hcju/isabelleps huggingface dataset:
cast_simps_7: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "Op (fst3 (castOp s X)) (snd3 (castOp s X)) (trd3 (castOp s X)) = X"
divides_imp_divides_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R - {\<zero>}"   and "a divides b" shows "a divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> b"
fail_env_def: fixes fail :: "'e" shows "fail_env fail = EnvT (\<lambda>r. fail)"
eval_vars_det\<^sub>B: fixes b :: "'BExp"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<forall>x\<in>bexp_vars b. mem\<^sub>1 x = mem\<^sub>2 x" shows "ev\<^sub>B mem\<^sub>1 b = ev\<^sub>B mem\<^sub>2 b"
append_eq_map_conv: fixes ys :: "'c list"   and zs :: "'c list"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "(ys @ zs = map f xs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
Worder_equiv: fixes E :: "('e, 'f) Order_scheme" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))"   and "\<forall>c\<in>carrier E. \<exists>d\<in>carrier D. ord_equiv (Iod E (segment E c)) (Iod D (segment D d))" shows "ord_equiv D E"
split_gallery_gf: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and D :: "'b set"   and Cs :: "'b set list" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> g \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "SimplicialComplex.maxsimpchain X (C # Cs @ [D])" shows "\<exists>As A B Bs. A \<in> g \<turnstile> ChamberComplex.chamber_system X \<and> B \<in> f \<turnstile> ChamberComplex.chamber_system X \<and> C # Cs @ [D] = As @ A # B # Bs"
wqo_on_Sigma: fixes P1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A1 :: "'b set"   and P2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A2 :: "'c set" assumes "wqo_on P1 A1"   and "wqo_on P2 A2" shows "wqo_on (prod_le P1 P2) (A1 \<times> A2)"
Un_absorb2: fixes B :: "'a set"   and A :: "'a set" assumes "B \<subseteq> A" shows "A \<union> B = A"
Rcons: fixes tr :: "('state, 'act, 'out) trans list"   and trn :: "('state, 'act, 'out) trans" assumes "valid2 tr"   and "tgtOf (last tr) = srcOf trn"   and "validTrans trn" shows "valid2 (tr ## trn)"
prj_simps_2: shows "C.seq \<nu>\<pi>.prj\<^sub>0 \<p>\<^sub>1[\<pi>.leg0 \<cdot> \<nu>\<pi>.prj\<^sub>0, \<rho>.leg1]"
geom_proc_filt_info: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and w :: "bool stream" assumes "f \<in> borel_measurable (G n)" shows "f w = f (pseudo_proj_True n w)"
map_of_None: fixes x :: "'c"   and ps :: "('c \<times> 'd) list" assumes "sorted (x # map fst ps)" shows "map_of ps x = None"
fv_formula_of_constraint: fixes t1 :: "Formula.trm"   and p :: "bool"   and c :: "mconstraint"   and t2 :: "Formula.trm" shows "fv (formula_of_constraint (t1, p, c, t2)) = fv_trm t1 \<union> fv_trm t2"
distinct_map_filter: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "distinct (map f xs)" shows "distinct (map f (filter P xs))"
get_child_nodes_def: shows "get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = (\<lambda>ptr. invoke CD.a_get_child_nodes_tups ptr ())"
linear_axioms: fixes A :: "(('a, 'b) vec, 'c) vec" shows "Vector_Spaces.linear (*s) (*s) ((*v) A)"
extensional_def: fixes A :: "'e set" shows "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
le_cInf_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and a :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "S \<noteq> {}"   and "preorder.bdd_below less_eq S" shows "less_eq a (Inf S) = (\<forall>x\<in>S. less_eq a x)"
