hcju/isabelleps huggingface dataset:
degree_add_less: fixes p :: "'a Polynomial.poly"   and n :: "nat"   and q :: "'a Polynomial.poly" assumes "Polynomial.degree p < n"   and "Polynomial.degree q < n" shows "Polynomial.degree (p + q) < n"
perm_struct_induct2: fixes p :: "perm"   and S :: "atom set"   and P :: "perm \<Rightarrow> bool" assumes "supp p \<subseteq> S"   and "P 0"   and "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b; a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"   and "\<And>p1 p2. \<lbrakk>P p1; P p2; supp p1 \<subseteq> S; supp p2 \<subseteq> S\<rbrakk> \<Longrightarrow> P (p1 + p2)" shows "P p"
assms_3: shows "i \<le> v \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>"
R2_def: shows "R2 = {(V, V'). (V, V') \<in> R2' \<and> set (PPV V) \<subseteq> set (PPc c2) \<and> set (PPV V') \<subseteq> set (PPc c2)}"
DirProds_iso: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and J :: "'d set"   and Gs :: "'c \<Rightarrow> ('e, 'f) monoid_scheme"   and Hs :: "'d \<Rightarrow> ('g, 'h) monoid_scheme" assumes "bij_betw f I J"   and "\<And>i. i \<in> I \<Longrightarrow> Gs i \<cong> Hs (f i)"   and "\<And>i. i \<in> I \<Longrightarrow> Group.group (Gs i)"   and "\<And>j. j \<in> J \<Longrightarrow> Group.group (Hs j)" shows "DirProds Gs I \<cong> DirProds Hs J"
exhaust: fixes y :: "pexpr2" obtains x11 :: "pexpr" and x12 :: "pexpr" where "y = PMul x11 x12"   | x21 :: "pexpr" and x22 :: "nat" where "y = PPow x21 x22"
field_long_division_theorem_shell: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list"   and b :: "'c list" assumes "domain R"   and "subfield K R"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "b \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "b \<noteq> \<zero>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub>" shows "\<exists>q r. q \<in> carrier (K [X]\<^bsub>R\<^esub>) \<and> r \<in> carrier (K [X]\<^bsub>R\<^esub>) \<and> p = b \<otimes>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> q \<oplus>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> r \<and> (r = \<zero>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> \<or> degree r < degree b)"
foldri_def: fixes l :: "'f list" shows "foldri l = foldli (rev l)"
Accessible_left_lang_eq: fixes M :: "'b dfa"   and q :: "hf" assumes "dfa M" shows "dfa.left_lang (dfa.Accessible_dfa M) q = dfa.left_lang M q"
COMP: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and p :: "L"   and K :: "Var \<Rightarrow> L"   and G :: "Var \<Rightarrow> L"   and q :: "L"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c2 :: "IMP"   and H :: "Var \<Rightarrow> L"   and \<Psi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "X \<rhd> c1 : Sec (Q p K) (EQ G q) (EQ K q) \<Phi>"   and "X \<rhd> c2 : Sec (Q p H) (EQ K q) (EQ H q) \<Psi>"   and "\<forall>x. LEQ (G x) (K x) \<or> LEQ p (K x)"   and "\<forall>x. LEQ (K x) (H x) \<or> LEQ p (H x)" shows "X \<rhd> Comp c1 c2 : Sec (Q p H) (EQ G q) (EQ H q) (\<lambda>(x, y). \<exists>z. \<Phi> (z, y) \<and> (\<forall>w. EQ K q z w \<longrightarrow> \<Psi> (x, w)))"
IH_4: fixes x4 :: "int"   and x :: "nat"   and pc' :: "nat" assumes "pc < array_length ins"   and "array_get ins pc = GotoI x4"   and "x = nat (int (pc + 1) + x4)"   and "pc < x"   and "x < pc_bound ins"   and "pc' \<in> set (exec' ins s x)" shows "pc' < pc_bound ins"
simps_2: fixes d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and i :: "nat"   and x :: "'b"   and xs :: "'b list" shows "trace d i (x # xs) = d x i # trace d (d x i) xs"
