hcju/isabelleps huggingface dataset:
pr_sign_parts: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and g :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Crypt (priSK CA) (Hash (pubAK g)) \<in> parts (A \<union> spies evs)" shows "\<exists>n. g = priAK n"
Least_mono: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "mono f"   and "\<exists>x\<in>S. \<forall>y\<in>S. x \<le> y" shows "(LEAST y. y \<in> f ` S) = f (LEAST x. x \<in> S)"
DSourcesA93_L1: shows "DSources level1 sA93 = {sA82}"
branch_instr_low_equal_sub1: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "'c \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s1)))))))) \<and> t2 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s2))))))))" shows "low_equal t1 t2"
h_eq: fixes k :: "nat"   and x :: "nat" shows "(\<Sum>n = 1..x. ln (real n) ^ k / real n) = ln (real x) ^ (k + 1) / real (k + 1) + stieltjes_gamma k + h k x"
a_mul_d: fixes x :: "'a" shows "ad x \<cdot> d x = (0::'a)"
nearly_healthy_wlp_SetPC: fixes p :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>a s. a \<in> supp (p s) \<Longrightarrow> nearly_healthy (wlp (f a))"   and "\<And>s. sound (p s)"   and "\<And>s. sum (p s) (supp (p s)) \<le> 1" shows "nearly_healthy (wlp (SetPC f p))"
normal_def: fixes \<phi> :: "'a fm" shows "normal \<phi> = (\<forall>a\<in>atoms \<phi>. anormal a)"
vp_split_snoc: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as n'"   and "as \<noteq> []" obtains a' :: "'a" and as' :: "'a list" where "as = as' @ [a']" and "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as' (sourcenode a')" and "valid_edge a'" and "n' = targetnode a'"
not_beta_reducible_imp_beta_norm_unchanged: fixes t :: "Core.term" assumes "\<not> beta_reducible t" shows "beta_norm t = Some t"
lower_plus: fixes A :: "'b interval"   and B :: "'b interval" shows "lower (A + B) = lower A + lower B"
eq: shows "(\<lambda>n. n + r) -` supp f = (\<lambda>n. n - r) ` supp f"
2: shows "regular (f \<sqinter> - (top * minarc h * f\<^sup>T\<^sup>\<star>) \<squnion> (f \<sqinter> top * minarc h * f\<^sup>T\<^sup>\<star>)\<^sup>T \<squnion> minarc h) \<and> regular (forest_components f) \<and> regular (minarc h\<^sup>T)"
removeAll_id: fixes x :: "'g"   and xs :: "'g list" assumes "x \<notin> set xs" shows "removeAll x xs = xs"
is_ff_dghm_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB> \<equiv> \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB> \<and> \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
swaprows_mat: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat n nc"   and "k < n"   and "l < n" shows "swaprows k l A = swaprows_mat n k l * A"
set_fmap_minus_insert: fixes x :: "'g \<times> 'h"   and F :: "('g \<times> 'h) set"   and F' :: "'g \<Rightarrow> 'h option" assumes "x \<notin> F"   and "insert x F = set_fmap F'" shows "F = set_fmap (F' -- x)"
slpa_get_proc: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list"   and as :: "'a list"   and n :: "'b"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.same_level_path_aux kind get_return_edges cs as"   and "CFG.upd_cs kind cs as = []"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "\<forall>c\<in>set cs. valid_edge c" shows "(if cs = [] then get_proc n else get_proc (last (CFG.sourcenodes sourcenode cs))) = get_proc n'"