generalise_output_directly_subsumes_original: fixes t' :: "transition"   and t :: "transition"   and r :: "nat"   and p :: "nat"   and v :: "value"   and s :: "nat"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and e2 :: "((nat \<times> nat) \<times> transition) fset" assumes "stored_reused t' t = Some (r, p)"   and "Outputs t ! p = L v"   and "\<forall>c1 c2 t. obtains s c1 e1 0 <> t \<and> obtains s' c2 e2 0 <> t \<longrightarrow> c2 $ r = Some v" shows "directly_subsumes e1 e2 s s' t' t"
pr1_in_hom': fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<guillemotleft>\<pp>\<^sub>1[a, b] : local.dom \<pp>\<^sub>0[a, b] \<rightarrow> a\<guillemotright>"
cut_less_Min_empty: fixes t :: "'a"   and I :: "'a set" assumes "t \<le> iMin I" shows "I \<down>< t = {}"
cblinfun_apply_code_def: shows "cblinfun_apply_code = (*\<^sub>V)"
nsgTr4: fixes N :: "'a set"   and a :: "'a"   and x :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "x \<in> N \<bullet> a" shows "\<rho> x \<in> N \<bullet> \<rho> a"
cod_eqI': fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "a \<in> VV.codomains f" shows "a = VV.cod f"
gauss_sum: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "comm_monoid_add.sum plus zero (semiring_1.of_nat one plus zero) {0..n} = divide (times (semiring_1.of_nat one plus zero n) (plus (semiring_1.of_nat one plus zero n) one)) (numeral.numeral one plus (num.Bit0 num.One))"
remove_vertex_Digraph: fixes v :: "'a" shows "Digraph (remove_vertex v)"
sup_loc_def: fixes P :: "'b prog"   and LT :: "ty err list"   and LT' :: "ty err list" shows "P \<turnstile> LT [\<le>\<^sub>\<top>] LT' \<equiv> Listn.le (sup_ty_opt P) LT LT'"
zero_zmultiset_def: shows "{#}\<^sub>z \<equiv> abs_zmultiset ({#}, {#})"
assms: fixes v :: "nat \<Rightarrow> 'a" shows "p \<lbrace> v \<rbrace> = (0::'a)"
exact_im_sub_ker: fixes L :: "('i, 'a) Module"   and M :: "('j, 'a) Module"   and N :: "('k, 'a) Module"   and f :: "'i \<Rightarrow> 'j"   and g :: "'j \<Rightarrow> 'k"   and z1 :: "'k \<Rightarrow> 'l"   and e :: "'l"   and Z :: "('m, 'a, 'n) Module_scheme"   and x :: "'j \<Rightarrow> 'm" assumes "R module L"   and "R module M"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N"   and "z1 \<in> mHom R N (Zm R e)"   and "R module Z"   and "exact4 R L f M g N z1 (Zm R e)"   and "x \<in> mHom R M Z"   and "compos L x f = mzeromap L Z" shows "(\<lambda>z\<in>carrier N. x (SOME y. y \<in> carrier M \<and> g y = z)) \<in> mHom R N Z"
fo_rg_inv: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "NPreflow c s t f" shows "Network.flow_of_cf c (residualGraph c f) = f"
nrows_diagonal_to_Smith: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "nrows (diagonal_to_Smith A bezout) = nrows A"
eq_0: fixes x :: "'p"   and y :: "'p" shows "(dist x y = 0) = (x = y)"
code: fixes x :: "quat" shows "- x = Quat (- quat.Re x) (- Im1 x) (- Im2 x) (- Im3 x)"
vbrelation_axioms: shows "vbrelation (F\<lparr>ArrVal\<rparr>)"
pigeonhole: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "card (f ` A) < card A" shows "\<not> inj_on f A"
N_uminus: fixes q :: "real"   and n :: "int" assumes "fbnorm q" shows "fbnorm.N q (- n) = fbnorm.N q n"
inca_to_ubx_simulation_axioms: shows "inca_to_ubx_simulation Finca_empty Finca_get Finca_add Finca_to_list Fubx_empty Fubx_get Fubx_add Fubx_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>"
