hcju/isabelleps huggingface dataset:
ideal_zero: fixes A :: "'a set"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "ideal A R add mult \<zero> \<one>" shows "\<zero> \<in> A"
ctr_1: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and g22 :: "'a \<Rightarrow> 'a" assumes "p a" shows "unfold_tllist p g1 g21 g22 a = TNil (g1 a)"
ordinal_utility_left: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq>[relation] y" shows "u y \<le> u x"
conseqPrePost: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P' :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q' :: "'f set"   and A' :: "'f set"   and P :: "'f set"   and Q :: "'f set"   and A :: "'f set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A'"   and "P \<subseteq> P'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
proper_rev_map_flip_signed: fixes xs :: "('b \<times> bool) list" assumes "proper_signed_list xs" shows "proper_signed_list (rev (map flip_signed xs))"
left_ideal_def: fixes I :: "'a set" shows "left_ideal I = (subgroup I \<and> (\<forall>x\<in>I. \<forall>r. r * x \<in> I))"
e_approx_32: shows "\<bar>exp 1 - 5837465777 / 2147483648\<bar> \<le> inverse (2 ^ 32)"
db: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j \<le> k" shows "d j \<lless> b k (j, i)"
FDERIV: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'a"   and f' :: "'d \<Rightarrow> 'a"   and x :: "'d"   and s :: "'d set"   and g :: "'d \<Rightarrow> 'b"   and g' :: "'d \<Rightarrow> 'b" assumes "bounded_bilinear prod"   and "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. prod (f x) (g x)) has_derivative (\<lambda>h. prod (f x) (g' h) + prod (f' h) (g x))) (at x within s)"
Esigma_prime_sum: fixes m :: "nat"   and k :: "nat" assumes "prime m"   and "1 \<le> k" shows "real (Esigma (m ^ k)) = real (m ^ (k + 1) - 1) / (real m - 1)"
encode_interfering_operator_exclusion_valid: fixes sv :: "sat_plan_variable"   and t :: "nat" assumes "sv \<in> atoms (encode_interfering_operator_exclusion (\<phi> prob_with_noop abs_prob ) t)"   and "t < Suc h" shows "valid_state_var sv"
simps_2: fixes lens_get :: "'c \<Rightarrow> 'd"   and lens_put :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and more :: "'e" shows "put\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_put"
simps_2: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and t1 :: "'d bt"   and t2 :: "'d bt" shows "bt_map f (Br a t1 t2) = Br (f a) (bt_map f t1) (bt_map f t2)"
card: fixes a :: "'a"   and b :: "'b" assumes "(a, b) \<in> X" shows "0 < card (Pair a -` X)"
polymul_properties_3: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> min n0 n1" shows "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
igSwapAbsIPresIGWlsAbsSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSwapAbsIPresIGWlsAbsSTR MOD \<equiv> \<forall>zs z1 z2 us s A. igWlsAbs MOD (us, s) (igSwapAbs MOD zs z1 z2 A) = igWlsAbs MOD (us, s) A"
wadjust_on_left_moving_B_Bk1: fixes m :: "nat"   and rs :: "nat" shows "wadjust_on_left_moving_B m rs (Oc # Oc # Oc \<up> rs @ Bk # Oc # Oc \<up> m, [Bk])"
f'a_nonneg_val: shows "0 \<le> val_Zp (f' \<bullet> a)"
exists_trace_for_any_i: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "\<exists>c'. trace c t c'" shows "trace c (take i t) (s c t i)"
multiset_eq_iff: fixes M :: "'d multiset"   and N :: "'d multiset" shows "(M = N) = (\<forall>a. count M a = count N a)"
propex\<^sub>2: fixes F :: "\<Pi>\<^sub>2" shows "\<exists>r. Some r = d\<^sub>2 F"
in_box_eucl_less: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x \<in> box a b) = (eucl_less a x \<and> eucl_less x b)"
mzeromap_def: fixes M :: "('e, 'g, 'h) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme" shows "mzeromap M N = (\<lambda>x\<in>carrier M. \<zero>\<^bsub>N\<^esub>)"
remove1_idem: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "remove1 x xs = xs"
