hcju/isabelleps huggingface dataset:
Map_dom: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Map (local.dom f) = []"
sound_o: fixes P :: "'c \<Rightarrow> real"   and f :: "'d \<Rightarrow> 'c" assumes "sound P" shows "sound (P \<circ> f)"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> 'univ"   and xa :: "('fsym, 'psym) atm" assumes "satA x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and T1 :: "'fsym trm" and T2 :: "'fsym trm" where "x = \<xi>'__" and "xa = Eq T1 T2" and "int \<xi>'__ T1 = int \<xi>'__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and p :: "'psym" and Tl :: "'fsym trm list" where "x = \<xi>" and "xa = Pr p Tl" and "intP p (map (int \<xi>) Tl)"
closure_minimal: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "closed T" shows "closure S \<subseteq> T"
intro: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and p :: "int"   and a :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP (padic_int p))"   and "a \<in> carrier (padic_int p)"   and "UP_cring.to_fun (padic_int p) f a \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"   and "eint 2 * padic_integers.val_Zp p (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) a) < padic_integers.val_Zp p (UP_cring.to_fun (padic_int p) f a)" shows "hensel_axioms p f a"
SCDNonInterferenceGraph: fixes prog :: "cmd"   and nx :: "w_node" assumes "While_CFGExit.inner_node prog nx" shows "NonInterferenceIntraGraph src trg knd (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) NewEntry (lift_Def (Defs prog) (_Entry_) (_Exit_) H L) (lift_Use (Uses prog) (_Entry_) (_Exit_) H L) id (PDG.PDG_BS src trg (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) (lift_Def (Defs prog) (_Entry_) (_Exit_) H L) (lift_Use (Uses prog) (_Entry_) (_Exit_) H L) (Postdomination.standard_control_dependence src trg (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) NewExit)) NewExit H L (LDCFG_node.Node (_Entry_)) (LDCFG_node.Node (_Exit_))"
distr_id2: fixes M :: "'b measure"   and N :: "'b measure" assumes "sets M = sets N" shows "distr N M (\<lambda>x. x) = N"
filterlim_Suc: shows "filterlim Suc sequentially sequentially"
w_sG: shows "{(x, w), (y, w), (w, x), (w, y)} \<subseteq> parcs sG"
l2_def: fixes a :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list" shows "l2 a b = LINT x|Pi\<^sub>M {..<length a} (\<lambda>d. lborel). \<Phi> a x * \<Phi> b x"
matrix_scalar_mult_def: fixes k :: "'c"   and A :: "(('c, 'd) vec, 'e) vec" shows "k *k A \<equiv> \<chi>i j. k * A $ i $ j"
point_indices_alt_def: fixes point_set :: "'c set"   and block_collection :: "'c set multiset"   and t :: "int" assumes "incidence_system point_set block_collection" shows "incidence_system.point_indices point_set block_collection t = {int (block_collection index ps) |ps. int (card ps) = t \<and> ps \<subseteq> point_set}"
comp_fun_commute: fixes y :: "name"   and x :: "name" shows "subst y (F y) \<circ> subst x (F x) = subst x (F x) \<circ> subst y (F y)"
c_1: shows "\<forall>rs\<in>ran \<Gamma>. no_call_to c rs"
Rep_matrix_inject: fixes x :: "'a matrix"   and y :: "'a matrix" shows "(Rep_matrix x = Rep_matrix y) = (x = y)"
transitive: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<equiv> y"   and "y \<equiv> z" shows "x \<equiv> z"
9: shows "mat (row_length (B1 \<circ> B2)) (length (B1 \<circ> B2)) (B1 \<circ> B2)"
kernel_basis_exists: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "\<exists>B. finite B \<and> vectorspace.basis class_ring (module_vec TYPE('b) nc\<lparr>carrier := mat_kernel A\<rparr>) B"
cong_power_nat_code: fixes b :: "nat"   and e :: "nat"   and x :: "nat"   and m :: "nat" shows "[b ^ e = x] (mod m) = (mod_exp b e m = x mod m)"
distinct3l_def: fixes l :: "Lines"   and m :: "Lines"   and n :: "Lines" shows "distinct3l l m n \<equiv> l \<noteq> m \<and> l \<noteq> n \<and> m \<noteq> n"
luentries_noneD: fixes s :: "('a, 'b) pointermap_scheme"   and a :: "'a" assumes "getentry s a = None"   and "pointermap_sane s" shows "a \<notin> set (entries s)"
