hcju/isabelleps huggingface dataset:
ball_imageD: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>f ` A. P x" shows "\<forall>x\<in>A. P (f x)"
respects_cut': fixes A :: "('a \<times> 'a) set"   and r :: "'a" shows "A \<subseteq> Prim.S r A \<times> Prim.S r A"
map_FGco_def: fixes l1 :: "'au \<Rightarrow> 'bh"   and co1 :: "'av \<Rightarrow> 'bi"   and co2 :: "'ax \<Rightarrow> 'bk"   and co3 :: "'ay \<Rightarrow> 'bl"   and co4 :: "'az \<Rightarrow> 'bm"   and co5 :: "'aw \<Rightarrow> 'bj"   and co6 :: "'bd \<Rightarrow> 'bp"   and contra1 :: "'bn \<Rightarrow> 'ba"   and contra2 :: "'bo \<Rightarrow> 'bb"   and contra3 :: "'bq \<Rightarrow> 'be"   and contra4 :: "'br \<Rightarrow> 'bf" shows "map_FGco l1 co1 co2 co3 co4 co5 co6 contra1 contra2 contra3 contra4 = map_F l1 co1 co5 (map_G co1 co2 co3 co4 contra1 contra2) co3 co6 contra1 contra3 contra4"
sum_choose_drop_zero: fixes n :: "nat" shows "(\<Sum>k = 0..Suc n. if k = 0 then 0 else Suc n - k choose (k - 1)) = (\<Sum>j = 0..n. n - j choose j)"
assms: shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"
cA: shows "char_poly A = (\<Prod>(n, a)\<leftarrow>n_as. [:- a, 1::'a:] ^ n)"
nd_fun_eq_iff: fixes f :: "'b nd_fun"   and g :: "'b nd_fun" shows "(f = g) = (\<forall>x. (f\<^sub>\<bullet>) x = (g\<^sub>\<bullet>) x)"
set_disconnected_nodes_defs: fixes document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" shows "local.a_set_disconnected_nodes document_ptr disc_nodes = put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update disc_nodes"
ph_exists: shows "space2 x y = c m * c m * time2 x y \<longrightarrow> (\<exists>p. Ph p \<and> m sees p at x \<and> m sees p at y)"
simps_2: fixes P :: "pi" shows "depth (\<tau>.P) = 1 + depth P"
fp_cop_F_worst: fixes x :: "'a"   and ds :: "'b set"   and y :: "'a" assumes "x \<in> Pdds'.cop ds"   and "y \<in> Pdds'.fp_cop_F ds"   and "Xd y = Xd x" shows "(x, y) \<in> Pd_shuffle_to_top (insert d ds') Y (Xd x)"
that: fixes ys :: "'a list"   and zs :: "'a list" assumes "xs = ys @ x # zs" shows "thesis"
soldAllocationIsAllocation: fixes a :: "(integer \<times> 'c set) set"   and \<Omega> :: "'c set"   and N :: "integer set" assumes "a \<in> soldAllocations N \<Omega>" shows "a \<in> allocationsUniverse"
add_scaleR_degen: fixes u :: "real"   and b :: "'a"   and v :: "real"   and a :: "'a" assumes "u *\<^sub>R b + v *\<^sub>R a = u *\<^sub>R a + v *\<^sub>R b"   and "u \<noteq> v" shows "a = b"
atLeastLessThan_eq_atLeastAtMost_diff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order less_eq less" shows "ord.atLeastLessThan less_eq less a b = ord.atLeastAtMost less_eq a b - {b}"
forall_nat_set_suc: fixes n :: "nat"   and P :: "'b \<Rightarrow> nat \<Rightarrow> bool" shows "{x. \<forall>m\<ge>n. P x m} = {x. P x n} \<inter> {x. \<forall>m\<ge>Suc n. P x m}"
prod_def: shows "local.prod \<equiv> \<lparr>g_V = igba.V \<times> sa.V, g_E = {((q, s), q', s'). local.igba.L q (local.sa.L s) \<and> (q, q') \<in> igba.E \<and> (s, s') \<in> sa.E}, g_V0 = igba.V0 \<times> sa.V0, igbg_num_acc = igba.num_acc, igbg_acc = \<lambda>(q, s). if s \<in> sa.V then igba.acc q else {}\<rparr>"
igSubstIGOpSTR_def: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSubstIGOpSTR MOD \<equiv> (\<forall>ys y y1 delta inp binp. igSubst MOD ys (igVar MOD ys y1) y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys (igVar MOD ys y1) y inp) (igSubstBinp MOD ys (igVar MOD ys y1) y binp)) \<and> (\<forall>ys y Y delta inp binp. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys Y y inp) (igSubstBinp MOD ys Y y binp))"