subst_rename': fixes y :: "name"   and t :: "trm"   and x :: "name"   and v :: "trm" assumes "y \<sharp> t" shows "t[x::=v] = ([(y, x)] \<bullet> t)[y::=v]"
map_simp: fixes f :: "'a" assumes "hom.arr f" shows "hom.map f = f"
polytope_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and p :: "'b set" assumes "linear f"   and "polytope p" shows "polytope (f ` p)"
is_sig_red_top_tail_cases: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "('c \<Rightarrow>\<^sub>0 'e) set"   and p :: "'c \<Rightarrow>\<^sub>0 'e" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg ord F p = (qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg (=) F p \<or> qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg ord_strict F p)"
sym: fixes x :: "'a"   and y :: "'a" assumes "x \<sim> y"   and "x \<in> A"   and "y \<in> A" shows "y \<sim> x"
all_def: fixes P :: "'b \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.all P \<equiv> \<lambda>t n. \<forall>y. P y t n"
eval_tm_upper_bound: fixes \<AA> :: "interp"   and "is" :: "int list" shows "eval_tm \<AA> is \<le> (2 ^ Length \<AA> - 1) * sum_list (concat (map (\<lambda>i. if 0 < i then [i] else []) is))"
primitive_part_prim: fixes p :: "'b poly" assumes "content p = (1::'b)" shows "primitive_part p = p"
countable_ordLeq: fixes A :: "'a set"   and B :: "'b set" assumes "|A| \<le>o |B|"   and "countable B" shows "countable A"
lub_eqI: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a set"   and l :: "'a" assumes "class.po below"   and "po.is_lub below M l" shows "po.lub below M = l"
convergent_ereal: fixes X :: "nat \<Rightarrow> 'c" shows "convergent X = (limsup X = liminf X)"
S'_alt: shows "S' = (SIGMA p:{p. prime p \<and> real p \<le> x}. {i. 1 < i \<and> real (p ^ i) \<le> x})"
in_combination_in_component_1: fixes r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option" assumes "r \<in> RID\<^sub>S (\<sigma>;;\<tau>)"   and "r \<notin> RID\<^sub>S \<sigma>" shows "r \<in> RID\<^sub>S \<tau>"
X_5: fixes i :: "'b" assumes "i \<in> J" shows "X i \<in> sets (M i)"
lexordp_simps_2: fixes xs :: "'a list" shows "(xs <lex \<epsilon>) = False"
finite_measure_subadditive_finite: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "finite I"   and "A ` I \<subseteq> PS.events" shows "PS.prob (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. PS.prob (A i))"
swapEnvIm_preserves_wls: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "wlsEnv rho" shows "wlsEnv (swapEnvIm xs x y rho)"
vec_last_def: fixes v :: "'b vec"   and n :: "nat" shows "vec_last v n \<equiv> vec n (\<lambda>i. v $ (dim_vec v - n + i))"
has_bochner_integral_odd_function: fixes f :: "real \<Rightarrow> 'b"   and x :: "'b" assumes "has_bochner_integral lborel (\<lambda>x. indicat_real {0..} x *\<^sub>R f x) x"   and "\<And>x. f (- x) = - f x" shows "has_bochner_integral lborel f (0::'b)"
context_equiv_tyenv_sec: fixes \<Gamma>\<^sub>2 :: "'Var \<Rightarrow> 'BExp set option"   and P\<^sub>2 :: "'BExp set"   and \<Gamma>\<^sub>1 :: "'Var \<Rightarrow> 'BExp set option"   and mem :: "'Var \<Rightarrow> 'Val"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "\<Gamma>\<^sub>2 =:\<^sub>P\<^sub>2 \<Gamma>\<^sub>1"   and "pred P\<^sub>2 mem"   and "\<forall>x\<in>dom \<Gamma>\<^sub>2. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma>\<^sub>2 x)) mem \<le> dma mem x" shows "\<forall>x\<in>dom \<Gamma>\<^sub>1. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma>\<^sub>1 x)) mem \<le> dma mem x"
l_set_val_get_child_nodes_axioms_def: fixes set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_child_nodes_axioms set_val_locs get_child_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