hyps: assumes "all_roots_real p" shows "proots_count p UNIV = degree p"
cod_pr0: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "cod (pr0 a b) = b"
range_def: fixes prime_set :: "nat set"   and index :: "nat \<times> nat" assumes "rsa_base prime_set" shows "local.range index \<equiv> {..<fst index}"
times_right_mono: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "z \<le> y" shows "x * z \<le> x * y"
acc_modi_bottom: fixes a :: "acc_modi" assumes "a \<le> Private" shows "a = Private"
S_list_image_crosses_walls: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "{} \<notin> set (wall_crossings (map (\<lambda>w. w `\<rightarrow> C0) (sums ss)))"
intros: fixes M :: "trm" assumes "\<And>M'. M \<longrightarrow>\<^sub>a M' \<Longrightarrow> SNa M'" shows "SNa M"
I3: shows "((\<lambda>x. c / max b \<bar>x - a\<bar> ^ n) has_integral F3 r - F3 x2) {x2..r}"
unitr_ex: fixes x :: "'a" shows "\<exists>e\<in>\<xi>. \<rho> x x e"
Y_int: shows "integrable M (\<lambda>y. (Y y)\<^sup>2)"
F: shows "map (inv q) (filter (\<lambda>x. x \<in> range q) xs') \<in> divergences Q"
mult_isol: fixes x :: "'a ppath set"   and y :: "'a ppath set"   and z :: "'a ppath set" assumes "x \<subseteq> y" shows "pp_prod z x \<subseteq> pp_prod z y"
compact: shows "compact (inv_chart c ` cball (apply_chart c x) e)"
f0_def: fixes f :: "'sf" shows "f0 f = $f f local.empty"
trans: fixes b :: "('a, 'b) Term.term"   and a :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "b \<sqsubseteq> a"   and "c \<sqsubseteq> b" shows "c \<sqsubseteq> a"
append_cols_mult_left: fixes A :: "'a mat"   and n :: "nat"   and a :: "nat"   and B :: "'a mat"   and b :: "nat"   and P :: "'a mat" assumes "A \<in> carrier_mat n a"   and "B \<in> carrier_mat n b"   and "P \<in> carrier_mat n n" shows "P * (A @\<^sub>c B) = P * A @\<^sub>c P * B"
assms_3: fixes xs :: "'a llist"   and ys :: "'b llist" assumes "X xs ys"   and "\<not> lnull xs"   and "\<not> lnull ys" shows "P (lhd xs) (lhd ys) \<and> (X (ltl xs) (ltl ys) \<or> llist_all2 P (ltl xs) (ltl ys))"
nodes_target: fixes p :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and r :: "(('e \<times> 'f) \<times> 'd) list" assumes "p \<in> nodes A"   and "path A r p" shows "target r p \<in> nodes A"
orthogonal_imp_lin_indpt_list: fixes gs :: "rat vec list" assumes "orthogonal gs"   and "set gs \<subseteq> Rn" shows "gs.lin_indpt_list gs"
weak_ring_morphism_is_iso: fixes f :: "('c multiset \<Rightarrow> 'a) list \<Rightarrow> 'd"   and I :: "('c multiset \<Rightarrow> 'a) list set" assumes "weak_ring_morphism f I (poly_ring L)" shows "(\<lambda>x. the_elem (f ` x)) \<in> ring_iso (poly_ring L Quot I) (image_ring f (poly_ring L))"
intros_6: fixes CT :: "nat \<Rightarrow> classDef option"   and ei :: "exp"   and ei' :: "exp"   and e0 :: "exp"   and m :: "nat"   and el :: "exp list"   and er :: "exp list" assumes "CT \<turnstile> ei \<rightarrow> ei'" shows "CT \<turnstile> MethodInvk e0 m (el @ ei # er) \<rightarrow> MethodInvk e0 m (el @ ei' # er)"
mono_cInf: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "mono f"   and "A \<noteq> {}"   and "bdd_below A" shows "f (\<Sqinter> A) \<le> \<Sqinter> (f ` A)"
prj_simps_18: shows "C.cod trg_trg_f.Prj\<^sub>0\<^sub>0 = f.apex"