elims_1: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and x :: "('b \<times> 'c) list \<times> 'a"   and xa :: "('b \<times> 'c) list \<times> 'a"   and y :: "bool" assumes "oalist_raw.oalist_eq_raw rep_key_order x xa = y" obtains xs :: "('b \<times> 'c) list" and ox :: "'a" and ys :: "('b \<times> 'c) list \<times> 'a" where "x = (xs, ox)" and "xa = ys" and "y = (xs = oalist_raw.sort_oalist_aux rep_key_order ox ys)"
iteration_zero: shows "(0::'a)\<^sup>\<infinity> = (1::'a)"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and G' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" assumes "group_homomorphism \<eta> G composition unit G' composition' unit'" shows "group_homomorphism_fundamental \<eta> G composition unit G' composition' unit'"
initializes: shows "\<exists>r. initial = \<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)) \<and> program_expr e"
esssup_zero_space: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ereal" assumes "emeasure M (space M) = 0"   and "f \<in> borel_measurable M" shows "esssup M f = - \<infinity>"
run_asset_ii_ex: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "pred_asset_ii n (run_owner_ii n s) (run_asset_ii n s)"
12: shows "[a] @ l =\<^sub>F l @ [a]"
\<alpha>_ide_simp: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>\<^sub>SC.ide a"   and "\<F>\<^sub>SC.ide b"   and "\<F>\<^sub>SC.ide c" shows "\<F>\<^sub>SC.assoc a b c = \<a>\<^sub>S[a, b, c]"
Gramian_determinant_div: fixes n :: "nat"   and fs :: "'a vec list"   and l :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs"   and "l < length fs" shows "gram_schmidt.Gramian_determinant n fs (Suc l) / gram_schmidt.Gramian_determinant n fs l = \<parallel>gram_schmidt_fs.gso n fs l\<parallel>\<^sup>2"
disjnt_subset2: fixes X :: "'d set"   and Y :: "'d set"   and Z :: "'d set" assumes "disjnt X Y"   and "Z \<subseteq> Y" shows "disjnt X Z"
empty_is_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "({} = f ` A) = (A = {})"
poly_subst_poly_subst: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and g :: "'e \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f (poly_subst g p) = poly_subst (\<lambda>x. poly_subst f (g x)) p"
M2: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a"   and r :: "'a"   and s :: "'a" assumes "class.interval meets \<I>"   and "meets p q"   and "meets r s" shows "meets p s \<oplus> ((\<exists>t. meets p t \<and> meets t s) \<oplus> (\<exists>t. meets r t \<and> meets t q))"
K_I_K_subseteq_K: fixes X :: "'a set" shows "closure (interior (closure X)) \<subseteq> closure X"
rationalizable_binary_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f"   and "\<And>B x y. \<lbrakk>B \<subseteq> A; y \<in> f B; x \<in> B; y \<in> B\<rbrakk> \<Longrightarrow> y \<in> f {x, y}"   and "\<And>B y. \<lbrakk>B \<subseteq> A; y \<in> B; \<forall>x\<in>B. y \<in> f {x, y}\<rbrakk> \<Longrightarrow> y \<in> f B" shows "rationalizable_binary_on A f"
WHILE_rule: fixes I :: "'c \<Rightarrow> bool"   and s :: "'c"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> f s \<le> SPEC I"   and "\<And>s. \<lbrakk>I s; \<not> b s\<rbrakk> \<Longrightarrow> \<Phi> s" shows "WHILE b f s \<le> SPEC \<Phi>"
smult_l_distr: fixes a :: "'a"   and b :: "'a"   and x :: "'e" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M2" shows "(a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M2\<^esub> x = a \<odot>\<^bsub>M2\<^esub> x \<oplus>\<^bsub>M2\<^esub> b \<odot>\<^bsub>M2\<^esub> x"
