hcju/isabelleps huggingface dataset:
add_commute: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b = b \<sqinter> a"
intvs_incr_l: fixes l :: "int"   and h :: "int" shows "{l + 1..<h} = {l..<h} - {l}"
iff_exI: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "(\<exists>x. P x) = (\<exists>x. Q x)"
chain_def: fixes A :: "'b set set"   and C :: "'b set set" shows "subset.chain A C = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x))"
coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<preceq> c" shows "a \<sqinter> b \<preceq> c"
sylvester_mat_sub_def: fixes m :: "nat"   and n :: "nat"   and p :: "'a poly"   and q :: "'a poly" shows "sylvester_mat_sub m n p q \<equiv> mat (m + n) (m + n) (\<lambda>(i, j). if i < n then if i \<le> j \<and> j - i \<le> m then coeff p (m + i - j) else (0::'a) else if i - n \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'a))"
nGt_le_conv: fixes x :: "'a"   and y :: "'a" shows "(cmp x y \<noteq> Gt) = cmp.le x y"
assms: shows "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>r new_character_data_ptr"
up_antitone: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less" shows "less_eq x y = ({ya. less_eq y ya} \<subseteq> {y. less_eq x y})"
get_env_bigger2: fixes x :: "char list"   and e :: "'b environment"   and a :: "char list"   and X :: "'b"   and b :: "char list"   and Y :: "'b" assumes "x \<in> env_dom (e\<lparr>a:X\<rparr>\<lparr>b:Y\<rparr>)"   and "x \<noteq> a"   and "x \<noteq> b" shows "e!x = e\<lparr>a:X\<rparr>\<lparr>b:Y\<rparr>!x"
inj_on_image_mem_iff: fixes f :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and a :: "'a"   and A :: "'a set" assumes "inj_on f B"   and "a \<in> B"   and "A \<subseteq> B" shows "(f a \<in> f ` A) = (a \<in> A)"
S_separates: fixes xs :: "'a list" assumes "v0 \<leadsto>xs\<leadsto>\<^bsub>remove_vertex v\<^esub> v1" shows "set xs \<inter> S \<noteq> {}"
isLam_obtain_fresh: fixes z :: "exp"   and c :: "'a" assumes "isLam z" obtains y :: "Vars.var" and e' :: "exp" where "z = Lam [y]. e'" and "atom y \<sharp> c"
dom_fmlookup_finite: fixes m :: "('a, 'b) fmap" shows "finite (dom (fmlookup m))"
from_nat_into_inj: fixes A :: "'b set"   and m :: "nat"   and n :: "nat" assumes "countable A"   and "m \<in> to_nat_on A ` A"   and "n \<in> to_nat_on A ` A" shows "(from_nat_into A m = from_nat_into A n) = (m = n)"
bres_fres_eq: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.ab_quantale times Inf Sup inf less_eq less sup bot top" shows "proto_near_quantale.bres Sup less_eq times x y = proto_near_quantale.fres Sup less_eq times y x"
L3_5v_lemma2: fixes s :: "'a"   and Ts :: "('a ann_com option \<times> 'a set) list"   and t :: "'a" assumes "(\<Omega>, s) -P*\<rightarrow> (Parallel Ts, t)"   and "All_None Ts" shows "False"
polyneg_polyneg: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "~\<^sub>p (~\<^sub>p p) = p"
minus_iarray_def: fixes A :: "'a iarray"   and B :: "'a iarray" shows "A - B = (let length_A = IArray.length A; length_B = IArray.length B; n = max length_A length_B; A' = IArray.of_fun (\<lambda>a. if a < length_A then A !! a else (0::'a)) n; B' = IArray.of_fun (\<lambda>a. if a < length_B then B !! a else (0::'a)) n in IArray.of_fun (\<lambda>a. A' !! a - B' !! a) n)"
sup_inf_selective: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = x \<and> x \<sqinter> y = y \<or> x \<squnion> y = y \<and> x \<sqinter> y = x"