SemComp: fixes s :: "Var \<Rightarrow> Val"   and c1 :: "IMP"   and n :: "nat"   and r :: "Var \<Rightarrow> Val"   and c2 :: "IMP"   and m :: "nat"   and t :: "Var \<Rightarrow> Val"   and k :: "nat" assumes " s , c1 \<rightarrow>\<^sub>n  r "   and " r , c2 \<rightarrow>\<^sub>m  t "   and "k = max n m + 1" shows " s , Comp c1 c2 \<rightarrow>\<^sub>k  t "
word_plus_mono_left: fixes y :: "'b word"   and z :: "'b word"   and x :: "'b word" assumes "y \<le> z"   and "x \<le> x + z" shows "y + x \<le> z + x"
sorted_remdups: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (remdups xs)"
one_uexpr_def: shows "1 = \<guillemotleft>1::'a\<guillemotright>"
dom_release_takeWhile: fixes sb :: "'b memref list"   and S :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" shows "dom (release (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S \<R>) = dom \<R> \<union> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
assms_5: shows "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s"
lemma_W1: fixes a :: "'a G"   and b :: "'a G" shows "(a l\<rightarrow> b) r\<rightarrow> b = (b l\<rightarrow> a) r\<rightarrow> a"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "bicategory V H \<a> \<i> src trg" shows "chosen_right_adjoints V H \<a> \<i> src trg"
s_S: shows "S = {x \<in> I. \<nexists>n. I \<rightarrow> n = x}"
diamond_edges_are_valid_1: shows "valid_path (snd diamond_top_left_edge)"
prems: shows "(i, j) \<in> G ^^ Suc n"
trss_spawn: fixes u :: "'a"   and p :: "'b"   and v :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and r :: "'a list"   and c :: "'a list multiset" assumes "(u, Spawn p, v) \<in> edges fg" shows "((u # r, c), LSpawn p, v # r, add_mset [entry fg p] c) \<in> trss fg"
SCND: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "set_mset \<Gamma> \<union> \<^bold>\<not> ` set_mset \<Delta> \<turnstile> \<bottom>"
T_A_A'_leq: fixes n :: "nat" assumes "n \<le> length paid_A'" shows "T_A n \<le> T_A' n"
image_ident: fixes Y :: "'d set" shows "(\<lambda>x. x) ` Y = Y"
member_def: fixes xs :: "'b odlist"   and x :: "'b" shows "member xs x \<equiv> List.member (toList xs) x"
a_pos: fixes n :: "nat" shows "0 \<le> a n"
l3_inv10I: fixes implem :: "chan \<Rightarrow> msg"   and x :: "l3_state" assumes "pfslvl3 implem"   and "secret x \<subseteq> payload" shows "x \<in> l3_inv10"
j: shows "degree (q * u + Polynomial.smult m r) < j"
trivial_limit_eventually: fixes net :: "'a filter"   and P :: "'a \<Rightarrow> bool" assumes "net = bot" shows "eventually P net"
of_complex_eq_iff: fixes x :: "complex"   and y :: "complex" shows "(of_complex x = of_complex y) = (x = y)"
treap_treap_of: fixes A :: "('a \<times> 'b) set" shows "treap (treap_of A)"
RDirProd_list_carrier_def': fixes Rs :: "('a, 'b) ring_scheme list" shows "carrier (RDirProd_list Rs) = carrier (DirProd_list Rs)"
w2_in: shows "w2 \<in> set (awalk_verts u q) \<union> set (awalk_verts w1 r)"
Ifun_star_n: fixes F :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'c" shows "star_n F \<star> star_n X = star_n (\<lambda>n. F n (X n))"
simps: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.tree_to_multiset (SkewBinomialHeapStruc.Node e a r ts) = {#(e, a)#} + SkewBinomialHeapStruc.queue_to_multiset ts"
iRelease_def: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iRelease I P Q \<equiv> (\<box> t I. Q t) \<or> (\<diamond> t I. P t \<and> (\<box> t' I \<down>\<le> t. Q t'))"
