hcju/isabelleps huggingface dataset:
simps_2: fixes uw :: "bool"   and ux :: "'e \<Rightarrow>\<^sub>0 'f"   and i' :: "nat"   and uy :: "'g"   and uz :: "'e \<Rightarrow>\<^sub>0 'f"   and j' :: "nat"   and va :: "'g"   and ps :: "(bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and i :: "nat"   and j :: "nat" shows "pair_in_list ((uw, (ux, i', uy), uz, j', va) # ps) i j = (i = i' \<and> j = j' \<or> i = j' \<and> j = i' \<or> pair_in_list ps i j)"
IH: shows "\<Union> {pcp_seq C S n |n. n \<le> m} = pcp_seq C S m"
prems_1: shows "\<F> (\<phi>1 M\<^sub>n \<phi>2) w \<subseteq> X"
gama2_cont_comp: shows "continuous_on ({0..1} - s' \<inter> {0..1}) (\<lambda>x. \<gamma>2 x \<bullet> b)"
lex_pp_plus_monotone': fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp"   and u :: "('a, 'b) pp" assumes "lex_pp s t" shows "lex_pp (u + s) (u + t)"
rep_eq: fixes x :: "'x"   and xa :: "'x fset" shows "(x |\<in>| xa) = (x \<in> fset xa)"
rebalance_middle_tree_inorder: fixes t :: "'a btree"   and sub :: "'a btree"   and rs :: "('a btree \<times> 'a) list"   and k :: "nat"   and ls :: "('a btree \<times> 'a) list"   and sep :: "'a" assumes "height t = height sub"   and "case rs of [] \<Rightarrow> True | (rsub, rsep) # list \<Rightarrow> height rsub = height t" shows "inorder (rebalance_middle_tree k ls sub sep rs t) = inorder (Node (ls @ (sub, sep) # rs) t)"
dvdI: fixes a :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and b :: "'k"   and k :: "'k" assumes "a = times b k" shows "dvd.dvd times b a"
is_leftderivation_def: fixes u :: "symbol list" shows "is_leftderivation u = leftderives [\<SS>] u"
termFSwImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" shows "termFSwImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSwapAll h hA MOD"
word_ao_dist: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "(x || y) && z = x && z || y && z"
char_root: fixes a :: "'a" assumes "a \<in> carrier G" shows "\<chi> a ^ ord a = 1"
queue_invar_def: fixes q :: "('b, 'c) BinomialTree list" shows "queue_invar q \<equiv> \<forall>t\<in>set q. tree_invar t"
dvd_eq_mod_eq_0: fixes a :: "'a"   and b :: "'a" shows "(a dvd b) = (b mod a = (0::'a))"
ascP_def: fixes f :: "'c list \<Rightarrow> 'c list" shows "ascP f = (\<forall>xs. f xs = f [] @ xs)"
Myhill_Nerode1: fixes A :: "'a list set" assumes "finite (UNIV // \<approx>A)" shows "regular A"
simps_1: fixes i :: "'c" shows "extended_to_set \<lbrakk>i\<rbrakk> = {i}"
If_else_FF: fixes t :: "tr"   and b :: "tr" shows "If b then t else FF = (b andalso t)"
acyclic_def: fixes r :: "('a \<times> 'a) set" shows "acyclic r = (\<forall>x. (x, x) \<notin> r\<^sup>+)"
group_add_hom_axioms: shows "group_add_hom (map_poly of_rat)"
icd: shows "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"
irreducible\<^sub>dD2: fixes p :: "'b poly"   and q :: "'b poly" assumes "irreducible\<^sub>d p"   and "degree q < degree p"   and "degree q \<noteq> 0" shows "\<not> q dvd p"
append_eq_append_conv_if: fixes xs\<^sub>1 :: "'b list"   and xs\<^sub>2 :: "'b list"   and ys\<^sub>1 :: "'b list"   and ys\<^sub>2 :: "'b list" shows "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) = (if length xs\<^sub>1 \<le> length ys\<^sub>1 then xs\<^sub>1 = ys\<^sub>1 \<down> length xs\<^sub>1 \<and> xs\<^sub>2 = ys\<^sub>1 \<up> length xs\<^sub>1 @ ys\<^sub>2 else xs\<^sub>1 \<down> length ys\<^sub>1 = ys\<^sub>1 \<and> xs\<^sub>1 \<up> length ys\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)"