nsub_RIs_iconfs: fixes es :: "'a exp list"   and sh :: "sheap" assumes "\<not> sub_RIs es" shows "iconfs sh es"
nth_derivative_differentiable: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and x :: "'b"   and v :: "'b" assumes "higher_differentiable_on S f (Suc n)"   and "x \<in> S" shows "(\<lambda>x. nth_derivative n f x v) differentiable at x"
backlinks_simp: shows "(e \<notin> set (backlinks (edgesL G))) = (e \<notin> backflows (set (edgesL G)))"
minus_fset: fixes x :: "'a fset"   and xa :: "'a fset" shows "fset (x |-| xa) = fset x - fset xa"
R: fixes s1 :: "real"   and s2 :: "real" assumes "0 \<le> s1"   and "s1 \<le> 1"   and "0 \<le> s2"   and "s2 \<le> 1" shows "discrete_evolution (rk2_increment p (\<lambda>t x. f (t, x))) (t + h) t (x t) + heun_remainder1 x f f' f'' t h s1 - heun_remainder2 p x f f'' t h s2 \<in> R"
intros_2: fixes vs :: "e list"   and lholed :: "Lholed"   and n :: "nat"   and es' :: "e list"   and l :: "Lholed"   and es'' :: "e list"   and k :: "nat"   and es :: "e list"   and lfilledk :: "e list" assumes "const_list vs"   and "lholed = LRec vs n es' l es''"   and "Lfilled k l es lfilledk" shows "Lfilled (k + 1) lholed es (vs @ [Label n es' lfilledk] @ es'')"
polyn_add_n1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "pol_coeff S (n, f)"   and "pol_coeff S (n, g)" shows "polyn_expr R X n (n, f) \<plusminus> polyn_expr R X n (n, g) = polyn_expr R X n (add_cf S (n, f) (n, g))"
DIVERGENCES_def: fixes P :: "('b event list \<times> 'b event set) set \<times> 'b event list set" shows "DIVERGENCES P = snd P"
exp_less_cancel: fixes x :: "real"   and y :: "real" assumes "exp x < exp y" shows "x < y"
downset_set_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "ord.downset_set less_eq X = {y. \<exists>x\<in>X. less_eq y x}"
proj2_Col_iff_not_invertible: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" shows "proj2_Col p q r = (\<not> invertible (vector [proj2_rep p, proj2_rep q, proj2_rep r]))"
LIM_equal: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "\<forall>x. x \<noteq> a \<longrightarrow> f x = g x" shows "f \<midarrow>a\<rightarrow> l = g \<midarrow>a\<rightarrow> l"
homotopic_with_imp_subset2: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "homotopic_with_canon P X Y f g" shows "g ` X \<subseteq> Y"
jozsa_transform_coeff_2: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "jozsa f n"   and "i < dim_row (jozsa.jozsa_transform f n) \<and> j < dim_col (jozsa.jozsa_transform f n)" shows "i = j + 1 \<and> odd i \<longrightarrow> jozsa.jozsa_transform f n $$ (i, j) = of_nat (f (i div 2))"
card_of_subchain_1: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat"   and Y :: "'a set" assumes "[f[f 0 ..]X]"   and "0 \<le> i"   and "i < j"   and "Y = f ` {i..j}" shows "card Y = card {i..j}"
P_def: fixes \<mu>' :: "'a set \<Rightarrow> ennreal" shows "P \<mu>' = ((\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>')"
in_snd_imageE: fixes y :: "'a"   and S :: "('b \<times> 'a) set" assumes "y \<in> snd ` S" obtains x :: "'b" where "(x, y) \<in> S"
set_induct: fixes x :: "'c"   and a :: "'c llist"   and P :: "'c \<Rightarrow> 'c llist \<Rightarrow> bool" assumes "x \<in> lset a"   and "\<And>z1 z2. P z1 (LCons z1 z2)"   and "\<And>z1 z2 xa. \<lbrakk>xa \<in> lset z2; P xa z2\<rbrakk> \<Longrightarrow> P xa (LCons z1 z2)" shows "P x a"
hyp3: shows "fst (snd (foldl Gauss_Jordan_column_k_det_P (1::'a, 0, A) [0..<Suc k])) = fst (snd (foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<Suc k]))"