Cons_path2: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and n' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n' \<in> set (predecessors g n)" shows "g \<turnstile> n'-n' # ns\<rightarrow>m"
intros: fixes x :: "'x1"   and m :: "'m1"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x' :: "'x1"   and m' :: "'m1" assumes "\<tau>move1 (x, m) ta (x', m')" shows "r1.init_fin_\<tau>move ((Running, x), m) (convert_TA_initial (convert_obs_initial ta)) ((Running, x'), m')"
m_concat_closed: fixes l :: "'a list" assumes "set l \<subseteq> carrier G" shows "m_concat l \<in> carrier G"
monom_term_car: fixes c :: "'a"   and x :: "'a"   and n :: "nat" assumes "c \<in> carrier R"   and "x \<in> carrier R" shows "c \<otimes> x [^] n \<in> carrier R"
rQuot_union: fixes a :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "rQuot a (A \<union> B) = rQuot a A \<union> rQuot a B"
cases: fixes x :: "('a \<times> ('b, 'c) fmap) list" obtains "x = []"   | a :: "'a \<times> ('b, 'c) fmap" and as :: "('a \<times> ('b, 'c) fmap) list" where "x = a # as"
productive_def: fixes g :: "'c \<Rightarrow> ('d, 'c) step" shows "productive g = (productive_on g = UNIV)"
Abs_Int_Fun_axioms: shows "Abs_Int_Fun num' plus' \<gamma>"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "methodDef list"   and a3 :: "nat" assumes "a1 \<turnstile>+ a2 OK IN a3" obtains CT :: "nat \<Rightarrow> classDef option" and C :: "nat" where "a1 = CT__" and "a2 = []" and "a3 = C__"   | CT :: "nat \<Rightarrow> classDef option" and m :: "methodDef" and C :: "nat" and ms :: "methodDef list" where "a1 = CT" and "a2 = m # ms" and "a3 = C" and "CT \<turnstile> m OK IN C" and "CT \<turnstile>+ ms OK IN C"
measurable_return1: fixes N :: "'a measure"   and x :: "'a"   and L :: "'b measure" shows "return N x \<rightarrow>\<^sub>M L = N \<rightarrow>\<^sub>M L"
SV_inv3_inductive_1: shows "init sv_TS \<subseteq> SV_inv3"
singleton_group_def: fixes a :: "'b" shows "singleton_group a = \<lparr>carrier = {a}, mult = \<lambda>x y. a, one = a\<rparr>"
proportionality_def: shows "vec.proportionality \<equiv> {(x, y). x \<noteq> 0 \<and> y \<noteq> 0 \<and> (\<exists>k. x = k *s y)}"
iIN_cut_greater: fixes n :: "nat"   and d :: "nat"   and t :: "nat" shows "[n\<dots>,d] \<down>> t = (if t < n then [n\<dots>,d] else if t < n + d then [Suc t\<dots>,n + d - Suc t] else {})"
ogets_def: fixes f :: "'a \<Rightarrow> 'b" shows "ogets f \<equiv> \<lambda>s. Some (f s)"
minus: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "Modules.additive f" shows "f (- x) = - f x"
ex_least_nat_less: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P n"   and "\<not> P 0" shows "\<exists>k. order.greater n k \<and> (\<forall>i. order.greater_eq k i \<longrightarrow> \<not> P i) \<and> P (Suc k)"
primitive_iff_content_eq_1: fixes f :: "'b poly" shows "Missing_Polynomial_Factorial.primitive f = (content f = (1::'b))"
simply_connected_eq_contractible_loop_all: fixes S :: "'c set" shows "simply_connected S = (S = {} \<or> (\<exists>a\<in>S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> homotopic_loops S p (linepath a a)))"
chain_inf_llist_if_infinite_chain_function: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "\<forall>i. r (f (Suc i)) (f i)" shows "\<not> lfinite (inf_llist f) \<and> chain r\<inverse>\<inverse> (inf_llist f)"
Subobjs_appendPath: 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"   and C :: "char list"   and Cs :: "char list list"   and Ds :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C Cs"   and "Subobjs P (last Cs) Ds"   and "wf_prog wf_md P" shows "Subobjs P C (Cs @\<^sub>p Ds)"