cod_char: shows "C'.cod = (\<lambda>i. if i \<in> S then \<psi> (cod (\<phi> i)) else n)"
d_def: shows "d = (LCM r\<in>supp f. snd (quotient_of r))"
sdp_SetPC: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> supp (P s) \<Longrightarrow> sub_distrib_pconj (p a)"   and "\<And>s. finite (supp (P s))"   and "\<And>s a. 0 \<le> P s a"   and "\<And>s. sum (P s) (supp (P s)) \<le> 1" shows "sub_distrib_pconj (SetPC p P)"
sol_lemma2: fixes ODE :: "('sf, 'sz) ODE"   and t :: "real"   and I :: "('sf, 'sc, 'sz) interp"   and x :: "'sz"   and y :: "'sz"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and sol :: "real \<Rightarrow> (real, 'sz) vec"   and b :: "(real, 'sz) vec" assumes "ORadmit ODE"   and "0 \<le> t"   and "\<And>\<nu>. (\<nu> \<in> fml_sem I (FUrename x y \<phi>)) = (Radj x y \<nu> \<in> fml_sem I \<phi>)"   and "(sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (sol 0, b) x \<in> fml_sem I \<phi>}" shows "((\<lambda>t. RSadj x y (sol t)) solves_ode (\<lambda>a. ODE_sem I (OUrename x y ODE))) {0..t} {xaa. mk_v I (OUrename x y ODE) (RSadj x y (sol 0), RSadj x y b) xaa \<in> fml_sem I (FUrename x y \<phi>)}"
assms_1: shows "A \<subseteq> U\<^sub>2"
distinct: shows "distinct (sorted_list_of_set (fst ` S))"
simps: fixes a1 :: "'a com"   and a2 :: "'a option"   and a3 :: "'a option" shows "Sem a1 a2 a3 = ((\<exists>f. a1 = Basic f \<and> a2 = None \<and> a3 = None) \<or> (\<exists>f s. a1 = Basic f \<and> a2 = Some s \<and> a3 = Some (f s)) \<or> (\<exists>s. a1 = Abort \<and> a2 = s \<and> a3 = None) \<or> (\<exists>c1 s s'' c2 s'. a1 = Seq c1 c2 \<and> a2 = s \<and> a3 = s' \<and> Sem c1 s s'' \<and> Sem c2 s'' s') \<or> (\<exists>b c1 c2. a1 = Cond b c1 c2 \<and> a2 = None \<and> a3 = None) \<or> (\<exists>s b c1 s' c2. a1 = Cond b c1 c2 \<and> a2 = Some s \<and> a3 = s' \<and> s \<in> b \<and> Sem c1 (Some s) s') \<or> (\<exists>s b c2 s' c1. a1 = Cond b c1 c2 \<and> a2 = Some s \<and> a3 = s' \<and> s \<notin> b \<and> Sem c2 (Some s) s') \<or> (\<exists>b x y c. a1 = While b x y c \<and> a2 = None \<and> a3 = None) \<or> (\<exists>s b x y c. a1 = While b x y c \<and> a2 = Some s \<and> a3 = Some s \<and> s \<notin> b) \<or> (\<exists>s b c s'' x y s'. a1 = While b x y c \<and> a2 = Some s \<and> a3 = s' \<and> s \<in> b \<and> Sem c (Some s) s'' \<and> Sem (While b x y c) s'' s'))"
fc_j_eq_j_inv: fixes h :: "'a"   and j :: "'a" assumes "forest h"   and "forest_components h * j = j" shows "forest_components h * (j \<sqinter> - choose_component (forest_components h) j) = j \<sqinter> - choose_component (forest_components h) j"
simps_11: fixes e :: "'a exp"   and M :: "char list"   and es :: "'a exp list"   and A :: "'a set option" shows "\<D> (e\<bullet>M(es)) A = (\<D> e A \<and> \<D>s es (A \<squnion> \<A> e))"
ref: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<in> lists \<langle>G\<rangle>" shows "concat (Ref G us) = concat us"
infinite_iff_countable_subset: fixes S :: "'b set" shows "infinite S = (\<exists>f. inj f \<and> range f \<subseteq> S)"
monoidI: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x" shows "Group.monoid G"
dg_prod_is_arrI: fixes f :: "V"   and a :: "V"   and b :: "V" assumes "vsv f"   and "\<D>\<^sub>\<circ> f = 3\<^sub>\<nat>"   and "vsv a"   and "\<D>\<^sub>\<circ> a = 3\<^sub>\<nat>"   and "vsv b"   and "\<D>\<^sub>\<circ> b = 3\<^sub>\<nat>"   and "\<And>i. i \<in>\<^sub>\<circ> 3\<^sub>\<nat> \<Longrightarrow> f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | i = 1\<^sub>\<nat> \<Rightarrow> \<BB> | otherwise \<Rightarrow> \<CC>\<^esub> b\<lparr>i\<rparr>" shows "f : a \<mapsto>\<^bsub>\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>3\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | i = 1\<^sub>\<nat> \<Rightarrow> \<BB> | otherwise \<Rightarrow> \<CC>\<^esub> b"