length_ge_1_conv: fixes l :: "'a list" shows "(Suc 0 \<le> length l) = (l \<noteq> [])"
less_le_trans: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" assumes "x <s y"   and "y \<le>s z" shows "x <s z"
map_of_zip_submap: fixes xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'b" assumes "length xs = length ys"   and "distinct xs" shows "(map_of (zip xs ys) \<subseteq>\<^sub>m Some \<circ> f) = (map f xs = ys)"
divide_eq_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = zero) = (a = zero \<or> b = zero)"
LHS_def: shows "LHS = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)"
pcr_cr_eq: shows "pcr_\<Pi>\<^sub>1 = cr_\<Pi>\<^sub>1"
nth_ge_len: fixes xs :: "'a list"   and n :: "nat" assumes "length xs \<le> n" shows "xs ! n = [] ! (n - length xs)"
to_fun_def: fixes f :: "nat \<Rightarrow> 'a" shows "to_fun f \<equiv> to_function R f"
bisim_final: fixes t :: "'t"   and x1 :: "'x1"   and m1 :: "'m1"   and x2 :: "'x2"   and m2 :: "'m2" assumes "t \<turnstile> (x1, m1) \<approx> (x2, m2)" shows "final1 x1 = final2 x2"
"1_4": assumes "n \<noteq> 0"   and "n \<noteq> 1"   and "a \<noteq> 1"   and "\<not> n < 0"   and "odd n"   and "\<not> a < 0"   and "a \<noteq> 0"   and "n \<le> a" shows "Jacobi (a mod n) n = jacobi_code (a mod n) n"
max_equal_flows: fixes t :: "real" assumes "t \<in> {t0--t_max}" shows "x t = y t"
A3: shows "A3 = mat nr2 (dim_col A) (\<lambda>(i, j). four_block_mat A B C D $$ (i + dim_row A, j))"
get_disconnected_nodes_locs_def: shows "get_disconnected_nodes_locs \<equiv> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes_locs"
mbufnt_take_add_induct': fixes f :: "event_data option list set list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xss :: "event_data option list set list list"   and buf :: "event_data option list set list list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list list"   and nts' :: "nat list"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and j' :: "nat"   and P' :: "String.literal \<Rightarrow> nat option"   and m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and U :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "mbufnt_take f z (mbufn_add xss buf) nts = (z', buf', nts')"   and "pred_mapping (\<lambda>x. x \<le> j) P"   and "pred_mapping (\<lambda>x. x \<le> j') P'"   and "rel_mapping (\<le>) P P'"   and "safe_regex m g r"   and "to_mregex r = (mr, \<phi>s)"   and "wf_mbufn' \<sigma> P V j n R r buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [progress_regex \<sigma> P r j..<j'] nts"   and "list_all3 list_all2 (map (\<lambda>\<phi> i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s) (map2 upt (map (\<lambda>\<phi>. Monitor.progress \<sigma> P \<phi> j) \<phi>s) (map (\<lambda>\<phi>. Monitor.progress \<sigma> P' \<phi> j') \<phi>s)) xss"   and "j \<le> j'"   and "U (progress_regex \<sigma> P r j) z"   and "\<And>k Xs z. \<lbrakk>progress_regex \<sigma> P r j \<le> k; Suc k \<le> progress_regex \<sigma> P' r j'; list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) k \<phi>)) \<phi>s Xs; U k z\<rbrakk> \<Longrightarrow> U (Suc k) (f Xs (\<tau> \<sigma> k) z)" shows "U (progress_regex \<sigma> P' r j') z'"
atLeast0_atMost_Suc: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {0..Suc n} = prod g {0..n} * g (Suc n)"
arrow_of_spans_hcomp: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "arrow_of_spans (\<cdot>) (\<nu> \<star> \<mu>)"