inject: fixes x1 :: "'b"   and x2 :: "'b stream"   and y1 :: "'b"   and y2 :: "'b stream" shows "(x1 ## x2 = y1 ## y2) = (x1 = y1 \<and> x2 = y2)"
winding_number_as_continuous_log: fixes p :: "real \<Rightarrow> complex"   and \<zeta> :: "complex" assumes "path p"   and "\<zeta> \<notin> path_image p" obtains q :: "real \<Rightarrow> complex" where "path q" and "pathfinish q - pathstart q = 2 * complex_of_real pi * \<i> * winding_number p \<zeta>" and "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp (q t)"
rel_set_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
assms_3: shows "injective (f \<sqinter> - e\<^sup>T \<sqinter> - p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e)"
t: shows "eval (Mn 2 (Cn 3 r_not [r_result])) [i, x] \<down>= t"
Dagsp_Dags_eq: fixes nodes :: "ref set"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref" shows "Dagsp nodes low high = (\<lambda>x. x \<in> Dags nodes low high)"
simps: fixes s :: "nat"   and lm :: "nat list"   and p :: "abc_inst list" shows "abc_notfinal (s, lm) p = (s < length p)"
basic_cong_6: fixes p :: "int" assumes "2 < p" shows "[- 1 \<noteq> 0] (mod p)"
upt_conv_Cons: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "[i..<j] = i # [Suc i..<j]"
int_simps_13: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<longrightarrow> P) = #True"
LIMSEQ_incseq_SUP: fixes X :: "nat \<Rightarrow> 'b" assumes "bdd_above (range X)"   and "incseq X" shows "X \<longlonglongrightarrow> \<Squnion> range X"
infdist_triangle: fixes x :: "'b"   and A :: "'b set"   and y :: "'b" shows "infdist x A \<le> infdist y A + dist x y"
Inf_def: fixes A :: "'a set" shows "Inf A = Finite_Set.fold inf top A"
assms_9: shows "pathstart f $ 1 < pathstart g $ 1"
hyps_2: fixes \<Gamma>\<^sub>2 :: "(name \<times> ty) list"   and u :: "trm"   and \<Gamma>\<^sub>1 :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and \<theta> :: "(name \<times> trm) list" assumes "\<Gamma>\<^sub>2 \<turnstile> u : S"   and "\<Gamma>\<^sub>1 @ \<Delta>\<^sub>1 @ \<Gamma>\<^sub>2 \<turnstile> t : T"   and "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>"   and "supp p \<sharp>* u" shows "\<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<turnstile> \<theta>\<lparr>t\<rparr> : T"
semantics_stateful_intro_process_one: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma>\<^sub>\<sigma> :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and state_upate :: "'b \<Rightarrow> final_decision \<Rightarrow> 'c \<Rightarrow> 'b"   and \<sigma>\<^sub>0 :: "'b"   and built_in_chain :: "char list"   and default_policy :: "action"   and p :: "'c"   and ps :: "'c list"   and ps_processed_old :: "('c \<times> final_decision) list"   and \<sigma>_old :: "'b"   and X :: "final_decision"   and \<sigma>' :: "'b"   and ps_processed :: "('c \<times> final_decision) list" assumes "semantics_stateful \<Gamma> \<gamma>\<^sub>\<sigma> state_upate \<sigma>\<^sub>0 (built_in_chain, default_policy) (p # ps) ps_processed_old \<sigma>_old"   and "\<Gamma>,\<gamma>\<^sub>\<sigma> \<sigma>_old,p\<turnstile> \<langle>[Rule MatchAny (Call built_in_chain), Rule MatchAny default_policy], Undecided\<rangle> \<Rightarrow> Decision X"   and "\<sigma>' = state_upate \<sigma>_old X p"   and "ps_processed = ps_processed_old @ [(p, X)]" shows "semantics_stateful \<Gamma> \<gamma>\<^sub>\<sigma> state_upate \<sigma>\<^sub>0 (built_in_chain, default_policy) ps ps_processed \<sigma>'"