assms_5: shows "insertion (nth_default 0 (L[var := B / C])) b = B"
wl_weight_correct: fixes wl :: "'V \<Rightarrow> 'W infty option"   and res :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and v :: "'V"   and w :: "'W infty" assumes "dinvarm (wl, res)"   and "wl v = Some w" shows "path_weight' (\<alpha>r res v) = w"
intersect_segment_xline': fixes prec :: "nat"   and p0 :: "real \<times> real"   and p1 :: "real \<times> real"   and x :: "real"   and m :: "real"   and M :: "real" assumes "intersect_segment_xline' prec (p0, p1) x = Some (m, M)" shows "closed_segment p0 p1 \<inter> {p. fst p = x} \<subseteq> {(x, m)..(x, M)}"
simps_38: shows "cospan_shape.comp cospan_shape.AA cospan_shape.AT = cospan_shape.Null"
similar_mat_witD2_5: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "B \<in> carrier_mat n n"
bi_unique_mod_ring_rel64_1: shows "bi_unique mod_ring_rel64"
gcd_dvd_prod: 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 a :: "'a"   and b :: "'a"   and k :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) (times k b)"
Op: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsInp delta inp"   and "wlsBinp delta binp"   and "liftAll2 phi (arOf delta) inp"   and "liftAll2 (\<lambda>(us, s). phiAbs (us, s)) (barOf delta) binp" shows "phi (stOf delta) (Op delta inp binp)"
assms: fixes b :: "'b"   and c :: "'b" shows "times_abc a (sup_b b c) = sup_c (times_abc a b) (times_abc a c)"
paths_stay_in_winning_region: fixes \<sigma>' :: "'a \<Rightarrow> 'a" assumes "strategy p \<sigma>'"   and "winning_strategy p \<sigma>' (P $ n)"   and "\<And>v. v \<in> winning_region p \<Longrightarrow> \<sigma>' v = \<sigma>' v" shows "lset P' \<subseteq> winning_region p"
m_inv_consistent: fixes H :: "'a set"   and x :: "'a" assumes "subgroup H G"   and "x \<in> H" shows "inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> x = inv x"
mk_mat_def: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'd" shows "local.mk_mat f = (\<chi>i j. f i j)"
abs_eq_iff: fixes x :: "('a, 'b, 'c, 'd) Tree"   and y :: "('a, 'b, 'c, 'd) Tree" shows "(abs_Tree\<^sub>p x = abs_Tree\<^sub>p y) = hull_relp x y"
Aim: shows "A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<noteq> (0::'a)"
s_inter_2: shows "\<phi> C1_differentiable_on {0..1} - s \<inter> {0..1}"
bound: fixes x :: "nat"   and xs :: "nat list"   and s :: "nat" assumes "B < x" shows "C (x # xs) s = False"
greatest_less_ma: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<le> ma"
simple_bochner_integral_diff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f"   and "simple_bochner_integrable M g" shows "simple_bochner_integral M (\<lambda>x. f x - g x) = simple_bochner_integral M f - simple_bochner_integral M g"
outputInduct: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'"   and "\<And>a b P C. F C (a{b}.P) a b P"   and "\<And>P a b P' c C. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C ([c\<frown>c]P) a b P'"   and "\<And>P a b P' c d C. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; \<And>C. F C P a b P'; c \<noteq> d\<rbrakk> \<Longrightarrow> F C ([c\<noteq>d]P) a b P'"   and "\<And>P a b P' Q C. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a b P'"   and "\<And>Q a b Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; \<And>C. F C Q a b Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a b Q'"   and "\<And>P a b P' Q C. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a b (P' \<parallel> Q)"   and "\<And>Q a b Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; \<And>C. F C Q a b Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a b (P \<parallel> Q')"   and "\<And>P a b P' x C. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; x \<sharp> C; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a b (<\<nu>x>P')"   and "\<And>P a b P' C. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; \<And>C. F C (P \<parallel> !P) a b P'\<rbrakk> \<Longrightarrow> F C (!P) a b P'" shows "F C P a b P'"