new_length: shows "length cms\<^sub>1' = length cms\<^sub>2'"
frefI: fixes P :: "'f \<Rightarrow> bool"   and R :: "('e \<times> 'f) set"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and S :: "('g \<times> 'h) set" assumes "\<And>x y. \<lbrakk>P y; (x, y) \<in> R\<rbrakk> \<Longrightarrow> (f x, g y) \<in> S" shows "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"
ZFType_def: shows "ZFType = Nat2nat \<circ> ZFTSnd"
extended_Gromov_product_le_liminf: fixes u :: "nat \<Rightarrow> 'a"   and xi :: "'a Gromov_completion"   and v :: "nat \<Rightarrow> 'a"   and eta :: "'a Gromov_completion"   and e :: "'a" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi"   and "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> eta" shows "extended_Gromov_product_at e xi eta \<le> liminf (\<lambda>x. ereal (Gromov_product_at e (u x) (v x)))"
l9_39: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and R :: "'p"   and D :: "'p"   and Q :: "'p" assumes "A B CTSP P R"   and "Coplanar A B C D"   and "D Out P Q" shows "A B CTSP Q R"
doubleton_eq_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a, b} = {c, d}) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
set_intros_2: fixes x2 :: "'c" shows "x2 \<in> set1_list_op (Delete x2)"
Ex1_def: fixes P :: "'b \<Rightarrow> bool" shows "Ex1 P \<equiv> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y = x)"
HStartBallot_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "HInv4c s q" shows "HInv4c s' q"
lfp_cs_unfold: fixes c :: "com"   and S :: "(char list \<Rightarrow> int) set" shows "Collecting.lfp c (step S) = step S (Collecting.lfp c (step S))"
assoc_unit_l: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "monoid_cancel G"   and "a \<sim>\<^bsub>G\<^esub> b"   and "b \<in> Units G"   and "a \<in> carrier G" shows "a \<in> Units G"
ucast_ucast_mask: fixes x :: "'b word" shows "ucast (ucast x) = x AND mask LENGTH('c)"
assms_4: shows "bn \<alpha> \<sharp>* subject \<alpha>"
rref_implies_rref_upt: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form A" shows "reduced_row_echelon_form_upt_k A k"
eq: fixes U :: "'a set"   and V :: "'a set"   and x :: "'b" assumes "U \<in> I"   and "V \<in> I"   and "V \<subseteq> U"   and "x \<in> \<FF> U" shows "(\<psi> V \<circ> \<rho> U V) x = \<psi> U x"
deterministic_invariant3p: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set" assumes "\<alpha>.deterministic I" shows "invariant3p \<alpha>.redT I"
slpa_induct: fixes xs :: "'edge list"   and ys :: "'edge list"   and P :: "'edge list \<Rightarrow> 'edge list \<Rightarrow> bool" assumes "same_level_path_aux xs ys"   and "\<And>cs. P cs []"   and "\<And>cs a as. \<lbrakk>intra_kind (kind a); same_level_path_aux cs as; P cs as\<rbrakk> \<Longrightarrow> P cs (a # as)"   and "\<And>cs a as Q r p fs. \<lbrakk>kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; same_level_path_aux (a # cs) as; P (a # cs) as\<rbrakk> \<Longrightarrow> P cs (a # as)"   and "\<And>cs a as Q p f c' cs'. \<lbrakk>kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; cs = c' # cs'; same_level_path_aux cs' as; a \<in> get_return_edges c'; P cs' as\<rbrakk> \<Longrightarrow> P cs (a # as)" shows "P xs ys"
RS1_DA: fixes xs :: "('b, 'c) Combinators list" shows "removeShadowRules1 (xs @ [DenyAll]) = [DenyAll]"
S5n_knowledge: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g"   and a :: "'e"   and \<phi> :: "('e, 'f) Kform" assumes "S5n M"   and "w \<in> worlds M"   and "M, w \<Turnstile> \<^bold>K\<^sub>a \<phi>" shows "M, w \<Turnstile> \<phi>"