soundI2: fixes b :: "real"   and P :: "'b \<Rightarrow> real" assumes "bounded_by b P"   and "nneg P" shows "sound P"
expl_cond_exp_borel: fixes Y :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> real" assumes "Y \<in> space M \<rightarrow> space N"   and "disc_fct Y"   and "\<forall>r\<in>range Y \<inter> space N. \<exists>A\<in>sets N. range Y \<inter> A = {r}" shows "expl_cond_expect M Y X \<in> borel_measurable (fct_gen_subalgebra M N Y)"
mydivcode: fixes k :: "nat"   and D :: "nat" assumes "0 < k"   and "k \<le> D" shows "mydiv D k = Suc (mydiv (D - k) k)"
in_renamed_in_unlabelled_global_1: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and r :: "'a"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "bij \<alpha>"   and "bij \<beta>" shows "(\<alpha> r \<in> RID\<^sub>G (\<R>\<^sub>G \<alpha> \<beta> s)) = (r \<in> RID\<^sub>G s)"
divide_eq_imp: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero"   and "b = times a c" shows "divide b c = a"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<lbrace>t \<^bold>\<otimes> u\<rbrace> = \<lbrace>t\<rbrace> \<otimes>\<^sub>D \<lbrace>u\<rbrace>"
coinduct: fixes X :: "'b llist \<Rightarrow> bool"   and x :: "'b llist"   and A :: "'b \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> x = LNil \<or> (\<exists>l a. x = a ## l \<and> (X l \<or> alllstsp A l) \<and> A a)" shows "alllstsp A x"
ap_sum_id: fixes c :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'e + 'f" shows "ap_sum c (pure id) x = x"
hcomp_arr_obj_2: fixes \<mu> :: "'b"   and a :: "'b" assumes "D.arr \<mu>"   and "D.obj a"   and "src\<^sub>D \<mu> = a" shows "\<r>\<^sub>D[D.cod \<mu>] \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[D.dom \<mu>] = \<mu>"
ideD_1: fixes a :: "'a" assumes "L.ide a" shows "L.arr a"
fls_divide_base_factor': fixes f :: "'b fls"   and g :: "'b fls" shows "fls_base_factor f / fls_base_factor g = fls_shift (fls_subdegree f - fls_subdegree g) (f / g)"
obs: shows "obsp \<sigma> k \<noteq> obsp \<sigma>' k'"
intros: fixes u :: "'a \<Rightarrow> 'b"   and R :: "('a \<Rightarrow> 'b) set"   and \<R> :: "('a \<Rightarrow> 'b) set set"   and R' :: "('a \<Rightarrow> 'b) set"   and t :: "'b" assumes "u \<in> R"   and "R \<in> \<R>"   and "R' \<in> \<R>"   and "(0::'b) \<le> t"   and "R' = [u \<oplus> t]\<^sub>\<R>" shows "R' \<in> Succ \<R> R"
alphaAbs_ex_equal_or_qAFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_ex_equal_or_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
adjacent_sym: fixes x :: "'b set"   and y :: "'b set" assumes "x \<sim> y" shows "y \<sim> x"
l_zero: fixes x :: "int" assumes "x \<in> carrier R" shows "\<zero> \<oplus> x = x"
list_dfs_def: fixes succ :: "'a \<times> 'b \<times> 'a \<Rightarrow> ('a \<times> 'b \<times> 'a) list"   and S :: "('a \<times> 'b \<times> 'a) list"   and start :: "('a \<times> 'b \<times> 'a) list" shows "list_dfs succ S start \<equiv> gen_dfs succ List.insert (\<lambda>x xs. x \<in> set xs) S start"
oalist_inv_update_by_raw: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and kv :: "'b \<times> 'c" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "oalist_raw.oalist_inv rep_key_order (oalist_raw.update_by_raw rep_key_order kv xs)"
invertible_left_inverse: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> G" shows "sub.inverse u \<cdot> u = \<one>"
SIod_Order: fixes D :: "'c Order"   and T :: "'c set" assumes "Order D"   and "T \<subseteq> carrier D" shows "Order (SIod D T)"