n_val: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "n_val K v x * Lv K v = v x"
induct: fixes x :: "color list"   and P :: "color list \<Rightarrow> bool" assumes "valid x"   and "P []"   and "\<And>xs. \<lbrakk>valid xs; P xs\<rbrakk> \<Longrightarrow> P (B # xs)"   and "\<And>xs n. \<lbrakk>valid xs; P xs; 3 \<le> n\<rbrakk> \<Longrightarrow> P (replicate n R @ xs)" shows "P x"
map_of_from_function_graph_is_some_if: fixes xs :: "'b list"   and x :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "set xs \<noteq> {}"   and "x \<in> set xs" shows "map_of (map (\<lambda>x. (x, f x)) xs) x = Some (f x)"
eval_all_or_nothing: fixes \<phi> :: "('a, 'p) Kform"   and ec :: "'s odlist" assumes "subjective agent \<phi>" shows "toSet (eval envVal ec \<phi>) = {} \<or> toSet (eval envVal ec \<phi>) = toSet ec"
inconsist_for_total: fixes i :: "nat"   and e :: "nat" assumes "total1 (\<phi> i)" shows "inconsist i e \<down>= (if \<exists>x<e_length e. \<phi> i x \<down>\<noteq> e_nth e x then LEAST x. x < e_length e \<and> \<phi> i x \<down>\<noteq> e_nth e x else e_length e)"
one_not_zero: shows "\<chi> 1 \<noteq> 0"
Mp_coeff: fixes f :: "int poly"   and i :: "nat" shows "coeff (q.Mp f) i = q.M (coeff f i)"
has_domain_iff_has_codomain: fixes f :: "'a \<times> 'a" shows "(VV.domains f \<noteq> {}) = (VV.codomains f \<noteq> {})"
int_pow_neg_int: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier \<G>" shows "x [^] - int n = inv (x [^] n)"
ewd: shows "Sig iS\<^bsub>CI T\<^esub> \<triangleright> Vx : A \<turnstile> e : B"
is_confluent_on_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "relation.is_confluent_on r A = (\<forall>a\<in>A. \<forall>b1 b2. r\<^sup>*\<^sup>* a b1 \<and> r\<^sup>*\<^sup>* a b2 \<longrightarrow> relation.cs r b1 b2)"
less_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "less_fun f g = (\<exists>k. f k < g k \<and> (\<forall>k'<k. f k' = g k'))"
homotopic_paths: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" shows "homotopic_paths s p q = (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and> h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and> (\<forall>x\<in>{0..1}. h (0, x) = p x) \<and> (\<forall>x\<in>{0..1}. h (1, x) = q x) \<and> (\<forall>t\<in>{0..1}. pathstart (h \<circ> Pair t) = pathstart p \<and> pathfinish (h \<circ> Pair t) = pathfinish p))"
left_null_space_def: fixes A :: "(('a, 'c) vec, 'b) vec" shows "left_null_space A = {x. x v* A = 0}"
simps_2: fixes t :: "varType"   and sz :: "nat"   and ar :: "integer iarray" shows "variable_inv (VArray t sz ar) = (varType_inv t \<and> sz \<le> max_array_size \<and> IArray.length ar = sz \<and> set (IArray.list_of ar) \<subseteq> {min_var_value..max_var_value})"
stutter_equiv_sym: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<tau> :: "nat \<Rightarrow> 'a" assumes "\<sigma> \<approx> \<tau>" shows "\<tau> \<approx> \<sigma>"
subdiv_ends_G_rev_2: shows "head G (rev_G uv) = u"
conditional_entropy_eq: fixes Y :: "'a \<Rightarrow> 'd"   and Py :: "'d \<Rightarrow> real"   and X :: "'a \<Rightarrow> 'e"   and Pxy :: "'e \<times> 'd \<Rightarrow> real" assumes "simple_distributed M Y Py"   and "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy" shows "\<H>(X | Y) = - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space M. Pxy (x, y) * log b (Pxy (x, y) / Py y))"