le_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Le C D" shows "B A Le C D"
card_state_set: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and s :: "('a, 'b) fmap" shows "Suc (length as) = card (state_set (state_list s as))"
simps_1: fixes uu :: "'d"   and ii :: "nat set"   and ni :: "nat set"   and m :: "'e" shows "inoclosed uu (ii\<not>ni:arrive(m)) = eq_newpkt m"
cnf_form_of_defs_3: fixes c :: "'b literal list" shows "disj_of_clause c \<equiv> \<^bold>\<Or>map form_of_lit c"
weight_bind_spmf: fixes x :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "weight_spmf (x \<bind> f) = integral\<^sup>L (measure_spmf x) (weight_spmf \<circ> f)"
simps_1: fixes step :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and heuristic :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> nat"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list" shows "step_augment step heuristic 0 var L F = list_conj (map fm.Atom L @ F)"
assms_2: shows "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
cases: fixes x :: "atom" obtains i :: "nat" and j :: "nat" where "x = Less i j"   | v :: "nat" and va :: "nat" where "x = Eq v va"
dlet4: fixes p :: "pat"   and ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and locs :: "locn \<times> locn"   and e :: "exp0" assumes "\<not> allDistinct (pat_bindings p [])" shows "evaluate_dec ck mn env s (Dlet locs p e) (s, Rerr (Rabort Rtype_error))"
hyps_2: assumes "es = [$Call j]" shows "s = s'"
Maclaurin_minus: fixes h :: "real"   and n :: "nat"   and diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "h < 0"   and "0 < n"   and "diff 0 = f"   and "\<forall>m t. m < n \<and> h \<le> t \<and> t \<le> 0 \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)" shows "\<exists>t>h. t < 0 \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n"
LPHIL0_rec: shows "LPHIL0 = (picks 0 (N - 1) \<rightarrow> picks 0 0 \<rightarrow> putsdown 0 0 \<rightarrow> putsdown 0 (N - 1) \<rightarrow> LPHIL0)"
null_def: shows "null = (THE n. \<forall>f. COMP n f = n \<and> COMP f n = n)"
isomorphic_transitive: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and h :: "'a \<times> 'a" assumes "VV.isomorphic f g"   and "VV.isomorphic g h" shows "VV.isomorphic f h"
leof_RES_UNIV: fixes m :: "'b nres" shows "m \<le>\<^sub>n RES UNIV"
lunit_hcomp_gen: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "(f \<star> \<l>[g \<star> h]) \<cdot> (f \<star> \<a>[trg g, g, h]) = f \<star> \<l>[g] \<star> h"
circ_circ_sup: fixes x :: "'a" shows "((1::'a) \<squnion> x)\<^sup>\<circ> = x\<^sup>\<circ>\<^sup>\<circ>"
bi_uniqueDl: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and z :: "'b" assumes "bi_unique A"   and "A x y"   and "A z y" shows "x = z"
image_subset_iff: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and B :: "'a set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
domTrans: fixes a :: "'a \<Rightarrow> 'b option"   and b :: "'a \<Rightarrow> 'c option"   and c :: "'a \<Rightarrow> 'd option" assumes "dom a \<subseteq> dom b"   and "dom b \<inter> dom c = {}" shows "dom a \<inter> dom c = {}"
admits_tupling_def: fixes I :: "'b set" shows "admits_tupling I \<equiv> \<exists>\<pi>. \<pi> \<in> (I \<rightarrow> Univ) \<inter> extensional' I \<rightarrow> Univ \<and> inj_on \<pi> ((I \<rightarrow> Univ) \<inter> extensional' I)"