map\<^sub>1_simps_3: fixes f :: "'a" assumes "C.ide f" shows "trg\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<cdot>\<^sub>D \<r>\<^sub>D[G f]) = G.map\<^sub>0 (trg\<^sub>C f)"
in_H_fcs_in_G_fcs: fixes a :: "'b"   and x :: "'b" assumes "a \<in> arcs H"   and "x \<in> H.face_cycle_set a" shows "x \<in> proj_arcs_H -` GM.face_cycle_set (proj_arcs_H a)"
assms: fixes x :: "'a"   and e :: "real" assumes "x \<in> B"   and "0 < e" shows "\<exists>y\<in>B. dist y x \<le> e \<and> y \<succeq>[P] x \<and> (x, y) \<notin> P"
mop_set_pick_alt: fixes x :: "'a set" shows "mop_set_pick x \<equiv> ASSERT (pre_set_pick x) \<bind> (\<lambda>_. IICF_Set.op_set_pick x)"
eq_funcs: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f \<in> A \<rightarrow> B"   and "g \<in> A \<rightarrow> B"   and "f = g"   and "x \<in> A" shows "f x = g x"
l7_21: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and P :: "'p" assumes "\<not> Col A B C"   and "B \<noteq> D"   and "Cong A B C D"   and "Cong B C D A"   and "Col A P C"   and "Col B P D" shows "P Midpoint A C \<and> P Midpoint B D"
simps_2: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and j :: "nat" shows "sig_crit' bs (Inr j) = is_syz_sig dgrad (term_of_pair (0::'a, j))"
f_1: fixes G :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "G \<in> (\<lambda>g. g \<union> trap) ` Gs" shows "flowsto (f G) T CX G"
eqvt_bound: fixes p :: "perm"   and x :: "'a" shows "p \<bullet> unpermute p x \<equiv> x"
int_pow_eq: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and m :: "int"   and n :: "int" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "(x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> m = x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> n) = (int (ord x) dvd n - m)"
assms: shows "ops' \<in> set (List_Supplement.embed \<pi>)"
intro: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "ergodic_qmpt M T"   and "fmpt M T" shows "ergodic_fmpt M T"
intros: fixes F1 :: "'fenv_std"   and F2 :: "'fenv_inca"   and H :: "'c"   and st :: "'d list" assumes "rel_fundefs (Fstd_get F1) (Finca_get F2)" shows "State F1 H st \<sim> State F2 H st"
one_idem': shows "(1::'a) + (1::'a) = (1::'a)"
Let_1: fixes x :: "Vars.var"   and ea :: "exp"   and aa :: "Arity"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" assumes "(x, ea) \<in> set \<Gamma>"   and "eq\<rho> (\<A>\<^bsub>aa\<^esub> ea) \<rho>1 \<rho>2" shows "eq aa (\<lbrakk> ea \<rbrakk>\<^bsub>\<rho>1\<^esub>) (\<lbrakk> ea \<rbrakk>\<^bsub>\<rho>2\<^esub>)"
set2list_prop: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "\<exists>xs. while (\<lambda>(Es, S). S \<noteq> {}) (\<lambda>(Es, S). let x = pick S in (x # Es, S - {x})) ([], A) = (xs, {}) \<and> set xs \<union> {} = A"
phiDefNodes_auxE: fixes n :: "'node"   and g :: "'g"   and v :: "'var"   and un :: "'node list"   and m :: "'node" assumes "n \<in> phiDefNodes_aux g v un m"   and "m \<in> set (\<alpha>n g)" obtains ns :: "'node list" where "g \<turnstile> n-ns\<rightarrow>m" and "\<forall>n\<in>set ns. v \<notin> defs g n" and "length (predecessors g n) \<noteq> 1" and "unvisitedPath un ns"
orthogonal_transformation_exists: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" assumes "norm a = norm b" obtains f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec" where "orthogonal_transformation f" and "f a = b"
less_natE: fixes m :: "nat"   and n :: "nat" assumes "m < n" obtains q :: "nat" where "n = Suc (m + q)"