hnfSummandsRemove: fixes P :: "pi"   and Q :: "pi" assumes "P \<in> summands Q"   and "uhnf Q" shows "summands Q - {P' |P'. P' \<in> summands Q \<and> P' \<equiv>\<^sub>e P} = summands Q - {P}"
simps_3: fixes \<Sigma> :: "'b list"   and n :: "nat"   and m :: "nat"   and uu :: "bool" shows "wf_atom \<Sigma> n (Arbitrary_Except m uu) = (m < n)"
pred_prod_inject: fixes P1 :: "'e \<Rightarrow> bool"   and P2 :: "'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f" shows "pred_prod P1 P2 (a, b) = (P1 a \<and> P2 b)"
ex_diff_cop: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "\<exists>E. Coplanar A B C E \<and> D \<noteq> E"
fmdom'_fmupd: fixes a :: "'a"   and b :: "'b"   and m :: "('a, 'b) fmap" shows "fmdom' (fmupd a b m) = insert a (fmdom' m)"
card_eq_UNIV: shows "(card S = card UNIV) = (S = UNIV)"
distinct_remove1_removeAll: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs" shows "remove1 x xs = removeAll x xs"
sufficiency: assumes "All admits_triangular_reduction"   and "\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'b} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'b})" shows "All admits_diagonal_reduction"
insert_Collect: fixes a :: "'d"   and P :: "'d \<Rightarrow> bool" shows "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
mtf_ub: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list"   and n :: "nat" assumes "MTF_Off as rs s0" shows "MTF_Off.t_mtf rs s0 n + MTF_Off.Phi as rs s0 (n + 1) - MTF_Off.Phi as rs s0 n \<le> 2 * MTF_Off.c_A as rs s0 n - 1 + MTF_Off.p_A as n - MTF_Off.f_A as rs s0 n"
wf_REV: fixes n :: "nat"   and r :: "'b rexp" assumes "local.wf n r" shows "local.wf n (REV r)"
independent_card_le_dim: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and Basis :: "'b set"   and B :: "'b set"   and V :: "'b set" assumes "finite_dimensional_vector_space scale Basis"   and "B \<subseteq> V"   and "\<not> module.dependent scale B" shows "card B \<le> vector_space.dim scale V"
acyclic_class_hierarchy_def: fixes P :: "'a prog" shows "acyclic_class_hierarchy P = (\<forall>(C, D, fs, ml)\<in>set (classes P). C \<noteq> Object \<longrightarrow> \<not> P \<turnstile> D \<preceq>\<^sup>* C)"
T_Abs: fixes T\<^sub>1 :: "POPLmarkRecord.type"   and \<Gamma> :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type" assumes "VarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2" shows "\<Gamma> \<turnstile> (\<lambda>:T\<^sub>1. t\<^sub>2) : T\<^sub>1 \<rightarrow> \<down>\<^sub>\<tau> 1 0 T\<^sub>2"
finite_borel_measurable_integrable: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "prob_space M"   and "f \<in> borel_measurable M"   and "finite (f ` space M)" shows "integrable M f"
int: fixes a :: "real"   and b :: "real" assumes "{a..b} \<subseteq> {0..}" shows "f integrable_on {a..b}"
simps_3: fixes x :: "int"   and y :: "int"   and xs :: "int list" shows "Challenge1A.take (x # y # xs) = (if x < y then take_incr (x # y # xs) else take_decr (x # y # xs))"
f_mono: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> M"   and "b \<in> M"   and "a \<subseteq> b" shows "f a \<le> f b"
closedin_subtopology_Un: fixes X :: "'c topology"   and T :: "'c set"   and S :: "'c set"   and U :: "'c set" assumes "closedin (subtopology X T) S"   and "closedin (subtopology X U) S" shows "closedin (subtopology X (T \<union> U)) S"
Abs_fresh_iff_2: fixes a :: "atom"   and bs :: "atom set"   and x :: "'a" shows "a \<sharp> [bs]res. x = (a \<in> bs \<or> a \<notin> bs \<and> a \<sharp> x)"