writes_union_right_I: fixes S' :: "('l, 'm, 'n) prog set"   and f :: "('l, 'm, 'o) prog"   and h :: "'l"   and h' :: "'l"   and S :: "('l, 'm, 'n) prog set" assumes "writes S' f h h'" shows "writes (S \<union> S') f h h'"
mor_comp: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and B1'' :: "'i set"   and B2'' :: "'j set"   and s1'' :: "'i \<Rightarrow> ('f, 'i, 'j) F1"   and s2'' :: "'j \<Rightarrow> ('f, 'i, 'j) F2"   and f' :: "'g \<Rightarrow> 'i"   and g' :: "'h \<Rightarrow> 'j" assumes "mor B1 B2 s1 s2 B1' B2' s1' s2' f g"   and "mor B1' B2' s1' s2' B1'' B2'' s1'' s2'' f' g'" shows "mor B1 B2 s1 s2 B1'' B2'' s1'' s2'' (f' \<circ> f) (g' \<circ> g)"
inv_op_closed1: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and h :: "'c" assumes "H \<lhd> G"   and "x \<in> carrier G"   and "h \<in> H" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> x \<in> H"
C\<^sub>1's: shows "\<^bold>\<forall>x xa xb. E (x \<cdot> xa) \<^bold>\<leftarrow> E (xa \<cdot> xb) \<^bold>\<and> E (x \<cdot> (xa \<cdot> xb))"
reduce_simple_not_eq: fixes es :: "e list"   and es' :: "e list" assumes "\<lparr>es\<rparr> \<leadsto> \<lparr>es'\<rparr>" shows "es \<noteq> es'"
NOT_mask_shifted_lenword: fixes len :: "nat" shows "~~ (mask len << LENGTH('a) - len) = mask (LENGTH('a) - len)"
smult_monom: fixes c :: "'b"   and x :: "nat \<Rightarrow>\<^sub>0 nat"   and c' :: "'b" shows "MPoly_Type.smult c (MPoly_Type.monom x c') = MPoly_Type.monom x (c * c')"
is_extensional: fixes f :: "'c" assumes "\<not> C.arr f" shows "C.map f = C.null"
Inter_comm: fixes r :: "'a language"   and s :: "'a language" shows "Inter r s = Inter s r"
causal_network_axioms_def: fixes history :: "nat \<Rightarrow> 'c event list" shows "causal_network_axioms history \<equiv> \<forall>m2 j m1. Deliver m2 \<in> set (history j) \<longrightarrow> network.hb history m1 m2 \<longrightarrow> node_histories.history_order history (Deliver m1) j (Deliver m2)"
chart_in_codomain: fixes x :: "'c"   and c :: "('c, 'd) chart" assumes "x \<in> domain c" shows "apply_chart c x \<in> codomain c"
simps_1: fixes x :: "'v" shows "capacity' (SOURCE, Inner x) = (if x = source \<Delta> then \<alpha> else 0)"
impl_of_ahm_update: fixes k :: "'a"   and v :: "'b"   and hm :: "('a, 'b) hashmap" shows "impl_of (ahm_update k v hm) = ArrayHashMap_Impl.ahm_update k v (impl_of hm)"
Eq: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> equality_axioms" shows "H \<turnstile> A"
align\<^sub>d_Ma: shows "local.aligned\<^sub>d e Ma"
rel_pmf_imp_rel_set: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a pmf"   and q :: "'b pmf" assumes "rel_pmf R p q" shows "rel_set R (set_pmf p) (set_pmf q)"
ordinal_rec0_def: fixes z :: "'a"   and s :: "ordinal \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" shows "ordinal_rec0 z s l \<equiv> wfrec {(x, y). x < y} (\<lambda>F x. if x = 0 then z else if \<exists>p. x = oSuc p then s (oPrec x) (F (oPrec x)) else THE y. \<forall>f. (\<forall>n. f n < oLimit f) \<and> oLimit f = x \<longrightarrow> l (\<lambda>n. F (f n)) = y)"
transcendental_def: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and x :: "'c" assumes "ring R" shows "ring.transcendental R K x = inj_on (\<lambda>p. ring.eval R p x) (carrier (K [X]\<^bsub>R\<^esub>))"