\<b>_decreasing: fixes X :: "'x set"   and i :: "nat"   and j :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "finite X"   and "i \<le> j" shows "\<b> ps j \<le> \<b> ps i"
new_compl: fixes n :: "'a"   and p :: "('a, 'b) form" assumes "new n p" shows "new n (compl p)"
Quotient_circline: shows "Quotient circline_eq_clmat abs_circline rep_circline cr_circline"
\<delta>_eq_\<epsilon>: shows "\<delta> = \<epsilon>"
lookup_not_eq_zero_eq_in_keys: fixes f :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" shows "(poly_mapping.lookup f k \<noteq> (0::'c)) = (k \<in> Poly_Mapping.keys f)"
freess_subset: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and xs :: "'b list"   and ys :: "'b list" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "set xs \<subseteq> set ys" shows "pre_term.freess frees xs |\<subseteq>| pre_term.freess frees ys"
assms_3: shows "bdd_above (ratfps_subdegree ` A)"
index_add_mat_3: fixes A :: "'c mat"   and B :: "'c mat" shows "dim_col (A + B) = dim_col B"
fsubsetD: fixes A :: "'a fset"   and B :: "'a fset"   and c :: "'a" assumes "A |\<subseteq>| B"   and "c |\<in>| A" shows "c |\<in>| B"
assms_3: fixes n :: "nat" assumes "1 \<le> n"   and "P n" shows "P (Suc n)"
prime_elemI: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "p \<noteq> zero"   and "\<not> dvd.dvd times p one"   and "\<And>a b. dvd.dvd times p (times a b) \<Longrightarrow> dvd.dvd times p a \<or> dvd.dvd times p b" shows "comm_semiring_1.prime_elem times one zero p"
cobounded2: fixes a :: "'b"   and b :: "'b" shows "a \<sqinter> b \<le> b"
B_3: fixes x :: "'a"   and y :: "'a" assumes "x \<in> {a--b}"   and "dist x y \<le> d" shows "B \<le> f y \<bullet> rot n"
outstanding_non_volatile_writes_unshared_axioms: shows "outstanding_non_volatile_writes_unshared \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
bfs_impl_correct: fixes succ :: "nat \<Rightarrow> nat dlist"   and src :: "nat"   and dst :: "nat" shows "bfs_impl succ src dst \<le> Graph.bfs_spec (ls.\<alpha> \<circ> succ) src dst"
wasm_deserialise_type: fixes bs :: "bytes"   and t :: "t" shows "typeof (wasm_deserialise bs t) = t"
assms_3: shows "coeff (rhs e) v \<noteq> 0"
that: fixes e' :: "nat \<Rightarrow> real" assumes "pdevs_val e X = pdevs_val e' (pdevs_of_list (map snd (list_of_pdevs X)))"   and "e' \<in> UNIV \<rightarrow> I" shows "thesis"
ins_eq: fixes ks :: "int list"   and l :: "int"   and x :: "int"   and y :: "int"   and S :: "nat option list \<times> int list" assumes "dioph_is_node ks l x"   and "dioph_is_node ks l y"   and "dioph_invariant ks l S"   and "\<not> dioph_memb y S" shows "dioph_memb x (dioph_ins y S) = (x = y \<or> dioph_memb x S)"
Mul_Redirect_Edge: fixes j :: "nat"   and n :: "nat" assumes "0 \<le> j \<and> j < n" shows "\<turnstile> Mul_Redirect_Edge j n  pre (Mul_Color_Target j n)"
suc_def: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "wo_rel r" shows "suc r A \<equiv> minim r (AboveS r A)"
orthogonal_independent: fixes S :: "(real, 'a) vec set"   and v :: "(real, 'a) vec" assumes "independent S"   and "v \<noteq> 0"   and "\<And>w. w \<in> S \<Longrightarrow> orthogonal v w" shows "independent (insert v S)"
alpha_tau: fixes x :: "'a"   and y :: "'a" shows "\<nu> (x \<cdot> \<tau> y) = (0::'a)"