IH: fixes i :: "nat"   and xs :: "nat list" assumes "len (fw m n k) i i xs < (0::'a)"   and "i \<le> n"   and "k \<le> n"   and "set xs \<subseteq> {0..n}" shows "\<exists>i\<le>n. \<exists>ys. set ys \<subseteq> set xs \<union> {0..k} \<and> len m i i ys < (0::'a)"
wf_converse: shows "wfP (\<rightarrow>)\<inverse>\<inverse> = wf ({(x, y). x \<rightarrow> y}\<inverse>)"
SNF_first_divides_all: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "Smith_normal_form_mat A"   and "A \<in> carrier_mat m n"   and "i < m"   and "j < n" shows "A $$ (0, 0) dvd A $$ (i, j)"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
derf: fixes x :: "'a" assumes "x \<in> S - K" shows "(f has_derivative (\<lambda>h. 0::'b)) (at x within S)"
vars_head_subseteq: fixes s :: "('b, 'a) tm" shows "vars_hd (head s) \<subseteq> vars s"
add_less_same_cancel1: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and b :: "'g"   and a :: "'g" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus b a) b = less a zero"
assms: shows "run_one_step d i (s, vs, ves, $Current_memory) = (s', vs', res)"
iso_lunit: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "local.iso \<l>[f]"
simps: fixes a1 :: "'item list"   and a2 :: "'item list" shows "(a1 =\<^sub>S a2) = (\<exists>a b u v. a1 = u @ [a] @ [b] @ v \<and> a2 = u @ [b] @ [a] @ v \<and> ind a b)"
vec_to_poly_dim_less: fixes x :: "rat vec" shows "dim_poly (vec_to_lpoly x) \<le> dim_vec x"
lemma_2_5_15: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(b l\<rightarrow> c) * (a l\<rightarrow> b) \<le> a l\<rightarrow> c"
assms: shows "FI_RESULT [(A, A)] (B * C * D) emp F"
assms_3: shows "rightPrincipal r (\<box> A) R"
SourcesEmptyDSources: fixes i :: "AbstrLevelsID"   and C :: "CSet" assumes "Sources i C = {}" shows "DSources i C = {}"
abcdN: shows "a\<^sup>2 + N * b\<^sup>2 = c\<^sup>2 + N * d\<^sup>2"
vproductE: fixes f :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct I A" obtains "vsv f" and "\<D>\<^sub>\<circ> f = I" and "\<forall>i\<in>elts I. f\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i"
At: shows "A\<^sup>T \<in> carrier_mat n m"
nat_pow_consistent: fixes x :: "'a"   and n :: "nat"   and H :: "'a set" shows "x [^]\<^bsub>G\<lparr>carrier := H'\<rparr>\<^esub> n = x [^]\<^bsub>G\<lparr>carrier := H', carrier := H\<rparr>\<^esub> n"
measure_negligible_finite_Union_image: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd set" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> lmeasurable"   and "pairwise (\<lambda>x y. negligible (f x \<inter> f y)) S" shows "Sigma_Algebra.measure lebesgue (\<Union> (f ` S)) = (\<Sum>x\<in>S. Sigma_Algebra.measure lebesgue (f x))"
assms_1: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
comm_monoid_list_axioms: shows "comm_monoid_list (+) (0::'a)"
nat_pow_0: fixes x :: "int" shows "x ^ 0 = 1"
Invariants_measurable_func: fixes f :: "'a \<Rightarrow> 'b"   and N :: "'b measure" assumes "f \<in> Invariants \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
bit_shiftl_word_iff: fixes w :: "'b word"   and m :: "nat"   and n :: "nat" shows "bit (w << m) n = (m \<le> n \<and> n < LENGTH('b) \<and> bit w (n - m))"
Tensor_in_tensor: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set"   and t :: "'b local.term"   and u :: "'b local.term" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "partial_magma.arr (free_monoidal_category.comp C) g"   and "t \<in> f"   and "u \<in> g" shows "t \<^bold>\<otimes> u \<in> free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g"
