hcju/isabelleps huggingface dataset:
fiX_def: fixes f :: "nat \<Rightarrow> real"   and p :: "nat"   and l :: "nat" shows "fiX f p l = (if \<bar>f p - f l\<bar> \<le> \<Delta> then f l else f p)"
v_pfxs: shows "valid_prefix (routing_match r) \<and> valid_prefixes rs"
h4e: shows "(\<not> eval (And (fm.Atom (Leq (A * B))) (fm.Atom (Eq (A\<^sup>2 - B\<^sup>2 * c)))) (L[var := sqrt Cv])) = eval (Or (fm.Atom (Less (- A * B))) (fm.Atom (Neq (A\<^sup>2 - B\<^sup>2 * c)))) (L[var := sqrt Cv])"
simps_2: fixes p :: "'psym"   and Tl :: "'fsym trm list" shows "wtA (Pr p Tl) = (wtPsym p \<and> list_all wt Tl \<and> parOf p = map tpOf Tl)"
getFresh: fixes x :: "NominalSequents.var"   and A :: "form"   and S :: "sequent"   and Ps :: "sequent list" shows "\<exists>y. y \<sharp> x \<and> y \<sharp> A \<and> y \<sharp> NominalSequents.set_of_seq S \<and> y \<sharp> set_of_prem Ps"
true_clss_mono: fixes DD :: "'b literal multiset set"   and CC :: "'b literal multiset set"   and I :: "'b set" assumes "DD \<subseteq> CC"   and "I \<Turnstile>s CC" shows "I \<Turnstile>s DD"
measure_induct: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
cong: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" shows "comm_monoid_add.Sum_any plus zero \<equiv> comm_monoid_add.Sum_any plus zero"
set_cases_2: fixes e :: "'b"   and a :: "('c, 'b) concrete_category.arr" assumes "e \<in> set2_arr a" obtains z1 :: "'c" and z2 :: "'c" where "a = MkArr z1 z2 e"
runiq_singleton_rel: fixes x :: "'a"   and y :: "'b" shows "runiq {(x, y)}"
pshowsp_list_def: fixes p :: "nat"   and xs :: "(char list \<Rightarrow> char list) list" shows "pshowsp_list p xs = shows_list_gen id ''[]'' ''['' '', '' '']'' xs"
lead_coeff_smult: fixes c :: "'a"   and p :: "'a poly" shows "lead_coeff (smult c p) = c * lead_coeff p"
f1: fixes a :: "'a" shows "Proj M *\<^sub>V a = P *\<^sub>V a"
ptensor_mat_def: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "ptensor_mat m1 m2 = st.tensor_mat m1 m2"
types_wellformed_mode_update: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and upd :: "'Var ModeUpd" assumes "types_wellformed \<Gamma>" shows "types_wellformed \<Gamma> \<oplus>\<^sub>\<S> upd"
Bij_one: fixes x :: "'e"   and X :: "'e set" assumes "x \<in> X" shows "\<one>\<^bsub>BijGroup X\<^esub> x = x"
wf12: shows "mat nr n2 (mat_mult nr m1 m2)"
small_big_trans': fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)"   and "g \<in> O[F](h)" shows "f \<in> O[F](h)"
transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun A (rel_fun (pcr_fset A) (=)) (\<in>) (|\<in>|)"
distinct_5: fixes x2 :: "nat" shows "Friend x2 \<noteq> Spy"
Var_eql_subst_Iff: fixes i :: "name"   and t :: "trm"   and H :: "fmla set"   and A :: "fmla" shows "insert (Var i EQ t) H \<turnstile> A(i::=t) IFF A"
gh_aezfun: shows "gh \<in> aezfun_set"
smallo_1_iff: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> real" assumes "eventually_nonneg F f" shows "((\<lambda>_. 1) \<in> o[F](f)) = filterlim f at_top F"
shiftr_bl_of: fixes n :: "nat" assumes "length (to_bl x) \<le> LENGTH('a)" shows "of_bl (to_bl x) >> n = of_bl (take (length (to_bl x) - n) (to_bl x))"
converse_Rel_components_2: fixes T :: "V" shows "T\<inverse>\<^sub>R\<^sub>e\<^sub>l\<lparr>ArrDom\<rparr> = T\<lparr>ArrCod\<rparr>"
Gamma_reflection_complex: fixes z :: "complex" shows "Gamma z * Gamma (1 - z) = complex_of_real pi / sin (complex_of_real pi * z)"
ds_ord_def: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (x = d x \<cdot> y)"