borel_sigma_sets_subset: fixes A :: "'c set set" assumes "A \<subseteq> sets borel" shows "sigma_sets UNIV A \<subseteq> sets borel"
ln_Gamma_bounds_1: fixes x :: "real" assumes "1 \<le> x" shows "ln (2 * pi / x) / 2 + x * ln x - x \<le> ln_Gamma x"
nc_s: shows "1\<^sub>\<sigma> \<sqinter> nc = 1\<^sub>\<sigma>"
eo: shows "\<forall>\<^sub>F t in at_top. norm (f t) \<le> M * exp (c * t)"
wf_dom_inA: fixes G :: "nat graph_rec"   and ss :: "nat list list"   and w :: "nat list" assumes "dom_sl G"   and "dom_sl.wf_dom G ss w" shows "ss \<in> list (length (cfg_doms.nodes G)) ((rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G)))"
less_eq_multiset_empty_right: fixes M :: "'a multiset" assumes "M \<noteq> {#}" shows "\<not> M \<le> {#}"
RPDs_Star: fixes r :: "mregex" shows "RPDs (MStar r) \<subseteq> {MStar r} \<union> MTimesL (MStar r) (RPDs r)"
mwb_UNIV_src_is_vwb_lens: fixes X :: "'d \<Longrightarrow> 'e" assumes "mwb_lens X"   and "\<S>\<^bsub>X\<^esub> = UNIV" shows "vwb_lens X"
base_a_1: shows "r = (\<Sum>i\<in>S. a i * Measure.measure M (A i))"
2: fixes a' :: "('a, 'b) fmap \<times> ('a, 'b) fmap" assumes "ListMem a' as" shows "action_needed_asses a' (state_succ s2 a) \<subseteq>\<^sub>f action_needed_asses a' (state_succ s1 a)"
v: shows "vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF (Fun f ss))"
carrier_coeff_def: fixes P :: "'d multiset \<Rightarrow> 'a" shows "carrier_coeff P = (\<forall>m. P m \<in> carrier R)"
emeasure_UN_countable: fixes I :: "'d set"   and X :: "'d \<Rightarrow> 'e set"   and M :: "'e measure" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M"   and "countable I"   and "disjoint_family_on X I" shows "emeasure M (\<Union> (X ` I)) = \<integral>\<^sup>+ i. emeasure M (X i) \<partial>count_space I"
fusion_lfp_eq: fixes H :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and G :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice" assumes "mono (H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "mono (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "dist_over_sup (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "\<And>x::'b::refinement_lattice. ((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)) x = ((H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> F) x" shows "(F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (lfp (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)) = lfp (H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"
dom_cod: fixes f :: "'a" assumes "A.arr f" shows "A.dom (A.cod f) = A.cod f"
FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1_dir2: fixes fs :: "nat list" assumes "0 < length fs" shows "FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m 0 (fs ! 0) \<sqsubseteq> FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m 1 fs"
arc_expanded: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.arc x = (matrix_idempotent_semiring.coreflexive (x \<odot> mtop \<odot> x\<^sup>t) \<and> matrix_idempotent_semiring.coreflexive (x\<^sup>t \<odot> mtop \<odot> x) \<and> matrix_bounded_idempotent_semiring.total (mtop \<odot> x))"
base: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "orbitp f x (f x)"
rel_restrict_alt_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" shows "rel_restrict R A = Restr R (- A)"
gauge_ball_dependent: fixes e :: "'c \<Rightarrow> real" assumes "\<forall>x. 0 < e x" shows "gauge (\<lambda>x. ball x (e x))"
rt_fresh_asD2: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1"