r_mod2: fixes n :: "nat" shows "eval r_mod2 [n] \<down>= n mod 2"
map_graph_def: fixes f :: "'c \<Rightarrow> 'd option" shows "map_graph f = {(x, y) |x y. (x, Some y) \<in> fun_graph f}"
hcomp_simps_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (H \<nu> \<mu>)" shows "trg (H \<nu> \<mu>) = trg \<nu>"
greek_less_lempty: fixes b :: "(accent \<times> 'a) list"   and r :: "('a \<times> 'a) set" shows "(([], b) \<in> greek_less r) = (b \<noteq> [])"
when_neq_zero: fixes zero :: "'c"   and a :: "'c"   and P :: "bool" shows "(zero.when zero a P \<noteq> zero) = (P \<and> a \<noteq> zero)"
Lim_component_ge_cart: fixes f :: "'a \<Rightarrow> (real, 'b) vec"   and l :: "(real, 'b) vec"   and net :: "'a filter"   and b :: "real"   and i :: "'b" assumes "(f \<longlongrightarrow> l) net"   and "net \<noteq> bot"   and "\<forall>\<^sub>F x in net. b \<le> f x $ i" shows "b \<le> l $ i"
simps_2: fixes lens_get :: "'e \<Rightarrow> 'f"   and lens_put :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and more :: "'g" shows "put\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_put"
assms_1: shows "f piecewise_C1_differentiable_on {a..c}"
sigma_sets_least_sigma_algebra: fixes A :: "'c set set"   and S :: "'c set" assumes "A \<subseteq> Pow S" shows "sigma_sets S A = \<Inter> {B. A \<subseteq> B \<and> sigma_algebra S B}"
U: shows "\<forall>S\<in>K. openin T (U S) \<and> S = f -` U S \<inter> A"
STEP: fixes x :: "'a"   and S1 :: "'a set"   and S2 :: "'a set"   and \<sigma> :: "'b" assumes "S\<^sub>0 = insert x (S1 \<union> S2)"   and "I S1 (insert x S2) \<sigma>"   and "c \<sigma>" shows "f x \<sigma> \<le> SPEC (I (insert x S1) S2)"
m1_inv4_rfreshI: fixes x :: "'a m1_state_scheme" assumes "\<And>Rb Rb' A A' B B' Kab. \<lbrakk>runs x Rb = Some (Resp, [A, B], [aKey Kab, END]); runs x Rb' = Some (Resp, [A', B'], [aKey Kab, END]); B \<notin> bad; A \<notin> bad; Kab \<notin> Domain (m1_state.leak x)\<rbrakk> \<Longrightarrow> Rb = Rb'" shows "x \<in> m1_inv4_rfresh"
lc_open_id: fixes u :: "expr"   and k :: "nat"   and t :: "expr" assumes "lc u" shows "open k t u = u"
spick_eq_pseudo_proj_True: fixes w :: "bool stream"   and n :: "nat" shows "spick w n True = pseudo_proj_True n w"
q_ends_1: shows "pathstart q = a - complex_of_real d"
cod_inv: fixes f :: "'a" assumes "C.iso f" shows "C.cod (C.inv f) = C.dom f"
exponential_eq_stretch: fixes l :: "real" assumes "0 < l" shows "exponential l = distr (exponential 1) borel ((*) (1 / l))"
security_false: assumes "\<And>e eP eV \<tau> \<pi>A i \<pi>' eV'. \<lbrakk>{$$} \<turnstile> e, eP, eV : \<tau>; \<lless>\<pi>A, eV\<ggreater> V\<rightarrow>i \<lless>\<pi>', eV'\<ggreater>\<rbrakk> \<Longrightarrow> \<exists>e' eP' \<pi> j \<pi>0 s s'. \<lless>[], e\<ggreater> I\<rightarrow>i \<lless>[], e'\<ggreater> \<and> \<lless>[], eP\<ggreater> P\<rightarrow>i \<lless>\<pi>, eP'\<ggreater> \<and> \<pi>A = \<pi> @ \<pi>' \<and> {$$} \<turnstile> e', eP', eV' : \<tau> \<or> j \<le> i \<and> \<lless>[], eP\<ggreater> P\<rightarrow>j \<lless>\<pi>0 @ [s], eP'\<ggreater> \<and> \<pi>A = \<pi>0 @ [s'] @ \<pi>' \<and> s \<noteq> s' \<and> hash s = hash s'"   and "hash (Inj1 Unit) = hash (Inj2 Unit)"   and "\<And>t. hash Unit = hash t \<Longrightarrow> t = Unit" shows "False"
inj_sgn_power: fixes n :: "nat" assumes "0 < n" shows "inj (\<lambda>y. sgn y * \<bar>y\<bar> ^ n)"
of_nat_mono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "of_nat m \<le> of_nat n"
fac_fac_ww_eq: shows "fac = fac_wrap"
oarrivemsg_simps_5: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and i1 :: "nat"   and i2 :: "nat" shows "oarrivemsg P \<sigma> (disconnect(i1, i2)) = True"
