hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "nat_int"   and xa :: "nat_int" shows "(x < xa) = (Rep_nat_int x \<subset> Rep_nat_int xa)"
subspace_sum: fixes A :: "'b set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'b" assumes "subspace A"   and "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" shows "sum f B \<in> A"
comp_apply: fixes f :: "'n \<Rightarrow> 'm"   and g :: "'o \<Rightarrow> 'n"   and x :: "'o" shows "(f \<circ> g) x = f (g x)"
minus_minus: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.group_add minus plus zero uminus" shows "uminus (uminus a) = a"
SUP_union: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and M :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (M ` (A \<union> B)) = sup (Sup (M ` A)) (Sup (M ` B))"
Rep_range: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "type_definition Rep Abs A" shows "range Rep = A"
subst_typ_rename_vars_cancel: fixes y :: "variable"   and T :: "typ"   and S :: "String.literal set"   and x :: "variable" assumes "y \<notin> fst ` tvsT T" shows "subst_typ [((y, S), Tv x S)] (subst_typ [((x, S), Tv y S)] T) = T"
ind_sheaf_is_sheaf: shows "sheaf_of_rings X (ind_topology.ind_is_open X is_open X) local.ind_sheaf ind_ring_morphisms b ind_add_str ind_mult_str ind_zero_str ind_one_str"
set_map0_1: fixes f1 :: "'l \<Rightarrow> 'n"   and f2 :: "'m \<Rightarrow> 'o" shows "F3set1 \<circ> F3map f1 f2 = (`) f1 \<circ> F3set1"
A: shows "\<forall>\<^sub>F n in sequentially. rGamma_series (1::'b) n = of_nat (Suc n) / of_nat n"
supp_alpha_supp_rel: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>" shows "supp t\<^sub>\<alpha> = supp_rel (=\<^sub>\<alpha>) (Formula.rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
lexord_linear: assumes "\<forall>a b. (a, b) \<in> lexord less_than_char \<or> a = b \<or> (b, a) \<in> lexord less_than_char" shows "(xs, ys) \<in> lexord (lexord less_than_char) \<or> xs = ys \<or> (ys, xs) \<in> lexord (lexord less_than_char)"
aux2: fixes a :: "'b"   and b :: "'b" shows "{a, b} = {b, a}"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b msllist \<Rightarrow> 'a"   and msllist :: "'b msllist" shows "P (case msllist of MSLNil \<Rightarrow> f1 | MSLCons x xa \<Rightarrow> f2 x xa) = ((msllist = MSLNil \<longrightarrow> P f1) \<and> (\<forall>x21 x22. msllist = MSLCons x21 x22 \<longrightarrow> P (f2 x21 x22)))"
mult_rel_class_of: fixes U :: "'c set"   and x :: "'d"   and V :: "'c set"   and y :: "'d"   and W :: "'c set" assumes "(U, x) \<in> Sigma {U. is_open\<^sub>Y U \<and> f x \<in> U} \<O>\<^sub>Y"   and "(V, y) \<in> Sigma {U. is_open\<^sub>Y U \<and> f x \<in> U} \<O>\<^sub>Y"   and "W \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "stfx.mult_rel (stfx.class_of U x) (stfx.class_of V y) = stfx.class_of W (mult_str\<^sub>Y W (\<rho>\<^sub>Y U W x) (\<rho>\<^sub>Y V W y))"
has_domain_char: fixes f :: "'a1 \<times> 'a2" shows "(domains f \<noteq> {}) = (C1.arr (fst f) \<and> C2.arr (snd f))"
obj_simps_3: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.obj a" shows "S.trg a = a"
generated_ringI_Basic: fixes A :: "'a set" assumes "A \<in> M" shows "A \<in> generated_ring"
nonzero_eq_divide_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero" shows "(a = divide b c) = (times a c = b)"