leq_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
assms_1: shows "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P f"
vals_meld: fixes xs :: "('d, 'c) bintree option list"   and ys :: "('d, 'c) bintree option list" shows "set (vals (meld xs ys)) = set (vals xs) \<union> set (vals ys)"
SD_preorderI: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> lotteries_on carrier"   and "q \<in> lotteries_on carrier"   and "\<And>x. x \<in> carrier \<Longrightarrow> measure_pmf.prob q (preferred_alts le x) \<le> measure_pmf.prob p (preferred_alts le x)" shows "SD le q p"
wt_subst_const_fv_type_eq: fixes t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and \<delta> :: "(('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat \<Rightarrow> (('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "\<forall>x\<in>fv t. \<exists>a. \<Gamma> (Var x) = Var a"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "\<forall>x\<in>fv (t \<cdot> \<delta>). \<exists>y\<in>fv t. \<Gamma> (Var x) = \<Gamma> (Var y)"
union_ac_2: fixes M :: "'b multiset"   and N :: "'b multiset" shows "M + N = N + M"
continuous_on_swap_args: fixes A :: "'b set"   and B :: "'c set"   and d :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "continuous_on (A \<times> B) (\<lambda>(x, y). d x y)" shows "continuous_on (B \<times> A) (\<lambda>(x, y). d y x)"
prems_3: shows "lst (s, x # xs) = fst ss2"
map_poly_hom_coeff_lift: fixes a :: "int" shows "Ring_Hom_Poly.of_int_poly [:a:] = [:of_int a:]"
nu_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "\<nu> x \<le> \<nu> y"
eq: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" shows "(key_compare (rep_key_order xa) x y = Eq) = (x = y)"
zero: fixes f :: "'b \<Rightarrow> 'c" assumes "additive f" shows "f (0::'b) = (0::'c)"
assms_2: shows "rk {A, B, \<gamma>} = 2"
has_bochner_integral_iff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" shows "has_bochner_integral M f x = (integrable M f \<and> integral\<^sup>L M f = x)"
lexordp_eq_conv_lexord: fixes xs :: "'d word list"   and ys :: "'d word list" shows "signed.lexordp_eq xs ys = (xs = ys \<or> signed.lexordp xs ys)"
ord_Suc_0_right: fixes n :: "nat" shows "ord n (Suc 0) = 1"
verified_def: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" shows "verified G = (\<forall>A. A \<in> G \<longrightarrow> G \<rhd> body : A)"
P_lnull_ltl_deadend_v0: assumes "lnull (ltl P)" shows "deadend v0"
Gromov_product_e_x_x: fixes e :: "'b"   and x :: "'b" shows "Gromov_product_at e x x = dist e x"
preserves_iso: fixes f :: "'b" assumes "F.B.iso f" shows "G.B.iso (G f)"
elims_3: fixes x :: "'a ns_constraint" assumes "\<not> is_leq_ns x" obtains p :: "linear_poly" and c :: "'a" where "x = GEQ_ns p c"
simps_4: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Ps :: "char list set"   and uu :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and uv :: "((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre ({(P', Ps)/uu/uv} CONSEQ C) Q = P'"
B0_def: shows "B0 = {x. jumpF f (at_left x) \<noteq> 0 \<and> a < x \<and> x \<le> c}"
lazy_def: fixes X :: "'v \<Rightarrow> int pmf" shows "lazy X \<equiv> eval (memo (sample_var X))"
supteq_supt_set_conv: shows "{\<unrhd>} = {\<rhd>}\<^sup>="
PUTList_DomMT: fixes list :: "('c \<Rightarrow> 'd decision option) list"   and a :: "'c \<Rightarrow> 'e option"   and x :: "'c" assumes "\<forall>y\<in>set list. dom a \<inter> dom y = {}"   and "x \<in> dom a" shows "x \<notin> dom (list2policy list)"