induct: fixes x1 :: "'Var \<Rightarrow> Sec option"   and x2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and x3 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and P :: "('Var \<Rightarrow> Sec option) \<Rightarrow> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "x2 \<R>\<^sup>3\<^bsub>x1\<^esub> x3"   and "\<And>\<Gamma> c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2 c \<Gamma>'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; \<turnstile> \<Gamma> {c} \<Gamma>'\<rbrakk> \<Longrightarrow> P \<Gamma>' \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"   and "\<And>\<Gamma> c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2 c \<Gamma>'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; \<turnstile> \<Gamma> {c} \<Gamma>'\<rbrakk> \<Longrightarrow> P \<Gamma>' \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"   and "\<And>\<Gamma> c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2 c \<Gamma>'. \<lbrakk>\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; P \<Gamma> \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>; \<turnstile> \<Gamma> {c} \<Gamma>'\<rbrakk> \<Longrightarrow> P \<Gamma>' \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>" shows "P x1 x2 x3"
square_integrable_diff: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S" shows "(\<lambda>x. f x - g x) square_integrable S"
RedWhile: fixes b :: "char list exp"   and c :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>while (b) c,s\<rangle> \<rightarrow> \<langle>if (b) (c;; while (b) c) else unit,s\<rangle>"
f': fixes y :: "'b" shows "y \<in> T \<longrightarrow> f' y \<in> S \<and> f (f' y) = y"
C1_differentiable_on_pair: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'b" assumes "f C1_differentiable_on S"   and "g C1_differentiable_on S" shows "(\<lambda>x. (f x, g x)) C1_differentiable_on S"
inf_add_right: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "z +\<^sub>h \<infinity>\<^sub>h = \<infinity>\<^sub>h"
fps_lr_inverse_mult_ring1_1: fixes x :: "'a"   and f :: "'a fps"   and y :: "'a"   and g :: "'a fps" assumes "x * f $ 0 = (1::'a)"   and "f $ 0 * x = (1::'a)"   and "y * g $ 0 = (1::'a)"   and "g $ 0 * y = (1::'a)" shows "fps_left_inverse (f * g) (y * x) = fps_left_inverse g y * fps_left_inverse f x"
bounded_linear: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_clinear f" shows "bounded_linear f"
2: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> topspace (product_topology X I)"   and "\<forall>i\<in>I. contractible_space (X i)" shows "contractible_space (product_topology X I)"
le_oexp': fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "1 < \<alpha>"   and "Ord \<beta>" shows "\<beta> \<le> \<alpha> \<up> \<beta>"
DynCastRed: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list" assumes "redp P E e s e' s'" shows "redp P E (Cast C e) s (Cast C e') s'"
1: shows "local.dom \<pp>\<^sub>0[a, b] = a \<otimes> b"
factorcount_exists: fixes a :: "'a" assumes "a \<in> carrier G" shows "\<exists>c. \<forall>as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> c = length as"
polys_inf_sign_thresholds: fixes ps :: "real poly set" assumes "finite ps" obtains l :: "real" and u :: "real" where "l \<le> u" and "\<And>p. \<lbrakk>p \<in> ps; p \<noteq> 0\<rbrakk> \<Longrightarrow> {x. l < x \<and> x \<le> u \<and> poly p x = 0} = {x. poly p x = 0}" and "\<And>p x. \<lbrakk>p \<in> ps; u \<le> x\<rbrakk> \<Longrightarrow> sgn (poly p x) = poly_inf p" and "\<And>p x. \<lbrakk>p \<in> ps; x \<le> l\<rbrakk> \<Longrightarrow> sgn (poly p x) = poly_neg_inf p"
is_group: shows "Group.group (reduced_homology_group (int p) (nsphere p))"
apply_updates_cons: fixes ra :: "nat"   and r :: "nat"   and u :: "(nat \<times> vname aexp) list"   and ia :: "value list"   and c :: "nat \<Rightarrow>f value option"   and a :: "vname aexp" assumes "ra \<noteq> r" shows "apply_updates u (join_ir ia c) c $ ra = apply_updates ((r, a) # u) (join_ir ia c) c $ ra"
Suffixes_altdef: fixes A :: "'a list set" shows "Suffixes A = (\<Union>w\<in>A. set (suffixes w))"