prems_5: shows "((\<mu> R\<^sub>r \<eta> \<in> old (fst n_ns) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst n_ns) \<union> new (fst n_ns) \<or> \<eta> \<in> old (fst n_ns) \<union> new (fst n_ns) \<and> \<mu> R\<^sub>r \<eta> \<in> next (fst n_ns)) \<and> (\<forall>nd\<in>snd n_ns. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst n_ns) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old (fst n_ns) \<union> new (fst n_ns))) \<and> (\<forall>nd\<in>snd n_ns. \<forall>nd'\<in>snd n_ns. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)"
disjnt_insert1: fixes a :: "'b"   and X :: "'b set"   and Y :: "'b set" shows "disjnt (insert a X) Y = (a \<notin> Y \<and> disjnt X Y)"
intY1_f_closed: fixes x :: "'a" assumes "x \<in> intY1" shows "f x \<in> intY1"
prj1_def: fixes f :: "'a"   and g :: "'a" shows "prj1 f g \<equiv> if cospan f g then fst (SOME x. limit_cone cospan_shape.comp (\<cdot>) (cospan_diagram.map (\<cdot>) f g) (local.dom (fst x)) (cospan_diagram.mkCone (\<cdot>) f g (fst x) (snd x))) else null"
has_partial_deriv: shows "\<forall>p\<in>D. has_partial_vector_derivative (\<lambda>x. F x \<bullet> j) i (F_j_i p) p"
wf_syscls_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "wf_syscls P \<equiv> sys_xcpts \<subseteq> set (map fst P)"
"case": fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(case WriterT x of WriterT x \<Rightarrow> f x) = f x"
map_fun_def: fixes f :: "'k \<Rightarrow> 'm"   and g :: "'n \<Rightarrow> 'l"   and h :: "'m \<Rightarrow> 'n" shows "map_fun f g h = g \<circ> h \<circ> f"
finite_filter: fixes S :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "finite S" shows "finite (Set.filter P S)"
f: fixes i :: "'n set" assumes "i \<in> \<D>" shows "f integrable_on i"
itrm_brackets_Cons_SomeE: fixes extra_rule :: "nat itrm \<Rightarrow> nat itrm \<Rightarrow> bool"   and itrm_bracket :: "nat \<Rightarrow> nat itrm \<Rightarrow> nat itrm option"   and v :: "nat"   and vs :: "nat list"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_abstraction extra_rule itrm_bracket"   and "itrm_abstraction.itrm_brackets itrm_bracket (v # vs) x = Some x'" obtains y' :: "nat itrm" where "itrm_abstraction.itrm_brackets itrm_bracket vs x = Some y'" and "itrm_bracket v y' = Some x'"
inverts_components: fixes a :: "'d" assumes "D.ide a" shows "D.inverse_arrows (\<eta> a) (\<eta>'.map a)"
perm_2: fixes pi :: "(name \<times> name) list"   and x1 :: "name"   and x2 :: "'d frame" shows "pi \<bullet> \<lparr>\<nu>x1\<rparr>x2 = \<lparr>\<nu>pi \<bullet> x1\<rparr>pi \<bullet> x2"
release_not_unshared_no_write_take: fixes \<S> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and \<R> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "non_volatile_writes_unshared \<S> (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"   and "release (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) (dom \<S>) \<R> a \<noteq> Some False"   and "a \<in> dom (share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S>)" shows "a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
onl_invariant_sterms: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'f, 'd, 'g) seqp"   and A :: "('h \<times> ('e, 'f, 'd, 'g) seqp, 'i) automaton"   and I :: "'i \<Rightarrow> bool"   and P :: "'h \<times> 'g \<Rightarrow> bool"   and \<xi> :: "'h"   and p :: "('e, 'f, 'd, 'g) seqp"   and p' :: "('e, 'f, 'd, 'g) seqp"   and l :: "'g" assumes "wellformed \<Gamma>"   and "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"   and "(\<xi>, p) \<in> reachable A I"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "P (\<xi>, l)"