expand: fixes inference :: "'a inference"   and inference' :: "'a inference" assumes "prems_of inference = prems_of inference' \<and> concl_of inference = concl_of inference'" shows "inference = inference'"
card_of_Un2: fixes A :: "'d set"   and B :: "'d set" shows "|A| \<le>o |B \<union> A|"
skelAbs_def: fixes A :: "('e, 'f, 'g, 'h, 'i) qAbs set" shows "skelAbs A \<equiv> qSkelAbs (pick A)"
td_gchain_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "td_gchain G n g = (if n = 0 then g 0 = carrier G \<and> g 0 = {\<one>\<^bsub>G\<^esub>} else d_gchain G n g \<and> g 0 = carrier G \<and> g n = {\<one>\<^bsub>G\<^esub>})"
is_finite_uminus: fixes a :: "('a, 'b) IEEE.float" shows "is_finite (- a) = is_finite a"
d_ad_comp: fixes x :: "'a" shows "! (d x) \<sqinter> x = bot"
subgraphs_of_union_2: fixes G :: "('c, 'd) pre_digraph"   and G' :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "wf_digraph G'"   and "compatible G G'" shows "subgraph G' (Digraph_Component.union G G')"
abs_convergent_prod_altdef: fixes f :: "nat \<Rightarrow> 'e" shows "abs_convergent_prod f = convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - (1::'e)))"
count_mset_set_2: fixes A :: "'d set"   and x :: "'d" assumes "infinite A" shows "count (mset_set A) x = 0"
h: shows "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)"
char_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "b \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "principal_char (G\<lparr>carrier := adjoin G H a\<rparr>) (a \<otimes>\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> b) = principal_char (G\<lparr>carrier := adjoin G H a\<rparr>) a * principal_char (G\<lparr>carrier := adjoin G H a\<rparr>) b"
finite_is_iface: fixes G :: "prog" shows "finite {I. is_iface G I}"
int_pow_one: fixes G :: "('c, 'd) monoid_scheme"   and z :: "int" assumes "Group.group G" shows "\<one>\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> z = \<one>\<^bsub>G\<^esub>"
downset_set_prop_var: fixes X :: "'a set" shows "ord.downset_set (\<lambda>x y. y \<le> x) X = \<Union> (ord.downset (\<lambda>x y. y \<le> x) ` X)"
continuous_map_paired: fixes Z :: "'d topology"   and X :: "'e topology"   and Y :: "'f topology"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "continuous_map Z (prod_topology X Y) (\<lambda>x. (f x, g x)) = (continuous_map Z X f \<and> continuous_map Z Y g)"
p_eq: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "p \<noteq> []" shows "list2policy (map Cp (rev p)) = Cp (list2FWpolicy p)"
that: fixes point :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip xs ys) (Some subs) = Some point"   and "raw_match' x y point = Some subs'" shows "thesis"
Grp_apply: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" shows "Grp A f x y = (y = f x \<and> x \<in> A)"
IH: fixes v1 :: "'a" assumes "subpath g v1 es v2"   and "e \<in> set es" shows "\<exists>es'. es = es' @ [e] \<and> e \<notin> set es'"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "R.arr f"   and "R.cod f = b" shows "R.comp b f = f"
H_def: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "H p x q = \<lparr>t p\<rparr>x\<lparr>x \<cdot> t q\<rparr>"
subgroup_generated_def: fixes G :: "('c, 'd) monoid_scheme"   and S :: "'c set" shows "subgroup_generated G S = G\<lparr>carrier := generate G (carrier G \<inter> S)\<rparr>"
stutter_extend_en_def: fixes en :: "'a \<Rightarrow> 'b set" shows "stutter_extend_en en \<equiv> \<lambda>s. let as = en s in if as = {} then {None} else Some ` as"
trace_simulates_def: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset" shows "trace_simulates e1 e2 = (\<exists>f. \<forall>t. trace_simulation f e1 0 <> e2 0 <> t)"
