Longrightarrow> z \<sqsubseteq> a" shows "compatible x y \<and> x \<squnion> y = z"
prv_exi_imp: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<chi>"   and "bprv (all x (imp \<phi> \<chi>))" shows "bprv (imp (exi x \<phi>) \<chi>)"
axioms_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "inverse_functors A B F G" shows "category A"
spr_trans: fixes t :: "'s Trace"   and ec :: "'s odlist" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = spr_jkbpCSt t" shows "set (local.spr_trans agent ec) = rel_ext (\<lambda>uu_. \<exists>t' s. uu_ = s \<and> t' \<leadsto> s \<in> SPR.jkbpC \<and> spr_jview agent t' = spr_jview agent t)"
rep_eq: fixes x :: "'b set_rbt" shows "RBT_Set2.keys x = RBT_Impl.keys (mapping_rbt.impl_of x)"
simps_1: fixes cs :: "'edge list" shows "same_level_path_aux cs [] = True"
fun_lub_def: fixes L :: "'h set \<Rightarrow> 'g"   and A :: "('f \<Rightarrow> 'h) set" shows "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
closed_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "closed A f = (\<forall>x\<in>A. \<forall>y\<in>A. x \<squnion>\<^bsub>f\<^esub> y \<in> A)"
assms_2: assumes "\<not> individually_rational_on ds X" shows "P"
f_def: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)" shows "??.Euler_Criterion.f p a x = {x, ??.Euler_Criterion.f_1 p a x}"
syzygy_module_listI: fixes s' :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and s :: "'t \<Rightarrow>\<^sub>0 'c" assumes "s' \<in> pmdl.syzygy_module (set bs)"   and "s = atomize_poly (idx_pm_of_pm bs s')" shows "s \<in> syzygy_module_list bs"
CatchMatch: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "'b"   and s' :: "'b"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> Abrupt s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
assms: shows "min_maxsimpchain (w # x # ys @ [z])"
map_poly_map_poly: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and p :: "'c poly" assumes "f (0::'b) = (0::'a)"   and "g (0::'c) = (0::'b)" shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p"
intro: fixes inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and hyps :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and vertices :: "'d fset"   and freshenLC :: "nat \<Rightarrow> 'e \<Rightarrow> 'e"   and renameLCs :: "('e \<Rightarrow> 'e) \<Rightarrow> 'f \<Rightarrow> 'f"   and lconsts :: "'f \<Rightarrow> 'e set"   and closed :: "'f \<Rightarrow> bool"   and subst :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and subst_lconsts :: "'g \<Rightarrow> 'e set"   and subst_renameLCs :: "('e \<Rightarrow> 'e) \<Rightarrow> 'g \<Rightarrow> 'g"   and anyP :: "'f"   and vidx :: "'d \<Rightarrow> nat"   and nodes :: "'a stream"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set"   and labelsIn :: "'a \<Rightarrow> 'b \<Rightarrow> 'f"   and labelsOut :: "'a \<Rightarrow> 'c \<Rightarrow> 'f"   and inst :: "'d \<Rightarrow> 'g"   and local_vars :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set" assumes "Instantiation inPorts outPorts hyps vertices freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx"   and "Well_Shaped_Graph nodes inPorts outPorts vertices nodeOf edges hyps"   and "Solution inPorts outPorts nodeOf hyps vertices labelsIn labelsOut freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx inst edges"   and "Well_Scoped_Instantiation freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP inPorts outPorts nodeOf hyps vertices vidx inst edges local_vars" shows "Scoped_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP inPorts outPorts nodeOf hyps nodes vertices labelsIn labelsOut vidx inst edges local_vars"
n_ge_2: shows "- fls_subdegree g \<le> int n"
ed_ou_witness_not_discrete: fixes X :: "ed_ou_witness set" shows "\<not> discrete X"
upset_set_prop: shows "\<Up> = \<Union> \<circ> (`) \<up>"