simps_2: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and v :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and va :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.getMinTree (t # v # va) = (if SkewBinomialHeapStruc.prio t \<le> SkewBinomialHeapStruc.prio (SkewBinomialHeapStruc.getMinTree (v # va)) then t else SkewBinomialHeapStruc.getMinTree (v # va))"
goedel_rosser_first_theEasyHalf: assumes "consistent" shows "\<not> prv \<phi>R"
oInv_equality: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and a :: "ordinal" assumes "normal F"   and "F x \<le> a"   and "a < F (oSuc x)" shows "oInv F a = x"
LeastI2_wellorder: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "P a"   and "\<And>a. \<lbrakk>P a; \<forall>b. P b \<longrightarrow> a \<le> b\<rbrakk> \<Longrightarrow> Q a" shows "Q (Least P)"
graph_restrict_simps_1: fixes G :: "('a, 'b) graph_rec_scheme"   and R :: "'a set" shows "g_V (graph_restrict G R) = g_V G"
insert_lclss_of_state_1: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "insert (C, New) (lclss_of_state (N, P, Q)) = lclss_of_state (N \<union> {C}, P, Q)"
add: shows "(\<Sum>x\<in>#A. x) = (\<Sum>x\<in>set_mset A. x * count A x)"
lemma_2_5_8_b: fixes a :: "'a"   and b :: "'a" shows "a * b \<le> b"
subst_lit: fixes \<sigma> :: "'b \<Rightarrow> 'c"   and v :: "'a" shows "\<sigma> \<dagger> \<guillemotleft>v\<guillemotright> = \<guillemotleft>v\<guillemotright>"
set_empty2: fixes xs :: "'e list" shows "({} = set xs) = (xs = [])"
TC_SC: fixes G :: "('a, 'b) form list" assumes "\<stileturn> G" shows "\<turnstile> map compl G"
refl: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and lc :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval" shows "((fst (fst lc), snd (fst lc)), snd lc) \<in> sifum_security_init.loc_reach dma eval lc"
emeasure_pmf_single_eq_zero_iff: fixes M :: "'b pmf"   and y :: "'b" shows "(emeasure (measure_pmf M) {y} = 0) = (y \<notin> set_pmf M)"
assms_1: shows "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2"
Mapping_upd_set_keys: fixes m :: "('a, 'b) mapping"   and f :: "'a \<Rightarrow> 'b"   and X :: "'a set" shows "Mapping.keys (upd_set m f X) = Mapping.keys m \<union> X"
Arity_ind: fixes P :: "Arity \<Rightarrow> bool"   and n :: "Arity" assumes "P 0"   and "\<And>n. P n \<Longrightarrow> P (inc\<cdot>n)" shows "P n"
FactGroup_pow: fixes a :: "'a"   and k :: "nat" assumes "a \<in> carrier G" shows "(H #> a) [^]\<^bsub>G Mod H\<^esub> k = H #> a [^] k"
divide_by_norm_code: fixes A :: "(('d, 'e) vec, 'c) vec" shows "($) (divide_by_norm A) = divide_by_norm_row A"
compl_unique: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "inf x y = bot"   and "sup x y = top" shows "uminus x = y"
p_def: shows "p = (THE p. (\<Sum>i<k. as ! i * bs ! i powr p) = 1)"
open_halfspace_lt: fixes a :: "'b"   and b :: "real" shows "open {x. a \<bullet> x < b}"
naturality: fixes f :: "'a" assumes "S.arr f" shows "\<ll>.map (S.cod f) \<cdot>\<^sub>S L f = S.map f \<cdot>\<^sub>S \<ll>.map (S.dom f)"
solves_ode_on_subset: fixes x :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "real set"   and Y :: "'c set"   and T :: "real set"   and X :: "'c set" assumes "(x solves_ode f) S Y"   and "T \<subseteq> S"   and "Y \<subseteq> X" shows "(x solves_ode f) T X"
spr_abs_eq: fixes a :: "'a"   and t :: "'s Trace"   and t' :: "'s Trace" assumes "spr_jview a t = spr_jview a t'" shows "spr_abs t = spr_abs t'"