hm_it_has_next_rule: fixes m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable"   and m' :: "'a \<Rightarrow> 'b option"   and it :: "nat \<times> ('a \<times> 'b) list \<times> ('a, 'b) hashtable" shows "<hm_is_it m ht m' it> hm_it_has_next it <\<lambda>r. hm_is_it m ht m' it * \<up> (r = (m' \<noteq> Map.empty))>"
conjugate_square_smaller_0: fixes x :: "'a" shows "\<not> x * conjugate x < (0::'a)"
ctr_transfer_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun R (rel_option R) Some Some"
argmax_sol_in_s: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "x \<in> arg_max_set f S" shows "x \<in> S"
Ball_code_1: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "Ball (Set_Monad xs) P = list_all P xs"
gm_def: shows "gm = (\<lambda>x. ennreal (- g x))"
assms_2: fixes x :: "'a" assumes "x \<in> set ys" shows "P x = Q x"
Nil_sp: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and v2 :: "'c \<times> nat" assumes "wf_sub_rel subs" shows "SubRel.subpath g v1 [] v2 subs = (sub_rel_of g subs \<and> v1 \<in> Graph.vertices g \<and> (v1 = v2 \<or> (v1, v2) \<in> subs))"
equivp_reflp_symp_transp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "equivp R = (reflp R \<and> symp R \<and> transp R)"
poly_induct3: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int"   and Q :: "(nat \<Rightarrow> nat \<Rightarrow> int) \<Rightarrow> bool" assumes "p \<in> carrier (UP Zp)"   and "\<And>p q. \<lbrakk>q \<in> carrier (UP Zp); p \<in> carrier (UP Zp); Q p; Q q\<rbrakk> \<Longrightarrow> Q (p \<oplus>\<^bsub>UP Zp\<^esub> q)"   and "\<And>a n. a \<in> carrier Zp \<Longrightarrow> Q (monom (UP Zp) a n)" shows "Q p"
invertible_right_inverse2: fixes u :: "'a"   and v :: "'a" assumes "invertible u"   and "u \<in> G"   and "v \<in> G" shows "u \<cdot> (inverse u \<cdot> v) = v"
simps_2: fixes n :: "nat"   and m1 :: "nat"   and m2 :: "nat" shows "rexp_of n (FLess m1 m2) = (if m1 = m2 then Zero else rexp.Inter (TIMES [Full, Atom (Arbitrary_Except m1 True), Full, Atom (Arbitrary_Except m2 True), Full]) (ENC n {m1, m2}))"
perfect_hiding_ind_cpa_def: fixes \<A> :: "('a \<Rightarrow> ((nat \<times> nat) \<times> 'c) spmf) \<times> ('a \<Rightarrow> 'c \<Rightarrow> bool spmf)" shows "ped_commit.perfect_hiding_ind_cpa \<A> \<equiv> ped_commit.hiding_advantage_ind_cpa \<A> = 0"
clauseTrueIffContainsTrueLiteral: fixes clause :: "Literal list"   and valuation :: "Literal list" shows "clauseTrue clause valuation = (\<exists>literal. literal el clause \<and> literalTrue literal valuation)"
cancel_separate_1_sup: fixes x :: "'a"   and y :: "'a" assumes "coreflexive (x * y)"   and "coreflexive (y * x)" shows "(x \<squnion> y)\<^sup>\<star> = x\<^sup>\<star> \<squnion> y\<^sup>\<star>"
spaces_def: shows "spaces = manyof [CHR '' '', CHR ''\<newline>'', CHR 0x09, CHR 0x0D] \<bind> (\<lambda>_. Parser_Monad.return ())"
step_thread_Some_SomeD: 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"   and update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and s :: "('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"   and t' :: "'t"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and \<sigma>' :: "'a" assumes "deterministic I"   and "step_thread update_state s t' = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"   and "s \<in> I" shows "\<exists>x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<and> actions_ok s t ta \<and> \<sigma>' = update_state ta"