in_FV_\<epsilon>D: fixes v :: "nat"   and bs1 :: "WS1S_Nameful.fo list"   and bs2 :: "WS1S_Nameful.so list"   and \<phi> :: "WS1S_Nameful.ws1s" assumes "v \<in> FV (\<epsilon> bs1 bs2 \<phi>) FO"   and "set (fos \<phi>) \<subseteq> set bs1"   and "set (sos \<phi>) \<subseteq> set bs2" shows "\<exists>y\<in>set bs1. v = index bs1 y"
ln_upper_9_def: fixes x :: "real" shows "ln_upper_9 x \<equiv> (6 * x ^ 4 + 481 * x ^ 3 + 1881 * x\<^sup>2 + 1281 * x + 131) * (x - 1) / (30 * (5 * x ^ 4 + 40 * x ^ 3 + 60 * x\<^sup>2 + 20 * x + 1))"
iterate_to_map_alt_def: fixes emp :: "unit \<Rightarrow> 'a"   and upd_dj :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" shows "iterate_to_map emp upd_dj it = it (\<lambda>_. True) (\<lambda>(k, v). upd_dj k v) (emp ())"
raw_match'_imp_raw_match'_on_map_le: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and lesubs :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'"   and "lesubs \<subseteq>\<^sub>m subs" shows "\<exists>lesubs'. raw_match' T U lesubs = Some lesubs' \<and> lesubs' \<subseteq>\<^sub>m subs'"
branch_sat: fixes branch :: "(('d, 'e) fm list \<times> 'e) list"   and M :: "('f, 'd) model"   and g :: "'e \<Rightarrow> 'f"   and w :: "'f" assumes "\<forall>(ps, i)\<in>set branch. \<forall>p. p on (ps, i) \<longrightarrow> M, g, w \<Turnstile> p" shows "M, g \<Turnstile>\<^sub>\<Theta> branch"
eq: shows "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
Left_Coset_def: fixes x :: "'a"   and H :: "'a set" shows "x \<cdot>| H = {x \<cdot> h |h. h \<in> H}"
distinct_snd_half_segments: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" shows "distinct (map snd (half_segments_of_aform X))"
monoid_vec_simps_1: fixes ty :: "'a itself"   and n :: "nat" shows "(\<otimes>\<^bsub>monoid_vec ty n\<^esub>) = (+)"
fun_le_refl: fixes t :: "(val \<times> val) list" shows "t \<lesssim> t"
eval_normalize: fixes p :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "a \<in> carrier R" shows "local.eval (local.normalize p) a = local.eval p a"
merge_sort_poly_def: shows "merge_sort_poly = msort (\<lambda>a b. fst a \<le> fst b)"
trans: fixes K :: "'a multiset"   and M :: "'a multiset"   and N :: "'a multiset" assumes "K < M"   and "M < N" shows "K < N"
cone_axioms: shows "\<chi>_cod_a.D.cone (l (A.dom a)) Dao\<chi>_dom_a.map"
arr_cod: fixes f :: "'a2" assumes "A2.arr f" shows "A2.arr (A2.cod f)"
downset_on_Union: fixes R :: "('b \<times> 'b) set set"   and p :: "('b \<times> 'b) set" assumes "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p" shows "downset_on (Field (\<Union> R)) p"
loop_orientation_top_split: fixes y :: "'a" shows "(y \<squnion> id y = top \<and> y \<sqinter> id y \<le> top) = (y \<sqinter> id y = top \<and> y \<squnion> id y = top)"
subgroupE_3: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H \<G>"   and "a \<in> H" shows "inv a \<in> H"
wtE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "U.wtE \<xi>" shows "F.wtE \<xi>"
gcd_greatest_iff: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times a (gcd b c) = (dvd.dvd times a b \<and> dvd.dvd times a c)"
closedin_connected_component_of: fixes X :: "'c topology"   and x :: "'c" shows "closedin X (connected_component_of_set X x)"
xy: fixes x :: "'a"   and c :: "'a"   and y :: "'a"   and a :: "'a" shows "(x - c = y - a) = (y = x + 1 *\<^sub>R (a - c))"
munionM_def: fixes m1 :: "'m"   and m2 :: "'m" shows "munionM m1 m2 \<equiv> bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A + B)))"