that: fixes yvec1 :: "name list"   and y :: "name"   and yvec2 :: "name list" assumes "yvec = yvec1 @ y # yvec2"   and "length xvec1 = length yvec1"   and "length xvec2 = length yvec2"   and "\<lparr>\<nu>*(xvec1 @ xvec2)\<rparr>M \<prec>' P = \<lparr>\<nu>*(yvec1 @ [(x, y)] \<bullet> yvec2)\<rparr>([(x, y)] \<bullet> N) \<prec>' ([(x, y)] \<bullet> Q)" shows "thesis"
simps_130: fixes x41 :: "('d, 'c) Term.term"   and x42 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (insert\<langle>x41,x42\<rangle>) = fv x41 \<union> fv x42"
RSpan_smult_closed: fixes r :: "'r"   and ms :: "'m list"   and n :: "'m" assumes "r \<in> R"   and "set ms \<subseteq> M"   and "n \<in> RSpan ms" shows "r \<cdot> n \<in> RSpan ms"
bigo_plus_absorb: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(g)" shows "f +o O(g) = O(g)"
Lim_transform_within_set: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and a :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "(f \<longlongrightarrow> l) (at a within S)"   and "\<forall>\<^sub>F x in at a. (x \<in> S) = (x \<in> T)" shows "(f \<longlongrightarrow> l) (at a within T)"
content_ball_pos: fixes r :: "real"   and c :: "'c" assumes "0 < r" shows "0 < content (ball c r)"
occuring_gens_in_element: fixes x :: "(bool \<times> 'd) list"   and gens :: "'d set" assumes "x \<in> carrier \<F>\<^bsub>gens\<^esub>" shows "x \<in> lists (UNIV \<times> gens)"
t_lLam: fixes x :: "name"   and t :: "llam"   and T1 :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and T2 :: "ty" assumes "x \<sharp> t"   and " (x, T1) # \<Gamma> \<turnstile> freshen t x : T2 " shows " \<Gamma> \<turnstile> lLam t : T1 \<rightarrow> T2 "
zip_arg_subterm_2: fixes s :: "('a, 'b) Term.term"   and t :: "('c, 'd) Term.term"   and X :: "('a, 'b) Term.term list"   and Y :: "('c, 'd) Term.term list"   and g :: "'c" assumes "(s, t) \<in> set (zip X Y)" shows "t \<sqsubset> Fun g Y"
max_fun_diff_max: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and x :: "'c" assumes "f \<noteq> g"   and "f \<in> rst.FINFUNC"   and "g \<in> rst.FINFUNC"   and "x \<in> {a \<in> Field t. f a \<noteq> g a}" shows "(x, t.max_fun_diff f g) \<in> t"
add_mono_thms_linordered_semiring_1: fixes i :: "'h"   and j :: "'h"   and k :: "'h"   and l :: "'h" assumes "i \<le> j \<and> k \<le> l" shows "i + k \<le> j + l"
sort_all_coeffs: fixes xs :: "(char list list \<times> int) list" assumes "\<forall>x\<in>mononoms xs. distinct x" shows "sort_all_coeffs xs \<le> \<Down> Id (shuffle_coefficients xs)"
face_cycle_succ_pred: fixes a :: "'b" shows "H.face_cycle_succ (H.face_cycle_pred a) = a"
locally_ringed_space_axioms: shows "locally_ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
Un_insert_left: fixes a :: "'b"   and B :: "'b set"   and C :: "'b set" shows "insert a B \<union> C = insert a (B \<union> C)"
simps_7: fixes I :: "('e, 'f, 'd) interp"   and c :: "'f"   and \<phi> :: "('e, 'f, 'd) formula" shows "fml_sem I (InContext c \<phi>) = Contexts I c (fml_sem I \<phi>)"
take_intsFrom_enumFrom: fixes n :: "int"   and i :: "int" shows "take\<cdot>(MkI\<cdot>n)\<cdot>[MkI\<cdot>i..] = [MkI\<cdot>i..MkI\<cdot>(n + i) - 1]"
nsome_eqvt: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" shows "pi \<bullet> nSome x = nSome (pi \<bullet> x)"
lookup_pair_map_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'c"   and k :: "'a"   and v :: "'b" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "snd (f (k, 0::'b)) = (0::'c)"   and "\<And>a b. comp (fst (f a)) (fst (f b)) = comp (fst a) (fst b)" shows "comparator.lookup_pair comp (map_pair f xs) (fst (f (k, v))) = snd (f (k, comparator.lookup_pair comp xs k))"
goal3_2: fixes x :: "nat"   and y :: "nat \<times> nat"   and xa :: "nat"   and ya :: "nat" assumes "(x, y) = pm_pth (dpm bdd) n"   and "(xa, ya) = y"   and "pointermap_sane (dpm bdd)"   and "pointermap_getmk (v, ni1, ni2) (dpm bdd) = (x1, dpm s')"   and "Rmi_g ya e bdd" shows "Rmi_g ya e s'"