wt: shows "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>t\<Colon>T"
that: fixes \<gamma> :: "'m \<Rightarrow> 'm set" assumes "gauge \<gamma>"   and "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; \<gamma> fine p\<rbrakk> \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e" shows "thesis"
\<psi>_inC: fixes f :: "'c" assumes "C.arr f" shows "HomC.\<psi> (C.dom f, C.cod f) (inC f) = f"
xor_left_inverse: 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> x \<oplus>\<^bsub>L\<^esub> y = y"
less_trans: fixes y :: "('d, 'e) Term.term"   and x :: "('d, 'e) Term.term"   and z :: "('d, 'e) Term.term" assumes "y \<rhd> x"   and "z \<rhd> y" shows "z \<rhd> x"
region_cover: fixes X :: "'d set"   and u :: "'d \<Rightarrow> real"   and k :: "'d \<Rightarrow> nat" assumes "\<forall>x\<in>X. 0 \<le> u x" shows "\<exists>R. R \<in> {region X I r |I r. valid_region X k I r} \<and> u \<in> R"
sets_pair_eq: fixes Ea :: "'d set set"   and A :: "'d measure"   and Ca :: "'d set set"   and Eb :: "'e set set"   and B :: "'e measure"   and Cb :: "'e set set" assumes "Ea \<subseteq> Pow (space A)"   and "sets A = sigma_sets (space A) Ea"   and "countable Ca"   and "Ca \<subseteq> Ea"   and "\<Union> Ca = space A"   and "Eb \<subseteq> Pow (space B)"   and "sets B = sigma_sets (space B) Eb"   and "countable Cb"   and "Cb \<subseteq> Eb"   and "\<Union> Cb = space B" shows "sets (A \<Otimes>\<^sub>M B) = sets (sigma (space A \<times> space B) {a \<times> b |a b. a \<in> Ea \<and> b \<in> Eb})"
total_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> x \<succeq>[r] y \<or> y \<succeq>[r] x)"
simps: fixes s' :: "'i"   and s :: "'i"   and t' :: "'i"   and t :: "'i"   and r' :: "'i"   and r :: "'i" shows "euclid_ext_aux_i s' s t' t r' r = (if r = arith_ops_record.zero ops then let c = arith_ops_record.divide ops (arith_ops_record.one ops) (arith_ops_record.unit_factor ops r') in ((arith_ops_record.times ops s' c, arith_ops_record.times ops t' c), arith_ops_record.normalize ops r') else let q = arith_ops_record.divide ops r' r in euclid_ext_aux_i s (arith_ops_record.minus ops s' (arith_ops_record.times ops q s)) t (arith_ops_record.minus ops t' (arith_ops_record.times ops q t)) r (arith_ops_record.modulo ops r' r))"
sconsts_list_comb: fixes f :: "'a"   and xs :: "'a list"   and S :: "name fset" shows "(consts (list_comb f xs) |\<subseteq>| S) = (consts f |\<subseteq>| S \<and> list_all (\<lambda>x. consts x |\<subseteq>| S) xs)"
intro: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "comm_semigroup times" shows "comm_semigroup_mult times"
th3_2: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec" assumes "B\<^sub>\<real> a b c" shows "B\<^sub>\<real> c b a"
imp_cut: fixes C :: "fmla"   and H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "insert C H \<turnstile> A IMP B"   and "{A} \<turnstile> C" shows "H \<turnstile> A IMP B"
atLeastatMost_empty': fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order less_eq less"   and "\<not> less_eq a b" shows "ord.atLeastAtMost less_eq a b = {}"
iso_block_img_alt_rep: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "image_mset ((`) bij_map) \<B> = \<B>'"
emsapss_decode_def: fixes M :: "bit list"   and EM :: "bit list"   and emBits :: "nat" shows "emsapss_decode M EM emBits = (if 2 ^ 64 \<le> length M \<or> 2 ^ 32 * 160 < emBits then False else emsapss_decode_help1 (sha1 M) EM emBits)"
map_minsky_id: fixes M :: "('a \<times> ('a, 'b) Op) set" shows "map_minsky id id M = M"