induct: fixes P :: "tm \<Rightarrow> bool"   and a0 :: "tm" assumes "P (tm.Bound 0)"   and "\<And>c t. P t \<Longrightarrow> P (CNP 0 c t)"   and "\<And>t. P t \<Longrightarrow> P (tm.Neg t)"   and "\<And>v c t. P t \<Longrightarrow> P (CNP (Suc v) c t)"   and "\<And>s t. \<lbrakk>P s; \<And>x xa. P t\<rbrakk> \<Longrightarrow> P (tm.Add s t)"   and "\<And>s t. \<lbrakk>P s; \<And>x xa. P t\<rbrakk> \<Longrightarrow> P (tm.Sub s t)"   and "\<And>c t. P t \<Longrightarrow> P (tm.Mul c t)"   and "\<And>v. P (CP v)"   and "\<And>va. P (tm.Bound (Suc va))" shows "P a0"
inj_on_image_inv_of_rat_range_rcut: shows "inj_on ((`) (inv real_of_rat)) (range rcut)"
iarray_to_vec_vec_to_iarray: fixes A :: "('a, 'b) vec" shows "iarray_to_vec (vec_to_iarray A) = A"
assms_2: shows "xs \<noteq> xs' \<or> {x, y} \<inter> {x', y'} = {}"
Red_I_without_red_F: fixes N :: "'g set" shows "Red_I_G (N - Red_F_G N) = Red_I_G N"
has_field_is_class: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C has F,b:T in D" shows "is_class P C"
C_in_red_F_N: shows "C \<in> local.Red_F_\<G> N"
that: fixes C :: "real" assumes "0 \<le> C"   and "\<And>x. x \<in> closed_segment 0 (of_nat n) \<Longrightarrow> cmod (\<Prod>k = 1..n. x - of_nat k) \<le> C" shows "thesis"
lookup_update: fixes S :: "'a Abs_State.st"   and x :: "char list"   and y :: "'a" shows "lookup (update S x y) = (lookup S)(x := y)"
rev_is_rev_conv: fixes xs :: "'d list"   and ys :: "'d list" shows "(rev xs = rev ys) = (xs = ys)"
column_QR_decomposition': fixes A :: "((real, 'a) vec, 'b) vec"   and k :: "'a" assumes "rank A = ncols A" shows "column k A = column k (Gram_Schmidt_matrix A) + (\<Sum>x\<in>{column i (fst (QR_decomposition A)) |i. i < k}. (x \<bullet> column k A / (x \<bullet> x)) *\<^sub>R x)"
comm_semiring_1_cancel_ow_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" shows "comm_semiring_1_cancel_ow U plus minus zero times one \<equiv> (comm_semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero) \<and> zero_neq_one_ow U one zero \<and> comm_monoid_mult_ow U times one \<and> comm_semiring_1_cancel_ow_axioms U minus times"
max_Var_floatariths_rk2_fas_err: fixes x0 :: "floatarith list"   and cx :: "floatarith list"   and rkp :: "floatarith"   and h :: "floatarith"   and s2 :: "floatarith"   and z :: "nat" assumes "0 < length ode_e"   and "max_Var_floatariths ode_e \<le> length ode_e"   and "length x0 = length ode_e"   and "length cx = length ode_e"   and "Max {max_Var_floatarith rkp, max_Var_floatariths x0, max_Var_floatarith h, max_Var_floatariths cx, max_Var_floatarith s2} \<le> z" shows "max_Var_floatariths (rk2_fas_err rkp x0 h cx s2) \<le> z"
cross3_simps_8: fixes a :: "'e"   and b :: "'e" shows "a + b = b + a"
fps_conv_radius_ln: fixes c :: "'a" shows "fps_conv_radius (fps_ln c) = (if c = (0::'a) then \<infinity> else 1)"
sinks_aux_cons: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and x :: "'b"   and xs :: "'b list" shows "sinks_aux I D U (x # xs) = (if \<exists>v\<in>U. (v, D x) \<in> I then sinks_aux I D (insert (D x) U) xs else sinks_aux I D U xs)"
Loss_def: fixes lose :: "nat \<Rightarrow> bool list"   and a :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and a2 :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list" shows "Loss lose a i2 a2 i \<equiv> \<forall>t. if lose t = [False] then a2 t = a t \<and> i t = i2 t else a2 t = [] \<and> i t = []"
hyperDiff1_def: fixes A :: "'b set option"   and a :: "'b" shows "A \<ominus> a \<equiv> case A of None \<Rightarrow> None | \<lfloor>A\<rfloor> \<Rightarrow> \<lfloor>A - {a}\<rfloor>"