feasible_sound: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "feasible t"   and "sound P" shows "sound (t P)"
cd_def: fixes x :: "nat"   and m :: "nat"   and n :: "nat" shows "cd x m n = (x dvd m \<and> x dvd n)"
hyps_3: assumes "f \<in> A \<rightarrow> carrier_vec n" shows "dim_vec (finsum V f A) = n"
ParamDefs_in_Def: fixes n :: "'node"   and V :: "'var" assumes "valid_node n"   and "V \<in> set (ParamDefs n)" shows "V \<in> Def n"
poly_mapping_tail_induct: fixes P :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "P 0"   and "\<And>p. \<lbrakk>p \<noteq> 0; P (tail p)\<rbrakk> \<Longrightarrow> P p" shows "P p"
WP1n_DA_notinSet: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong p" shows "DenyAll \<notin> set (tl p)"
assms_2: shows "set_mset r|\<sigma>| \<subseteq> r \<down>s S"
alw_ev_sfilter_mono: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and Q :: "'b \<Rightarrow> bool" assumes "alw (ev (holds P)) xs"   and "\<And>x. P x \<Longrightarrow> Q x" shows "pred_stream Q (sfilter P xs)"
normalized_poly_def: fixes p :: "(char list multiset \<times> int) multiset" shows "normalized_poly p = (distinct_mset (fst `# p) \<and> 0 \<notin># snd `# p)"
conclusions_closed: fixes c :: "'form" assumes "c \<in> set conclusions" shows "closed c"
map_simp: fixes f :: "'a" assumes "arr f" shows "\<Omega>.map f = \<I>"
KBasic_8: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>& \<psi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
restrict_vals_nonequiv: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc (Suc n)) vv" shows "vals_nonequiv K (Suc n) vv"
closed_vimage_vec_nth: fixes S :: "'a set"   and i :: "'b" assumes "closed S" shows "closed ((\<lambda>x. x $ i) -` S)"
get_attribute_pure: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and element_ptr :: "(_) element_ptr"   and k :: "char list"   and h :: "(_) heap" assumes "l_get_attribute type_wf get_attribute get_attribute_locs" shows "pure (get_attribute element_ptr k) h"
d_def: shows "d = Cn 1 r_pdec1 [Cn 1 r_parallel [r_const (encode g), r_const (encode f), recf.Id 1 0]]"
tr_split: shows "take (length w1) tr @ drop (length w1) tr = tr"
invariant_1_sub_interval_1: fixes p :: "int poly"   and l :: "rat"   and r :: "rat"   and l' :: "rat"   and r' :: "rat" assumes "invariant_1 (p, l, r)"   and "root_cond (p, l', r') (real_of_1 (p, l, r))"   and "l \<le> l'"   and "r' \<le> r" shows "invariant_1 (p, l', r')"
is_isomorphic_to_concrete_category: shows "isomorphic_categories J' (concrete_category.COMP (Collect J'.ide) J'.hom (\<lambda>C B A. J'))"
H2: shows "H2 \<in> carrier_mat (dim_row H) (dim_col H)"
\<pi>_no_tokens: fixes k :: "nat"   and I :: "item set" shows "\<pi> k {} I = limit (\<lambda>I. Complete k (Predict k I)) I"
measure_lborel_cbox_eq: fixes l :: "'a"   and u :: "'a" shows "content (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
adverse_at_01_1: fixes z :: "nat"   and i :: "nat"   and j :: "nat" shows "local.adverse z i j 0 \<down>= i"
simps: fixes as :: "('c \<times> 'd atom) set"   and bi :: "((nat \<Rightarrow> 'd option) \<times> (nat \<Rightarrow> 'd option)) \<times> (nat \<Rightarrow> 'c) \<times> (nat \<Rightarrow> 'c)" shows "as \<Turnstile>\<^sub>i bi = (\<forall>I v. (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as \<longrightarrow> (I, v) \<Turnstile>\<^sub>i\<^sub>b bi)"