index_one_mat_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "1\<^sub>m n $$ (i, j) = (if i = j then 1::'a else (0::'a))"
classes_above_set: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "classes_above P C = classes_above P' C"
ivl_disj_int_15: fixes l :: "'a"   and m :: "'a"   and u :: "'a" shows "{l..<m} \<inter> {m..u} = {}"
retraction_is_epi: fixes g :: "'a" assumes "retraction g" shows "epi g"
interfree_Mutator_Collector: shows "interfree_aux (Some Mutator, {}, Some Collector)"
qle_trans: fixes qle :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "quasi_order qle"   and "qle x y"   and "qle y z" shows "qle x z"
acc_induct: fixes a :: "'k"   and r :: "('k \<times> 'k) set"   and P :: "'k \<Rightarrow> bool" assumes "a \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>x \<in> Wellfounded.acc r; \<forall>y. (y, x) \<in> r \<longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
coprime_power_right_iff: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "coprime a (b ^ n) = (coprime a b \<or> n = 0)"
depthTransition_2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "hnf P"   and "P \<longmapsto> a<x> \<prec> P'" shows "depth P' < depth P"
Quotient_equiv_abs1: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "Quotient R Abs Rep T"   and "R x y" shows "T x (Abs y)"
hyps_3: shows "alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}"
select_convs_2: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
except_Un_plus_Int: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and U :: "'a set"   and V :: "'a set" shows "except p (U \<union> V) + except p (U \<inter> V) = except p U + except p V"
ia_ncols_iarray: shows "ia < ncols_iarray (matrix_to_iarray A)"
assms_2: fixes a :: "'a" assumes "a \<in> A" shows "f a mod b = (0::'b)"
extract_var_single': fixes v :: "nat"   and v' :: "nat"   and n :: "nat"   and a :: "'b" assumes "v \<noteq> v'" shows "extract_var (monom (Poly_Mapping.single v n) a) v' = monom (Poly_Mapping.single v n) (monom 0 a)"
extract_None_iff: fixes P :: "'k \<Rightarrow> bool"   and xs :: "'k list" shows "(List.extract P xs = None) = (\<not> (\<exists>x\<in>set xs. P x))"
dghm_id_components_1: fixes \<CC> :: "V" shows "dghm_id \<CC>\<lparr>ObjMap\<rparr> = vid_on (\<CC>\<lparr>Obj\<rparr>)"
iso_verts_tail: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and e :: "'d" assumes "iG.digraph_isomorphism hom"   and "e \<in> arcs (app_iso hom G)" shows "iso_tail hom (iso_arcs hom e) = iso_verts hom (tail (app_iso hom G) e)"
interior_of_union_frontier_of: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of S \<union> X frontier_of S = X closure_of S"
gcd_impl_primitive: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly" assumes "subresultant_prs_gcd F n \<delta> f k \<beta> G1 G2"   and "primitive_part G1 = G1"   and "primitive_part G2 = G2" shows "gcd_impl_primitive G1 G2 = gcd G1 G2"
paths_weight_def: shows "paths_weight \<equiv> \<lambda>(paths', P_new'). Digraph.distance (remove_vertex v0) (last P_new') v1"
finite_ordering_inj: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "finite X"   and "TernaryOrdering.ordering f ord X"   and "\<forall>a b c. ord a b c \<longrightarrow> a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c"   and "m < card X"   and "n < card X"   and "f m = f n" shows "m = n"
join_True_code: fixes A :: "'b option list set"   and B :: "'b option list set" shows "join A True B = (\<Union>a\<in>A. \<Union>b\<in>B. set_option (join1 (a, b)))"