has_integral_implies_lebesgue_measurable_real: fixes f :: "'c \<Rightarrow> real"   and I :: "real"   and \<Omega> :: "'c set" assumes "(f has_integral I) \<Omega>" shows "(\<lambda>x. f x * indicat_real \<Omega> x) \<in> borel_measurable lebesgue"
A2aRedundant: fixes x :: "i" assumes "AxiomsSet6" shows "cod (dom x) \<cong> dom x"
eq_into_sim_actions: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "E = E'" shows "E [\<approx>] E'"
outer_measure_space_outer_measure: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "increasing M f" shows "outer_measure_space (Pow \<Omega>) (outer_measure M f)"
merge_thms_1: fixes F :: "assn" shows "F \<or>\<^sub>A F \<Longrightarrow>\<^sub>t F"
return_returns_heap: fixes h :: "'x"   and x :: "'z"   and h' :: "'x" shows "h \<turnstile> return x \<rightarrow>\<^sub>h h' = (h = h')"
gR_add_isol: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "D z y"   and "x \<preceq>\<^sub>R y" shows "z \<oplus> x \<preceq>\<^sub>R z \<oplus> y"
representation_ne_zero: fixes basis :: "'b set"   and v :: "'b"   and b :: "'b" assumes "crepresentation basis v b \<noteq> 0" shows "b \<in> basis"
smult_one: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "(1::'a) \<cdot>\<^sub>v x = x"
real_parts_3: fixes p :: "int poly"   and x :: "complex" assumes "p \<noteq> 0"   and "poly (complex_of_int_poly p) x = 0" shows "Re x \<in> real_of_3 ` set (real_parts_3 p)"
3: shows "rat_poly_times A (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) A) = A - A ^ 5"
polys_rel_in_dom_inD: fixes A :: "(nat, int mpoly) fmap"   and B :: "int mpoly multiset"   and x12 :: "nat" assumes "(A, B) \<in> polys_rel"   and "x12 \<in># dom_m A" shows "the (fmlookup A x12) \<in># B"
le: fixes x__ :: "'a"   and x'__ :: "'c"   and x :: "'a"   and x' :: "'c" assumes "(x__, x'__) \<in> A"   and "(x, x') \<in> A" shows "(le x__ x, x'__ \<le> x') \<in> bool_rel"
right_iff_right_inv: fixes \<mu> :: "'a"   and \<sigma> :: "'a" assumes "local.iso \<mu>" shows "right \<sigma> \<mu> = right \<sigma> (local.inv \<mu>)"
sums_zero_iff_shift: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and s :: "'a" assumes "\<And>i. i < n \<Longrightarrow> f i = (0::'a)" shows "(\<lambda>i. f (i + n)) sums s = f sums s"
\<sigma>_3: shows "subst_range \<sigma> \<subseteq> (\<lambda>c. Fun c []) ` {c. \<Gamma> (Fun c []) = Var a \<and> public c \<and> arity c = 0} - T"
rr: shows "local.random_scheduler \<sigma> s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"
assms: fixes s :: "'state"   and s1 :: "'state"   and vl :: "'value list"   and vl1 :: "'a" assumes "reachNT s"   and "reach s1"   and "\<Delta>e s vl s1 vl1" shows "vl \<noteq> [] \<and> exit s (hd vl)"
ends_in: shows "{tail G a, head G a} \<subseteq> verts G"
content_cbox_cases: fixes a :: "'c"   and b :: "'c" shows "content (cbox a b) = (if \<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i then \<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i else 0)"
distinct_map_nth: fixes ls :: "'a list"   and inds :: "nat list" assumes "distinct ls"   and "distinct inds"   and "\<And>j. j \<in> set inds \<Longrightarrow> j < length ls" shows "distinct (map ((!) ls) inds)"
SReal_def: shows "\<real> \<equiv> {x. \<exists>r. x = hypreal_of_real r}"
four_block_mat_def: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and D :: "'a mat" shows "four_block_mat A B C D = (let nra = dim_row A; nrd = dim_row D; nca = dim_col A; ncd = dim_col D in Matrix.mat (nra + nrd) (nca + ncd) (\<lambda>(i, j). if i < nra then if j < nca then A $$ (i, j) else B $$ (i, j - nca) else if j < nca then C $$ (i - nra, j) else D $$ (i - nra, j - nca)))"