kmp1_def: fixes s :: "'b list"   and t :: "'b list" shows "kmp1 s t \<equiv> ASSERT (s \<noteq> []) \<bind> (\<lambda>_. let i = 0; j = 0; pos = None in compute_\<ff>s_SPEC (butlast s) \<bind> (\<lambda>\<ff>s. WHILE\<^sub>T\<^bsup>I_outer s t\<^esup> (\<lambda>(i, j, pos). i + length s \<le> length t \<and> pos = None) (\<lambda>(i, j, pos). ASSERT (i + length s \<le> length t) \<bind> (\<lambda>_. WHILE\<^sub>T\<^bsup>I_in_na s t i\<^esup> (\<lambda>(j, pos). t ! (i + j) = s ! j \<and> pos = None) (\<lambda>(j, pos). let j = j + 1 in if j = length s then RETURN (j, Some i) else RETURN (j, None)) (j, pos) \<bind> (\<lambda>(j, pos). if pos = None then ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. let i = i + (j - \<ff>s ! j + 1); j = max 0 (\<ff>s ! j - 1) in RETURN (i, j, None)) else RETURN (i, j, Some i)))) (i, j, pos) \<bind> (\<lambda>(uu_, uu_, pos). RETURN pos)))"
trg_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "hseq \<mu> \<nu>" shows "trg (\<mu> \<star> \<nu>) = trg \<mu>"
suc_less_plus_3: fixes n :: "nat"   and x :: "nat" shows "Suc n < max (n + 3) x"
sets_density: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" shows "sets (density M f) = sets M"
SentP_fresh_iff: fixes a :: "atom"   and x :: "tm" shows "a \<sharp> SentP x = a \<sharp> x"
zero_polynom_imp_zero_coeffs: fixes c :: "nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "\<And>w. (\<Sum>i\<le>n. c i * w ^ i) = (0::'a)"   and "k \<le> n" shows "c k = (0::'a)"
lt_def: shows "lt = {(a, b). b < a \<and> a \<le> 111}"
map_to_sorted_list_def: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and M :: "'f \<Rightarrow> 'g option" shows "map_to_sorted_list R M \<equiv> it_to_sorted_list (key_rel R) (map_to_set M)"
VSigmaD1: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" assumes "\<langle>a, b\<rangle> \<in> elts (VSigma A B)" shows "a \<in> elts A"
no_steps_final: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and v :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate"   and w :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate" assumes "\<Gamma> \<turnstile> v \<rightarrow>\<^sup>* w"   and "final v" shows "w = v"
compute_int_floor_fl: fixes m :: "int"   and e :: "int" shows "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div 2 ^ nat (- e))"
AsB: shows "\<Union> (As ` Field r) \<subseteq> B"
adjoint_funcs_match: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and f :: "char"   and r :: "trm"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real" assumes "SFuncs \<sigma> f = Aterm r" shows "Funcs (USubst.adjoint \<sigma> I \<omega>) f = (\<lambda>d. term_sem (repc I dotid d) r \<omega>)"
in_closure_iff_infdist_zero: fixes A :: "'b set"   and x :: "'b" assumes "A \<noteq> {}" shows "(x \<in> closure A) = (infdist x A = 0)"
DSourcesA1_L0: shows "DSources level0 sA1 = {}"
that: fixes h :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<Rightarrow> ('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "\<And>p q. h (p + q) = h p + h q"   and "\<And>p q. h (p * q) = h p * h q"   and "\<And>p. h (h p) = h p"   and "range h = P[X]" shows "thesis"
diff_add_eq_diff_diff_swap_ennreal: fixes x :: "ennreal"   and y :: "ennreal"   and z :: "ennreal" shows "x - (y + z) = x - y - z"
If2_def: fixes Q :: "tr"   and x :: "'a"   and y :: "'a" shows "If2 Q x y = If Q then x else y"
map_\<Gamma>_cong: fixes \<sigma>\<^sub>1 :: "'c trace"   and \<sigma>\<^sub>2 :: "'c trace"   and f\<^sub>1 :: "'c set \<Rightarrow> 'd set"   and f\<^sub>2 :: "'c set \<Rightarrow> 'd set" assumes "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"   and "\<And>x. f\<^sub>1 x = f\<^sub>2 x" shows "map_\<Gamma> f\<^sub>1 \<sigma>\<^sub>1 = map_\<Gamma> f\<^sub>2 \<sigma>\<^sub>2"
Greatest_equality: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq y x" shows "order.Greatest less_eq P = x"
impl_of: fixes x :: "('a, 'b) hashmap" shows "impl_of x \<in> {hm. ArrayHashMap_Impl.ahm_invar hm}"
size_filter_unsat_elem: fixes x :: "'b"   and M :: "'b multiset"   and P :: "'b \<Rightarrow> bool" assumes "x \<in># M"   and "\<not> P x" shows "size (filter_mset P M) < size M"
basis_max_indep: fixes X :: "'a set"   and x :: "'a" assumes "\<E>.basis X"   and "x \<in> carrier - X" shows "\<not> indep_in carrier (insert x X)"