\<phi>_eq_0: fixes x :: "real"   and i :: "int"   and l :: "nat" assumes "x < 0 \<or> 1 < x"   and "0 < i"   and "i < 2 ^ Suc l" shows "\<phi> (l, i) x = 0"
inf_prefix_correctR: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "i < length (xs \<sqinter> ys)" shows "(xs \<sqinter> ys) ! i = ys ! i"
Mprefix_refines_Mndet_F: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "mndet A P \<sqsubseteq>\<^sub>F Mprefix A P"
B5: shows "trans (Rel \<union> _Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q) \<union> Rel')"
UnE: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
assertion_commute: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "y \<in> conjunctive.conjunctive inf inf times" shows "times y x = times (mbt_algebra.wpt one times inf top (times y x)) y"
d_OUT_restrict_current_outside: fixes x :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_OUT (g \<upharpoonleft> \<Gamma> / f) x = 0"
is_aligned_no_wrap': fixes ptr :: "'b word"   and sz :: "nat"   and off :: "'b word" assumes "is_aligned ptr sz"   and "off < 2 ^ sz" shows "ptr \<le> ptr + off"
wtC_def: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and arOf :: "'f \<Rightarrow> 'e list"   and resOf :: "'f \<Rightarrow> 'e"   and parOf :: "'g \<Rightarrow> 'e list" assumes "Signature TYPE('e) wtFsym wtPsym" shows "Signature.wtC wtFsym wtPsym arOf resOf parOf \<equiv> list_all (Signature.wtL wtFsym wtPsym arOf resOf parOf)"
A: fixes P' :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" shows "(\<forall>n<length ws. P' (ws ! n) (drop (Suc n) ws)) \<longrightarrow> x # xs @ ws \<cong> {ws, zs, P'} \<longrightarrow> \<not> P' x (xs @ ws)"
final: shows "nfa.final \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr> \<subseteq> nfa.states \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr>"
assms_1: shows "\<forall>b'\<in>contours_in_ve ve. Q b'"
rank_new_mat: shows "vec_space.rank (dim_row (take_rows_from_matrix (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))))) (take_rows_from_matrix (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))))) = dim_col (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))"
rbt_insert2: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "rbt t" shows "rbt (RBT_Set.insert x t)"
simps: fixes v :: "'a"   and n :: "nat" shows "time (TM v n) = n"
iIN_cut_ge: fixes n :: "nat"   and d :: "nat"   and t :: "nat" shows "[n\<dots>,d] \<down>\<ge> t = (if t < n then [n\<dots>,d] else if t \<le> n + d then [t\<dots>,n + d - t] else {})"
perm_3: fixes pi :: "(name \<times> name) list"   and x3 :: "ty"   and x1 :: "name"   and x2 :: "trm" shows "pi \<bullet> (\<lambda>x1:x3. x2) = (\<lambda>pi \<bullet> x1:pi \<bullet> x3. pi \<bullet> x2)"
assms: fixes args :: "nat list"   and n :: "nat" assumes "P args" shows "abc_notfinal (abc_steps_l (0, args) p n) p"
monom_mult_uminus_right: fixes c :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.monom_mult c t (- p) = - MPoly_Type_Class.punit.monom_mult c t p"
axioms_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Ide :: "'b \<Rightarrow> bool" assumes "full_inclusion_functor C Ide" shows "category C"
hyps_2: assumes "getCn state' \<noteq> 1" shows "let state'a = applyExplainUIP (applyExplain (getCl state') state') in getM state'a = getM (applyExplain (getCl state') state') \<and> getF state'a = getF (applyExplain (getCl state') state') \<and> getQ state'a = getQ (applyExplain (getCl state') state') \<and> getWatch1 state'a = getWatch1 (applyExplain (getCl state') state') \<and> getWatch2 state'a = getWatch2 (applyExplain (getCl state') state') \<and> getWatchList state'a = getWatchList (applyExplain (getCl state') state') \<and> getConflictFlag state'a = getConflictFlag (applyExplain (getCl state') state') \<and> getConflictClause state'a = getConflictClause (applyExplain (getCl state') state') \<and> getSATFlag state'a = getSATFlag (applyExplain (getCl state') state') \<and> getReason state'a = getReason (applyExplain (getCl state') state')"