ESem_fresh_cong: fixes ESem :: "'d \<Rightarrow> ('e \<Rightarrow> 'f) \<rightarrow> 'f"   and e :: "'d"   and \<rho> :: "'e \<Rightarrow> 'f"   and \<rho>' :: "'e \<Rightarrow> 'f" assumes "has_ignore_fresh_ESem ESem"   and "\<rho> f|` fv e = \<rho>' f|` fv e" shows "ESem e\<cdot>\<rho> = ESem e\<cdot>\<rho>'"
assms_1: shows "\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e"
assms_2: shows "\<pi> l \<noteq> \<pi> k"
divide_and_conquer: fixes P :: "'b list \<Rightarrow> bool"   and zs :: "'b list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>xs ys. \<lbrakk>xs \<noteq> []; ys \<noteq> []; P xs; P ys\<rbrakk> \<Longrightarrow> P (xs @ ys)" shows "P zs"
adjoint_eqI: fixes F :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and G :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "\<And>x y. (F *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)" shows "F = G*"
jump_poly_mult: fixes p' :: "real poly"   and q :: "real poly"   and p :: "real poly"   and x :: "real" assumes "p' \<noteq> 0" shows "jump_poly (p' * q) (p' * p) x = jump_poly q p x"
forks_transitions_def: fixes N :: "nat"   and n :: "nat"   and fs :: "nat list" assumes "DiningPhilosophers N" shows "DiningPhilosophers.forks_transitions N n fs = (\<Union>i<n. DiningPhilosophers.fork_transitions N i (fs ! i))"
dec_clock: fixes st :: "'a state" shows "clock (dec_clock st) = clock st - 1"
lincomb_del2: fixes S :: "'a vec set"   and v :: "'a vec"   and a :: "'a vec \<Rightarrow> 'a" assumes "finite S"   and "S \<subseteq> carrier_vec n"   and "True"   and "v \<in> S" shows "lincomb a S = a v \<cdot>\<^sub>v v + lincomb a (S - {v})"
uminus_vec_def: shows "uminus \<equiv> \<lambda>x. \<chi>i. - x $ i"
ExecMsgEq: shows "\<forall>i<length execMsg - 1. execMsg ! i = exec' ! i"
Fun_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o \<AA>" shows "Fun (mkIde A) = (\<lambda>x\<in>A. x)"
induct: fixes P :: "('a \<times> 'b) list \<Rightarrow> bool"   and a0 :: "('a \<times> 'b) list" assumes "P []"   and "\<And>p ps. P (AList.delete (fst p) ps) \<Longrightarrow> P (p # ps)" shows "P a0"
permutes_inv: fixes p :: "'e \<Rightarrow> 'e"   and S :: "'e set" assumes "p permutes S" shows "inv p permutes S"
integral_blinfun_apply: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and h :: "'b \<Rightarrow>\<^sub>L 'c" assumes "f integrable_on s" shows "integral s (\<lambda>x. blinfun_apply h (f x)) = blinfun_apply h (integral s f)"
is_aligned_after_mask: fixes k :: "'b word"   and m :: "nat"   and n :: "nat" assumes "is_aligned k m"   and "m \<le> n" shows "is_aligned (k AND mask n) m"
mod_div_decomp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo" obtains q :: "'b" and r :: "'b" where "q = divide a b" and "r = modulo a b" and "a = plus (times q b) r"
length_tms_of_elem_even: fixes n :: "nat"   and ap :: "abc_inst list" assumes "n < length ap" shows "is_even (length (tms_of ap ! n))"
proves_eq_intr: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and B :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "term_ok \<Theta> B"   and "typ_of B = Some propT"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> (A \<longmapsto> B) \<longmapsto> (B \<longmapsto> A) \<longmapsto> mk_eq A B"
convex_onD_Icc': fixes x :: "real"   and y :: "real"   and f :: "real \<Rightarrow> real"   and c :: "real" assumes "convex_on {x..y} f"   and "c \<in> {x..y}" shows "f c \<le> (f y - f x) / (y - x) * (c - x) + f x"
FEXISTS_1: fixes n :: "nat" assumes "wf_formula n \<phi>" shows "lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<phi> = lang n (rexp_of n \<phi>)"