eq_for_lvar_def: fixes t :: "(nat \<times> linear_poly) list"   and v :: "nat" shows "eq_for_lvar t v \<equiv> t ! eq_idx_for_lvar t v"
akra_bazzi_term_ceiling_add: fixes b :: "real"   and x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and c :: "real" assumes "0 < b"   and "b < 1"   and "real x\<^sub>0 \<le> b * real x\<^sub>1 + c"   and "c + 1 \<le> (1 - b) * real x\<^sub>1" shows "akra_bazzi_term x\<^sub>0 x\<^sub>1 b (\<lambda>x. nat \<lceil>b * real x + c\<rceil>)"
mult_poly_raw_def: fixes p :: "(char list list \<times> int) list"   and q :: "(char list list \<times> int) list" shows "PAC_Polynomials_Operations.mult_poly_raw p q = foldl (\<lambda>b x. map (mult_monomials x) q @ b) [] p"
sqrt_gt: shows "0 < sqrt (b\<^sup>2 - 4 * a * c)"
size: fixes as :: "'a list" shows "size (fset_of_list as) = length (remdups as)"
scp1: shows "simplicial_chain (p - Suc 0) (standard_simplex p) (chain_boundary p (frag_of (restrict id (standard_simplex p))))"
FactGroup_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" shows "G Mod H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
vars_def: fixes p :: "'b mpoly" shows "vars p = \<Union> (keys ` keys (mapping_of p))"
not_iFROM_iTILL_subset: fixes n' :: "nat"   and n :: "nat" shows "\<not> [n'\<dots>] \<subseteq> [\<dots>n]"
filter_insort: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "sorted xs" shows "filter P (insort x xs) = (if P x then insort x (filter P xs) else filter P xs)"
periodic: fixes x :: "real" shows "f (x + (b - a)) = f x"
nth_upto: fixes i :: "int"   and k :: "nat"   and j :: "int" assumes "i + int k \<le> j" shows "[i..j] ! k = i + int k"
types_wellformed_update: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and t :: "'BExp set"   and x :: "'Var" assumes "types_wellformed \<Gamma>"   and "type_wellformed t" shows "types_wellformed (\<Gamma>(x \<mapsto> t))"
srtc_conversion: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "relation.srtc r a b = ((a, b) \<in> {(x, y). r x y}\<^sup>\<leftrightarrow>\<^sup>*)"
unit_imp_det_non_zero: fixes A :: "'a mat"   and n :: "nat"   and b :: "'b" assumes "A \<in> Units (ring_mat TYPE('a) n b)" shows "det A \<noteq> (0::'a)"
map_iterator_foldli_conv: fixes iti :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option" shows "map_iterator iti m = (\<exists>l0. distinct (map fst l0) \<and> m = map_of l0 \<and> iti = foldli l0)"
simps_6: fixes D :: "char list" shows "is_relevant_class (New D) = (\<lambda>P C. True)"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "f (addA x y) = addB (f x) (f y)"
Qg_i: fixes G :: "('c, 'd) Group_scheme"   and N :: "'c set" assumes "Group G"   and "G \<triangleright> N" shows "\<forall>x\<in>set_rcs G N. c_top G N (c_iop G N x) x = N"
PA1: fixes p :: "'a pfm" assumes "ptautology p" shows "\<turnstile>\<^sub>! p"
map_parametrize_idx: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b list" shows "map f l = map (\<lambda>i. f (l ! i)) [0..<length l]"
eliminate_swap_store_rid: fixes r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and r' :: "'a" assumes "r \<notin> RID\<^sub>S \<sigma>"   and "r' \<notin> RID\<^sub>S \<sigma>" shows "\<R>\<^sub>S (id(r := r', r' := r)) id \<sigma> = \<sigma>"
dim_gen_eigenspace_jordan_matrix: fixes n_as :: "(nat \<times> 'a) list"   and ev :: "'a"   and k :: "nat" shows "local.dim_gen_eigenspace (jordan_matrix n_as) ev k = sum_list (map (min k) (map fst (filter (\<lambda>(n, e). e = ev) n_as)))"