diff_add_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_monoid_diff plus minus zero" shows "minus a (plus a b) = zero"
map_poly_id: fixes p :: "'b poly" shows "map_poly id p = p"
invimage_mono: fixes r' :: "V"   and r :: "V"   and A' :: "V"   and A :: "V" assumes "r' \<subseteq>\<^sub>\<circ> r"   and "A' \<subseteq>\<^sub>\<circ> A" shows "r' -`\<^sub>\<circ> A' \<subseteq>\<^sub>\<circ> r -`\<^sub>\<circ> A"
eint_ord_simps_5: fixes q :: "eint" shows "(\<infinity> \<le> q) = (q = \<infinity>)"
open_Collect_neg: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.topological_space open"   and "topological_space.closed open {x. P x}" shows "open {x. \<not> P x}"
prems_2: shows "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t (\<langle>assign: t' \<doteq> t''\<rangle>\<^sub>s\<^sub>t # S) \<union> V"
distinct_left_right: fixes l :: "'b tree"   and z :: "'b"   and b :: "bool"   and r :: "'b tree"   and x :: "'b"   and y :: "'b" assumes "all_distinct (Node l z b r)"   and "x \<in> set_of l"   and "y \<in> set_of r" shows "x \<noteq> y"
eq: shows "\<exists>\<^sub>F s in (\<lambda>s. s \<bullet> (1::'a)) going_to at_top. eval_fds f s = eval_fds g s"
atLeastatMost_psubset_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "({a..b} \<subset> {c..d}) = ((\<not> a \<le> b \<or> c \<le> a \<and> b \<le> d \<and> (c < a \<or> b < d)) \<and> c \<le> d)"
cod_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "cod f = (if arr f then MkIde (Cod f) else null)"
hlam_def: fixes A :: "hf"   and B :: "hf"   and C :: "hf"   and F :: "hf" shows "hlam A B C F = \<lbrace>XG \<^bold>\<in> A * hexp B C. \<forall>YZ. (YZ \<^bold>\<in> hsnd XG) = (is_hpair YZ \<and> \<langle>\<langle>hfst XG, hfst YZ\<rangle>, hsnd YZ\<rangle> \<^bold>\<in> F)\<rbrace>"
surj_dual: fixes dual :: "'b \<Rightarrow> 'b" assumes "class.dual dual" shows "surj dual"
closed_insert: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and a :: "'b" assumes "class.t1_space open"   and "topological_space.closed open S" shows "topological_space.closed open (insert a S)"
NSLIM_add: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and l :: "'d"   and g :: "'c \<Rightarrow> 'd"   and m :: "'d" assumes "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l"   and "g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m" shows "(\<lambda>x. f x + g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l + m"
conv_order: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x\<^sup>T \<le> y\<^sup>T)"
prob_component_measure: fixes p :: "real"   and x :: "bool stream"   and i :: "nat" assumes "0 \<le> p"   and "p \<le> 1" shows "emeasure (measure_pmf (bernoulli_pmf p)) {x !! i} = ennreal (prob_component p x i)"
unit_div: fixes a :: "'a"   and b :: "'a" assumes "is_unit a"   and "is_unit b" shows "is_unit (a div b)"
one_ennreal_def: shows "1 \<equiv> e2ennreal 1"
map2_val_compat_id: shows "tc.map2_val_compat id"
1: shows "coreflexive (((v \<sqinter> - F * e * top \<sqinter> top * e\<^sup>T * F)[[top]]) * (v \<sqinter> - F * e * top \<sqinter> top * e\<^sup>T * F))"
load_tmps_read_tmps_distinct: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "load_tmps is \<inter> read_tmps sb = {}"
open_centered_ball_chartE: fixes x :: "'d"   and U :: "'d set"   and e :: "real"   and x0 :: "'d" assumes "x \<in> U"   and "open U"   and "U \<subseteq> manifold_eucl.dest.carrier"   and "0 < e" obtains c :: "('d, 'd) chart" where "c \<in> manifold_eucl.dest.atlas" and "x \<in> domain c" and "apply_chart c x = x0" and "domain c \<subseteq> U" and "codomain c = ball x0 e"