Cons_1: fixes i :: "nat"   and k :: "nat" assumes "i \<le> k" shows "prefixToLevel_aux M' l i = prefixToLevel_aux M' (l + (k - i)) k"
1: shows "[\<^bold>\<not>\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<^bold>\<not>\<phi> \<alpha>) in v]"
inf_not_of_complex: fixes x :: "complex" shows "\<infinity>\<^sub>h \<noteq> of_complex x"
square_diff_one_factored: 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 x :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "minus (times x x) one = times (plus x one) (minus x one)"
cut_bound_max': fixes n :: "nat"   and F :: "'b formula" shows "max n (cut_bound' n F) = cut_bound' n F"
card_Union_disjoint: fixes C :: "'a set set" assumes "disjoint C"   and "\<And>A. A \<in> C \<Longrightarrow> finite A" shows "card (\<Union> C) = sum card C"
supp_union_fset: fixes S :: "'a fset"   and T :: "'a fset" shows "supp (S |\<union>| T) = supp S \<union> supp T"
simps: fixes a :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a" shows "(a \<in> greachable A w V p) = (a = p \<or> (\<exists>q aa. a = aa \<and> q \<in> greachable A w V p \<and> (case q of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> aa \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V)))"
poly_subst_id: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and f :: "'b \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>x. x \<in> indets p \<Longrightarrow> f x = monomial (1::'c) (monomial 1 x)" shows "poly_subst f p = p"
is_red_singleton_trans: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and g :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red {f} p"   and "lt g adds\<^sub>t lt f"   and "g \<noteq> 0" shows "is_red {g} p"
Nadd_normNum1: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "normNum x +\<^sub>N y = x +\<^sub>N y"
induct: fixes P :: "'c match_expr list \<Rightarrow> bool"   and a0 :: "'c match_expr list" assumes "P []"   and "\<And>m ms. P ms \<Longrightarrow> P (m # ms)" shows "P a0"
bet_col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet A B C"
z: shows "tmf_ivl_cs prec ord I xs f = Some z"
poly_lift_hom_X_minus_nat_pow_smult: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "poly_lift_hom R S \<phi> (b \<odot>\<^bsub>UP R\<^esub> X_poly_minus R a [^]\<^bsub>UP R\<^esub> n) = \<phi> b \<odot>\<^bsub>UP S\<^esub> X_poly_minus S (\<phi> a) [^]\<^bsub>UP S\<^esub> n"
ListSum_eq: fixes V :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>v. v \<in> set V \<Longrightarrow> f v = g v" shows "ListSum V f = ListSum V g"
assms_1: fixes r :: "'a" assumes "r \<in> deps oper" shows "r \<in> fst ` set xs \<and> r < oid"
relpowp_imp_rtranclp: fixes n :: "nat"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "(P ^^ n) x y" shows "P\<^sup>*\<^sup>* x y"
iter_mono_emeasure1: fixes n :: "nat"   and s :: "'a" shows "emeasure (measure_spmf (fuse.iter n s)) {s. \<not> guard s} \<le> emeasure (measure_spmf (fuse.iter (Suc n) s)) {s. \<not> guard s}"
irreducibleD2: fixes C :: "'a set"   and K :: "'a \<Rightarrow> 'a pmf"   and a :: "'a"   and b :: "'a" assumes "C \<in> UNIV // MC_syntax.communicating K"   and "a \<in> C"   and "(a, b) \<in> MC_syntax.communicating K" shows "b \<in> C"
map_update: fixes m :: "'m"   and a :: "'v"   and b :: "('v \<times> nat) list" assumes "M_invar m" shows "M_lookup (M_update a b m) = M_lookup m(a \<mapsto> b)"
atLeastLessThan_add_Un: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j" shows "{i..<j + k} = {i..<j} \<union> {j..<j + k}"
assms: fixes x :: "'t \<Rightarrow>\<^sub>0 'b" assumes "x \<in> set bs" shows "Inl (spp_of p, spp_of x) \<notin> app_pair spp_of ` set ps"
D_det: fixes P :: "'a process"   and Q :: "'a process" shows "D (P [+] Q) = D P \<union> D Q"
int_shiftr_lt_0: fixes i :: "int"   and n :: "nat" shows "(i >> n < 0) = (i < 0)"