isSucc_pred_diff: fixes i :: "'a" assumes "isSucc i" shows "pred i \<noteq> i"
measure_nonneg: fixes M :: "'b measure"   and A :: "'b set" shows "0 \<le> Sigma_Algebra.measure M A"
gen_iapath_sd_path: fixes e :: "'a \<times> 'a"   and w :: "'a"   and V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "e \<in> parcs G"   and "w \<notin> pverts G"   and "V \<subseteq> pverts G"   and "gen_iapath V u p v" shows "pre_digraph.gen_iapath (with_proj (subdivide G e w)) V u (sd_path e w p) v"
simps_1: shows "domain Init = {xpkE, xskE, xsk}"
nodes_le_order: shows "order nodes_le ((rev \<circ> sorted_list_of_set) ` Pow (set nodes))"
pure_def: fixes R :: "('b \<times> 'a) set" shows "pure R \<equiv> \<lambda>a c. \<up> ((c, a) \<in> R)"
RelStay: fixes P :: "pi"   and Q :: "pi"   and a :: "name" assumes "(P, Q) \<in> Rel" shows "([a\<frown>a]P, Q) \<in> Rel"
distinct_3: shows "SizeOp \<noteq> ToLowerCaseOp"
base_x_7: shows "\<Union> (A ` R) = UNIV"
integral_snd: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) (\<lambda>(x, y). f x y)" shows "LBINT y. LBINT x. f x y = LINT (x, y)|(lborel \<Otimes>\<^sub>M lborel). f x y"
src_hcomp: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "B.hseq \<mu> \<nu>" shows "src\<^sub>B (\<mu> \<star>\<^sub>B \<nu>) = src\<^sub>B \<nu>"
one_side_not_col123: 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 X :: "'a"   and Y :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OS Bet A B X Y" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B X"
list_greater_non_memb: fixes xs :: "'a list"   and oid :: "'a" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x < oid"   and "oid \<in> set xs" shows "False"
frontier_interiors: fixes s :: "'c set" shows "frontier s = - interior s - interior (- s)"
Span_contains_spanset_append_left: fixes ms :: "'v list"   and ns :: "'v list" assumes "set ms \<subseteq> V"   and "set ns \<subseteq> V" shows "set ms \<subseteq> RSpan (ms @ ns)"
ID3: fixes f :: "'a \<rightarrow> 'b" shows "ID oo f = f"
simps_3: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n\<^sub>s :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<not> Option.is_none (Blockchain.devBC active cmp bc mining t n nid n\<^sub>s)" shows "Blockchain.devExt active cmp bc mining t n nid n\<^sub>s 0 = bc (cmp (the (Blockchain.devBC active cmp bc mining t n nid n\<^sub>s)) (t n\<^sub>s))"
smallest_prime_between_None: fixes m :: "nat"   and n :: "nat" shows "(smallest_prime_between m n = None) = (\<forall>q. m \<le> q \<and> q \<le> n \<longrightarrow> \<not> prime q)"
monoid_mult_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "monoid_mult_hom hom \<equiv> one_hom hom \<and> semigroup_mult_hom hom"
strict_implies_order: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subset># b" shows "a \<subseteq># b"
Lambert_W_times_ln_self: fixes x :: "real" assumes "exp (- 1) \<le> x" shows "Lambert_W (x * ln x) = ln x"
waiting_def: fixes w :: "'b wait_set_status option" shows "waiting w = (\<exists>w'. w = \<lfloor>InWS w'\<rfloor>)"
i: shows "9 * 2 ^ (2 * 7 - 1) - 1 = 73727"
zero_not_ind_mod: shows "\<not> \<Phi>.principal.induced_modulus 0"
free: shows "(x, \<tau>) \<notin> fv (mk_eq f g) \<union> FV \<Gamma>"
xor_comm: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<oplus>\<^bsub>L\<^esub> y = y \<oplus>\<^bsub>L\<^esub> x"