sq_mtx_eq_iff_1: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" shows "(A = B) = (\<forall>i j. A $$ i $ j = B $$ i $ j)"
induct: fixes P :: "Num.num \<Rightarrow> bool"   and num :: "Num.num" assumes "P num.One"   and "\<And>x. P x \<Longrightarrow> P (num.Bit0 x)"   and "\<And>x. P x \<Longrightarrow> P (num.Bit1 x)" shows "P num"
has_bochner_integral_reflect_real: fixes b :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a"   and i :: "'a" shows "has_bochner_integral (lebesgue_on {- b..- a}) (\<lambda>x. f (- x)) i = has_bochner_integral (lebesgue_on {a..b}) f i"
FL_Infer_of_def: fixes \<iota> :: "'a literal multiset inference" shows "FL_Infer_of \<iota> = {Infer Cls (D, New) |Cls D. \<iota> = Infer (map fst Cls) D}"
abs_venv_singleton: fixes v :: "nat \<times> char list"   and b :: "contour"   and d :: "d" shows "|[(v, b) \<mapsto> d]| = { (v, |b|) := |d|}."
vertex_retraction: fixes v :: "'a" assumes "v \<in> f ` \<Union> X" shows "f v = v"
star2: fixes x :: "'a ppath set" shows "pp_star (pp_one \<union> x) = pp_star x"
left_lemma: fixes c :: "nat" shows "rec_exec rec_left [c] = left c"
mult_commute: fixes a :: "'a"   and b :: "'a" shows "a * b = b * a"
frontier_interior_subset: fixes S :: "'a set" shows "frontier (interior S) \<subseteq> frontier S"
face_distance_le: fixes C :: "'b set"   and F :: "'b set"   and D :: "'b set" assumes "codomain.chamber C"   and "order.greater_eq C F" shows "order.greater_eq (codomain.chamber_distance C D) (codomain.face_distance F D)"
convex_rel_interior_closure_aux: fixes a :: "real"   and b :: "real"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "0 < a"   and "0 < b"   and "(a + b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y" obtains e :: "real" where "0 < e" and "e < 1" and "z = y - e *\<^sub>R (y - x)"
that: fixes err :: "'a" assumes "\<bar>err\<bar> \<le> tdev' p (trunc_err_pdevs p xs)"   and "pdevs_val e (trunc_pdevs p xs) = pdevs_val e xs + err" shows "thesis"
distinct_disc_3: fixes "term" :: "'b local.term" assumes "is_Prim\<^sub>0 term" shows "\<not> is_Hcomp term"
division: shows "(\<lambda>n. {real_of_int n..real_of_int (n + 1)}) ` {a..<b} division_of {real_of_int a..real_of_int b}"
map_le_same_dom_eq: fixes m\<^sub>0 :: "'c \<Rightarrow> 'd option"   and m\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "m\<^sub>0 \<subseteq>\<^sub>m m\<^sub>1"   and "dom m\<^sub>0 = dom m\<^sub>1" shows "m\<^sub>0 = m\<^sub>1"
simps_2: fixes xs :: "nat set"   and p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list" shows "omit xs (p # ps) = (if length ps \<in> xs then omit xs ps else p # omit xs ps)"
sel_1: fixes x :: "complex" shows "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
frag_cmul_eq_0_iff: fixes k :: "int"   and c :: "'c \<Rightarrow>\<^sub>0 int" shows "(frag_cmul k c = 0) = (k = 0 \<or> c = 0)"
assms: shows "\<not> u \<rightarrow>\<^sup>* v"
deny: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and a :: "action"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p"   and "a = Drop \<or> a = Reject" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
coboundedI2: fixes b :: "'c multiset"   and c :: "'c multiset"   and a :: "'c multiset" assumes "b \<subseteq># c" shows "a \<inter># b \<subseteq># c"
head_arev: fixes a :: "'b" assumes "a \<in> arcs H" shows "head H (rev_H' a) = tail H a"
bounded_linear_image: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "bounded S"   and "bounded_linear f" shows "bounded (f ` S)"