urel32_inverse: fixes pp :: "uint32"   and x :: "uint32"   and x' :: "int" assumes "p = int_of_uint32 pp"   and "p \<le> 65535"   and "urel32 x x'" shows "urel32 (inverse_p32 pp x) (inverse_p p x')"
cond_spmf_def: fixes p :: "'a spmf"   and A :: "'a set" shows "cond_spmf p A = (if set_spmf p \<inter> A = {} then return_pmf None else cond_pmf p (Some ` A))"
map_cong: fixes x :: "'a \<times> 'b"   and ya :: "'a \<times> 'b"   and f1 :: "'a \<Rightarrow> 'c"   and g1 :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'd"   and g2 :: "'b \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z1. z1 \<in> fsts ya \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> snds ya \<Longrightarrow> f2 z2 = g2 z2" shows "map_prod f1 f2 x = map_prod g1 g2 ya"
cases: fixes a1 :: "'f set"   and a2 :: "'f set" assumes "a1 \<rhd> a2" obtains M :: "'f set" and N :: "'f set" where "a1 = M" and "a2 = N" and "M - N \<subseteq> Red_F N"
assms_1: fixes x :: "'a"   and s :: "'s" assumes "x \<in> S s" shows "bd_cts (wp (a x))"
NI_unrelated_total_def: shows "NI_unrelated_total \<equiv> \<forall>execs a n. realistic_executions execs \<longrightarrow> (let s_f = run_total n s0 execs in output_f s_f a = output_f (run_total n s0 (purge execs (current s_f))) a \<and> current s_f = current (run_total n s0 (purge execs (current s_f))))"
antisym_conv3: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
simps_3: fixes uv :: "'b extT" shows "(uv \<le> extT.Top) = True"
scc_of_is_scc: fixes E :: "('c \<times> 'c) set"   and v :: "'c" shows "is_scc E (scc_of E v)"
simps_2: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and x :: "nat"   and y :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isPath c u ((x, y) # p) v = (u = x \<and> (x, y) \<in> Graph.E c \<and> Graph.isPath c y p v)"
Limsup_lessD: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'b"   and y :: "'b" assumes "Limsup F f < y" shows "\<forall>\<^sub>F x in F. f x < y"
rep_eq: fixes x :: "nat \<Rightarrow> 'a"   and xa :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "insertion_aux x xa = insertion_fun x (poly_mapping.lookup xa)"
bibd_axioms: shows "bibd \<V> \<B> \<k> \<Lambda>"
matrix_to_iarray_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "matrix_to_iarray A = IArray (map (vec_to_iarray \<circ> ($) A \<circ> mod_type_class.from_nat) [0..<CARD('c)])"
stable_on_CD_on: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_on Pd Ch ds X" shows "Contracts.CD_on Pd ds X = X"
3: shows "- v\<^sup>T \<sqinter> v\<^sup>T * w\<^sup>\<star> \<le> - - v\<^sup>T * (w \<sqinter> - v\<^sup>T) * w\<^sup>\<star>"
\<tau>move2_CASD1: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and e3 :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3)) pc xcp"   and "pc < length (compE2 e1)" shows "\<tau>move2 P h stk e1 pc xcp"
closedin_closure_of: fixes X :: "'b topology"   and S :: "'b set" shows "closedin X (X closure_of S)"
CC': shows "C \<cdot>\<^sub>l\<^sub>s lmbd = C'"
othrow_def: fixes e :: "'e" shows "othrow e = K (Some (Inl e))"
assms_2: shows "t \<in> existence_ivl t0 x0"
eventually_at_filter: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a"   and s :: "'a set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<forall>\<^sub>F x in topological_space.nhds open a. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x)"
distinct_24: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm"   and x31 :: "'b ptrm"   and x32 :: "'b ptrm" shows "PFn x41 x42 x43 \<noteq> PApp x31 x32"
conv_radius_altdef: fixes f :: "nat \<Rightarrow> 'c" shows "conv_radius f = liminf (\<lambda>n. inverse (ereal (root n (norm (f n)))))"
noex: shows "\<forall>f\<in>Collect P. |vertices f| \<le> 4"