intros_1: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" shows "r\<^sup>*\<^sup>* a a"
moebius_ocircline_comp: fixes M1 :: "moebius"   and M2 :: "moebius"   and H :: "ocircline" shows "moebius_ocircline (moebius_comp M1 M2) H = moebius_ocircline M1 (moebius_ocircline M2 H)"
has_domain_iff_arr: fixes f :: "'a \<times> 'a" shows "(CC.domains f \<noteq> {}) = CC.arr f"
po_on_imp_transp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "po_on P A" shows "transp_on P A"
prefD: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>p v" shows "\<exists>z. v = u \<cdot> z"
elims: fixes x :: "game option"   and y :: "game option" assumes "Loopo x = y" obtains \<alpha> :: "game" where "x = Agame \<alpha>" and "y = Agame \<alpha>**"   | "x = undefg" and "y = undefg"
star_inductl: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<cdot> y \<le> y"
Max_degree_coeff_pCons: fixes a :: "'c poly"   and p :: "'c poly poly" shows "Max {degree (coeff (pCons a p) i) |i. i \<le> degree (pCons a p)} = max (degree a) (Max {degree (coeff p x) |x. x \<le> degree p})"
of_nat_Suc: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (Suc m) = plus one (semiring_1.of_nat one plus zero m)"
diff_intersect_left_idem: fixes M :: "'k multiset"   and N :: "'k multiset" shows "M - M \<inter># N = M - N"
simps: fixes a1 :: "pi"   and a2 :: "pi" shows "a1 \<approx> a2 = (\<exists>P Q. a1 = P \<and> a2 = Q \<and> P \<leadsto><weakBisim> Q \<and> Q \<approx> P)"
cc_restr_below_arg: fixes S :: "Vars.var set"   and G :: "CoCalls" shows "G G|` S \<sqsubseteq> G"
pdevs_of_centered_ivl_def: fixes r :: "'a" shows "pdevs_of_centered_ivl r = inner_scaleR_pdevs r One_pdevs"
simps_4: fixes h1 :: "'b tree"   and h2 :: "'b tree" shows "exec Merge [h1, h2] = merge h1 h2"
coprime_crossproduct': fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and d :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize"   and "b \<noteq> zero"   and "unit_factor b = unit_factor d"   and "algebraic_semidom.coprime times one a b"   and "algebraic_semidom.coprime times one c d" shows "(times a d = times b c) = (a = c \<and> b = d)"
fps_shift_mult_right_noncomm: fixes n :: "nat"   and g :: "'a fps"   and h :: "'a fps" assumes "n \<le> subdegree g" shows "fps_shift n (g * h) = fps_shift n g * h"
Union_image_single_mset: fixes m :: "'a multiset" shows "(\<Sum>x\<in>#m. {#x#}) = m"
degree_slist_zero: fixes xs :: "(nat, 'd) slist"   and n :: "nat"   and j :: "nat" assumes "degree_slist xs = n"   and "n \<le> j" shows "pdevs_apply (Pdevs xs) j = (0::'d)"
a: shows "(nres_of (inter_aform_plane_lv CARD('n) optns xi si), inter_aform_plane optns (eucl_of_list_aform xi) (Sctn (eucl_of_list (normal si)) (pstn si))) \<in> \<langle>lv_aforms_rel\<rangle>nres_rel"
axioms_1: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "monad_alt return bind alt" shows "monad return bind"
r: fixes x :: "'b" assumes "x \<in> V" shows "\<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
gmw_base_axioms: shows "gmw_base S1_14_OT R1_14_OT S2_14_OT R2_14_OT protocol_14_OT adv_14_OT"
simps_4: fixes va :: "agent" shows "invKey (shrK va) = shrK va"
recvmsg_def: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b seq_action" shows "recvmsg P a \<equiv> case a of receive m \<Rightarrow> P m | _ \<Rightarrow> True"