d_IN_restrict_current_le: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_IN (g \<upharpoonleft> \<Gamma> / f) x \<le> d_IN g x"
arrow_of_spans_of_maps_axioms: shows "arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg r\<^sub>0 (local.dom \<omega>) s\<^sub>0 s\<^sub>1 w'"
UNIV_eqvt: fixes p :: "perm" shows "p \<bullet> UNIV = UNIV"
assms_2: fixes x :: "'b"   and x' :: "'b" assumes "S x \<and> T x'"   and "x = x'" shows "s = t"
homI: fixes G :: "('d, 'e) monoid_scheme"   and h :: "'d \<Rightarrow> 'f"   and H :: "('f, 'g) monoid_scheme" assumes "\<And>x. x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" shows "h \<in> hom G H"
maxim_equality: fixes A :: "'c set"   and x :: "'c" assumes "t.isMaxim A x" shows "t.maxim A = x"
lappend_lnull2: fixes ys :: "'b llist"   and xs :: "'b llist" assumes "lnull ys" shows "xs $ ys = xs"
effect_newE: fixes n :: "nat"   and x :: "'b"   and h :: "heap"   and h' :: "heap"   and r :: "'b array" assumes "effect (Array.new n x) h h' r" obtains "r = fst (Array.alloc (replicate n x) h)" and "h' = snd (Array.alloc (replicate n x) h)" and "Array.get h' r = replicate n x" and "Array.present h' r" and "\<not> Array.present h r"
safe_imp_no_defection: fixes s :: "v_state"   and v :: "val"   and S :: "process set" assumes "safe s (next_round s) v" shows "no_defection s (const_map v S) (next_round s)"
leg1_simps_3: shows "src (tab\<^sub>1 f \<star> TfHgh.p\<^sub>1) = src (tab\<^sub>0 (g \<star> h) \<star> TfHgh.p\<^sub>0)"
span_subsetI: fixes ws :: "'a vec set"   and n :: "nat"   and us :: "'a vec set" assumes "ws \<subseteq> carrier_vec n"   and "us \<subseteq> LinearCombinations.module.span class_ring (module_vec TYPE('a) n) ws" shows "LinearCombinations.module.span class_ring (module_vec TYPE('a) n) us \<subseteq> LinearCombinations.module.span class_ring (module_vec TYPE('a) n) ws"
reciprocal_zero_iff: fixes z :: "complex_homo" shows "(reciprocal z = 0\<^sub>h) = (z = \<infinity>\<^sub>h)"
assms_6: fixes y :: "'b" assumes "y \<in> T" shows "h (k y) = y"
NotInDom_Fls: fixes d :: "tm"   and d' :: "tm"   and r :: "tm"   and A :: "fm" shows "{HPair d d' IN r, NotInDom d r} \<turnstile> A"
simply_connected_imp_holomorphic_log: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f holomorphic_on S"   and "simply_connected S"   and "locally path_connected S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" obtains g :: "complex \<Rightarrow> complex" where "g holomorphic_on S" and "\<And>z. z \<in> S \<Longrightarrow> f z = exp (g z)"
decompose_small_step: fixes sh :: "('g, 'h, 'k, 'i, 'j) system_state"   and p :: "'k"   and \<alpha> :: "('g, 'i) seq_label"   and ps' :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j"   and sys :: "('g, 'h, 'k, 'i, 'j, 'l) pre_system_scheme" assumes "GST sh p \<rightarrow>\<^bsub>\<alpha>\<^esub> ps'"   and "reachable_state sys sh" obtains c :: "('g, 'h, 'i, 'j) com" and cs :: "('g, 'h, 'i, 'j) com list" and aft :: "'j \<Rightarrow> 'h set" where "(c, aft) \<in> fragments (PGMs sys p) {}" and "atC c \<subseteq> atCs (cPGM (GST sh p))" and "aft (cLST (GST sh p)) = atCs (cPGM ps')" and "(c # cs, cTKN (GST sh p), cLST (GST sh p)) \<rightarrow>\<^bsub>\<alpha>\<^esub> ps'" and "\<forall>l\<in>atC c. cTKN ps' = Some l"
ccpo_Sup_below_iff: fixes Y :: "'b set"   and x :: "'b" assumes "Complete_Partial_Order.chain ordb Y" shows "ordb (lubb Y) x = (\<forall>y\<in>Y. ordb y x)"
some_geodesic_endpoints_2: fixes y :: "'b"   and x :: "'b"   and S :: "'b set" shows "y \<in> {x--S--y}"