det30_minus_scaled3: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and x :: "real" shows "coll 0 a (b - x *\<^sub>R a) = coll 0 a b"
Int_UN_distrib: fixes B :: "'a set"   and A :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "B \<inter> \<Union> (A ` I) = (\<Union>i\<in>I. B \<inter> A i)"
Solved_Task_axioms: shows "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"
get_disconnected_document_locs_def: shows "get_disconnected_document_locs \<equiv> l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_document_locs get_disconnected_nodes_locs"
Union_Un_distrib: fixes A :: "'e set set"   and B :: "'e set set" shows "\<Union> (A \<union> B) = \<Union> A \<union> \<Union> B"
assms_3: shows "\<^bold>g [^] fst z \<otimes> g' [^] snd z = a \<otimes> h [^] e"
linear_span_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "clinear f" shows "cspan (f ` S) = f ` cspan S"
closure_involutive_aux': fixes Z :: "('c \<Rightarrow> real) set" shows "\<exists>U\<subseteq>\<R>. Closure\<^sub>\<alpha> Z = \<Union> U"
point_rep_number_inv: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and x :: "'b" assumes "incidence_system point_set block_collection" shows "int (size (filter_mset ((\<notin>) x) block_collection)) = int (size block_collection) - block_collection rep x"
pr_ideal_axioms: shows "pr_ideal R I (+) (\<cdot>) \<zero> \<one>"
gauss_jordan_single_def: fixes A :: "'b mat" shows "gauss_jordan_single A = fst (gauss_jordan A (0\<^sub>m (dim_row A) 0))"
SE2: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(F \<and> G)"
sup_bot_left: fixes x :: "'a multiset" shows "\<Empt> \<union># x = x"
strict_order_OrdSum_def: fixes a :: "('b, 'c) OrdSum"   and b :: "('b, 'c) OrdSum" shows "a < b \<equiv> a \<le> b \<and> \<not> b \<le> a"
Abs_inverse': fixes r :: "int"   and a :: "'a word" assumes "r \<in> uints LENGTH('a)"   and "word_of_int r = a" shows "uint a = r"
cc_nr: fixes C :: "'a literal multiset"   and DD :: "'a literal multiset multiset" assumes "C \<in># CC"   and "set_mset DD \<subseteq> N"   and "\<forall>I. I \<Turnstile>m DD \<longrightarrow> I \<Turnstile> C" shows "\<exists>D. D \<in># DD \<and> \<not> D < C"
sdp_Seq: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sub_distrib_pconj a"   and "sub_distrib_pconj b"   and "healthy (wp a)"   and "healthy (wp b)"   and "nearly_healthy (wlp b)" shows "sub_distrib_pconj (a ;; b)"
FF_def: shows "\<tau>.F.FF \<equiv> \<lambda>\<mu>\<nu>. if C.VV.arr \<mu>\<nu> then (F (fst \<mu>\<nu>), F (snd \<mu>\<nu>)) else D.VV.null"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_after_left_moving (as, lm) (s, l, r) ires = (\<exists>rn. l = Bk # ires \<and> r = Bk # <lm> @ Bk \<up> rn)"
Ball_fold: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "finite A" shows "Ball A P = Finite_Set.fold (\<lambda>k s. s \<and> P k) True A"
right_unique_alt_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "right_unique R = (R\<inverse>\<inverse> OO R \<le> (=))"
e_cex: shows "\<not> INTERP N \<Turnstile> \<Sum>\<^sub># (mset Cs) + D"
nat_pow_one: fixes n :: "nat" shows "1 ^ n = 1"
prems_1: shows "set (x # xs) \<subseteq> carrier G"
index_tensor_gate_unitary2: fixes m :: "nat"   and G1 :: "complex Matrix.mat"   and n :: "nat"   and G2 :: "complex Matrix.mat"   and i :: "nat"   and j :: "nat" assumes "gate m G1"   and "gate n G2"   and "i < dim_row (1\<^sub>m (dim_col G1 * dim_col G2))"   and "j < dim_col (1\<^sub>m (dim_col G1 * dim_col G2))" shows "((G1 \<Otimes> G2) * G1 \<Otimes> G2\<^sup>\<dagger>) $$ (i, j) = 1\<^sub>m (dim_col G1 * dim_col G2) $$ (i, j)"