elims_3: fixes x :: "rat \<times> rat"   and xa :: "real" assumes "\<not> in_interval x xa" obtains l :: "rat" and r :: "rat" and x :: "real" where "xb__ = (l, r)" and "xa = x" and "\<not> (real_of_rat l \<le> x \<and> x \<le> real_of_rat r)"
assoc_in_hom'_1: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<guillemotleft>\<a>[\<mu>, \<nu>, \<tau>] : src \<tau> \<rightarrow> trg \<mu>\<guillemotright>"
elims_2: fixes x :: "heap \<times> nat set" assumes "one_assn_raw x" obtains h :: "heap" and as :: "nat set" where "x = (h, as)" and "as = {}"
restrict_restrict_spmf: fixes p :: "'h spmf"   and A :: "'h set"   and B :: "'h set" shows "p \<upharpoonleft> A \<upharpoonleft> B = p \<upharpoonleft> (A \<inter> B)"
msgDelivery: fixes t :: "nat \<Rightarrow> cnf"   and sid :: "'kid"   and n :: "nat"   and E :: "PROB set"   and n'' :: "nat"   and e :: "PROB"   and m :: "SOL" assumes "t \<in> arch"   and "\<parallel>sid\<parallel>\<^bsub>t n\<^esub>"   and "sub E = ksrp (\<sigma>\<^bsub>sid\<^esub>t n)"   and "n \<le> n''"   and "\<nexists>n' E'. n \<le> n' \<and> n' \<le> n'' \<and> \<parallel>sid\<parallel>\<^bsub>t n'\<^esub> \<and> unsub E' = ksrp (\<sigma>\<^bsub>sid\<^esub>t n') \<and> e \<in> E'"   and "e \<in> E"   and "(e, m) = bbcs (\<sigma>\<^bsub>the_bb\<^esub>t n'')"   and "\<parallel>sid\<parallel>\<^bsub>t n''\<^esub>" shows "(e, m) \<in> kscs (\<sigma>\<^bsub>sid\<^esub>t n'')"
assms_3: fixes k :: "'a" assumes "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)" shows "f k (lookup_raw xs k) (lookup_raw ys k) = Some Eq"
simps_10: fixes a :: "'a"   and r :: "'b rexp" shows "lderiv a (Pr r) = Pr (PLUS (map (\<lambda>a'. lderiv a' r) (embed a)))"
join_leftcomm: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> (y \<squnion>\<^bsub>L\<^esub> z) = y \<squnion>\<^bsub>L\<^esub> (x \<squnion>\<^bsub>L\<^esub> z)"
s_cf_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and p :: "'e" shows "s_cf R S X p = (if p = \<zero>\<^bsub>R\<^esub> then (0, \<lambda>j. \<zero>\<^bsub>S\<^esub>) else SOME c. pol_coeff S c \<and> p = polyn_expr R X (fst c) c \<and> snd c (fst c) \<noteq> \<zero>\<^bsub>S\<^esub>)"
nz: shows "(0::'a) < \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
sum_distl: fixes Y :: "'c set"   and x :: "'c" assumes "finite Y" shows "x \<cdot> \<Sum> Y = \<Sum> {x \<cdot> y |y. y \<in> Y}"
sem_eq: fixes \<nu>' :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "dsafe \<theta>"   and "is_interp I" shows "dterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> = dterm_sem (local.adjoint I \<sigma> \<nu>') \<theta>"
if_true: fixes n :: "i32"   and tf :: "tf"   and e1s :: "b_e list"   and e2s :: "b_e list" assumes "\<not> int_eq n 0" shows "\<lparr>[$C ConstInt32 n, $b_e.If tf e1s e2s]\<rparr> \<leadsto> \<lparr>[$Block tf e1s]\<rparr>"
integrable_negligible: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "negligible S" shows "f integrable_on S"
below_refl: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.po below" shows "below x x"
dvdm_imp_degree_le: fixes p :: "int poly"   and q :: "int poly" assumes "pl.dvdm p q"   and "monic p"   and "pl.Mp q \<noteq> 0"   and "1 < p ^ l" shows "degree p \<le> degree q"
divide_zero_right: fixes z :: "complex_homo" assumes "z \<noteq> 0\<^sub>h" shows "z :\<^sub>h 0\<^sub>h = \<infinity>\<^sub>h"
induct: fixes P1 :: "('c, 'd) Term.term \<Rightarrow> bool"   and P2 :: "('c, 'd) Term.term list \<Rightarrow> bool"   and term_list :: "('c, 'd) Term.term list" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (App x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 term_list"
vals_add_Cons: fixes x :: "('d, 'c) bintree option"   and xs :: "('d, 'c) bintree option list" shows "set (vals (add x xs)) = set (vals (x # xs))"
assms_2: shows "n' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
smap2_alt: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and s1 :: "'c stream"   and s2 :: "'d stream"   and s :: "'b stream" shows "(smap2 f s1 s2 = s) = (\<forall>n. f (s1 !! n) (s2 !! n) = s !! n)"
nat_infinite_iff: fixes N :: "nat set" shows "infinite N = (\<exists>f. N = range f \<and> strict_mono f)"
Union_image_insert: fixes f :: "'b \<Rightarrow> 'a set"   and a :: "'b"   and B :: "'b set" shows "\<Union> (f ` insert a B) = f a \<union> \<Union> (f ` B)"
simps_6: fixes bs :: "real list"   and a :: "MIR.num" shows "Ifm bs (Ge a) = (0 \<le> Inum bs a)"
rtrancl_imp_relpow: fixes p :: "'b \<times> 'b"   and R :: "('b \<times> 'b) set" assumes "p \<in> R\<^sup>*" shows "\<exists>n. p \<in> R ^^ n"
linorder_not_less: fixes x :: "'d"   and y :: "'d" shows "(\<not> x < y) = (y \<le> x)"
id_eq_prop_prop_5_a: fixes v :: "i"   and F :: "\<Pi>\<^sub>2"   and G :: "\<Pi>\<^sub>2" shows "[F \<^bold>= G \<^bold>\<rightarrow> G \<^bold>= F in v]"
ltln_to_ltlr_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>r ltln_to_ltlr \<phi> = w \<Turnstile>\<^sub>n \<phi>"
uv_edge_cases: obtains "(u, v) \<in> E" and "(v, u) \<notin> E"   | "(v, u) \<in> E" and "(u, v) \<notin> E"
independent_length_le: fixes K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "subfield K R"   and "R.independent K Us"   and "R.independent K Vs"   and "set Us \<subseteq> R.Span K Vs" shows "length Us \<le> length Vs"
subgraphE: fixes H :: "('e, 'f) pre_digraph"   and G :: "('e, 'f) pre_digraph" assumes "subgraph H G" obtains "verts H \<subseteq> verts G" and "arcs H \<subseteq> arcs G" and "compatible G H" and "wf_digraph H" and "wf_digraph G"
hom_zero: fixes d :: "nat" shows "monom (0::'b) d = 0"
rep_eq: fixes x :: "'a['b, 'c]"   and xa :: "'a['b, 'c]" shows "(x < xa) = (mag (fromQ x) < mag (fromQ xa))"
T': shows "T' \<in> M' \<rightarrow>\<^sub>M M"
list_assn_simps_3: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "hn_ctxt (list_assn P) [] [] = emp"
not_dvd_M: shows "\<not> p ^ Suc (multiplicity p m) dvd card M"
horiz_line_integral_zero: shows "one_chain_line_integral F {j} (two_chain_horizontal_boundary two_chain) = 0"
ARR_iff_has_domain: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.ARR C f = (partial_magma.domains (free_monoidal_category.comp C) f \<noteq> {})"
basis_in_subI: fixes \<E> :: "'a set"   and \<E>' :: "'a set"   and B :: "'a set" assumes "\<E> \<subseteq> carrier"   and "\<E>' \<subseteq> \<E>"   and "\<E> \<subseteq> carrier"   and "basis_in \<E>' B" shows "\<exists>B'\<subseteq>\<E> - \<E>'. basis_in \<E> (B \<union> B')"
assms_2: shows "(r', s') \<in> R \<times> S"
diff_self: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.group_add minus plus zero uminus" shows "minus a a = zero"
tarski_absolute_axioms_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_absolute_axioms C B \<equiv> (\<exists>a b c. \<not> B a b c \<and> \<not> B b c a \<and> \<not> B c a b) \<and> (\<forall>p q a b c. p \<noteq> q \<and> C a p a q \<and> C b p b q \<and> C c p c q \<longrightarrow> B a b c \<or> B b c a \<or> B c a b)"
prems_1: fixes ys1 :: "'a list"   and ys2 :: "'a list" assumes "ys1 \<in> listset (x # xs1)"   and "ys2 \<in> listset xs2"   and "ys = ys1 @ ys2" shows "thesis"