lossless_def: fixes \<A> :: "('b \<Rightarrow> (('plain \<times> 'plain) \<times> 'c) spmf) \<times> ('commit \<Rightarrow> 'c \<Rightarrow> bool spmf)" shows "lossless \<A> = ((\<forall>pk. lossless_spmf (fst \<A> pk)) \<and> (\<forall>commit \<sigma>. lossless_spmf (snd \<A> commit \<sigma>)))"
contra4: fixes s :: "real \<times> real"   and r :: "real \<times> real"   and q :: "real \<times> real"   and p :: "real \<times> real" assumes "distinct4 s r q p"   and "Counterclockwise_2D_Arbitrary.lex q p"   and "Counterclockwise_2D_Arbitrary.lex p r"   and "Counterclockwise_2D_Arbitrary.lex r s"   and "ccw r q s"   and "ccw r s p"   and "ccw r q p" shows "False"
infsI_sset: fixes w :: "'a stream"   and A :: "'a set" assumes "sset w \<subseteq> A" shows "infs A w"
mapping_of_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b" shows "mapping_of (MPoly_Type.monom m a) = Poly_Mapping.single m a"
Re_image_box: fixes a :: "complex"   and b :: "complex" assumes "Re a < Re b"   and "Im a < Im b" shows "Re ` box a b = {Re a<..<Re b}"
existence_ivl_maximal_segment: fixes x :: "real \<Rightarrow> 'a"   and t0 :: "real"   and t :: "real"   and x0 :: "'a" assumes "(x solves_ode f) {t0--t} X"   and "x t0 = x0"   and "{t0--t} \<subseteq> T" shows "t \<in> existence_ivl t0 x0"
Fvars_\<phi>T: shows "Fvars \<phi>T = {}"
ordinal_rec0_limit_ordinal: fixes x :: "ordinal"   and z :: "'a"   and s :: "ordinal \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "limit_ordinal x" shows "ordinal_rec0 z s l x = (THE y. \<forall>f. (\<forall>n. f n < oLimit f) \<and> oLimit f = x \<longrightarrow> l (\<lambda>n. ordinal_rec0 z s l (f n)) = y)"
Act: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" assumes "P \<rightarrow> \<langle>\<alpha>,P'\<rangle>" shows "P \<rightarrow>\<^sub>S \<langle>S_action.Act \<alpha>,P'\<rangle>"
toCard_pred_def: fixes A :: "'c set"   and r :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "toCard_pred A r f \<equiv> inj_on f A \<and> f ` A \<subseteq> Field r \<and> Card_order r"
assms_3: shows "\<forall>\<^sub>F x in F. g1 x = g2 x"
LIM_zero: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'c)) F"
lb3: shows "real_of_float (lapprox_rat prec 1 3) \<le> 1 / 3"
downset_faithful: fixes x :: "'a"   and y :: "'a" assumes "ord.downset (\<lambda>x y. y \<le> x) x \<subseteq> ord.downset (\<lambda>x y. y \<le> x) y" shows "y \<le> x"
simps_323: fixes x18 :: "float"   and x17 :: "nat" shows "Num x18 \<noteq> Var x17"
assms_2: shows "L-lipschitz_on {b..c} g"
lim_inverse_n': shows "(\<lambda>n. 1 / real n) \<longlonglongrightarrow> 0"
Ints_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<int>" shows "sum f A \<in> \<int>"
assms_1: shows "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|"
\<Sigma>_factor: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and j :: "int"   and l :: "int" shows "\<Sigma> (\<lambda>k. c * f k) j l = c * \<Sigma> f j l"
Gram_Schmidt_upt_k_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "Gram_Schmidt_upt_k A k = foldl Gram_Schmidt_column_k A [0..<Suc k]"
select_convs_1: fixes basepoint :: "'b Point"   and direction :: "'b Vector"   and more :: "'c" shows "basepoint \<lparr>basepoint = basepoint, direction = direction, \<dots> = more\<rparr> = basepoint"
ncols_def: fixes A :: "(('c, 'd) vec, 'e) vec" shows "ncols A = CARD('d)"
Cnext_solve_sporadicon: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and \<Psi> :: "'c TESL_atomic list"   and K\<^sub>1 :: "clock"   and \<tau> :: "'c tag_const"   and K\<^sub>2 :: "clock"   and \<Phi> :: "'c TESL_atomic list" shows "{\<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>, K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>} \<subseteq> \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi>"
assms_8: shows "l \<circ> h \<circ> k = k \<circ> h \<circ> l"
simps_1: fixes ps :: "('c \<times> 'd) list" shows "removeKeyList [] ps = ps"
prems_2: fixes xa :: "nat" assumes "xa \<in> {..<k}"   and "x\<^sub>0 \<le> (ts ! xa) x" shows "gc2 * f2' ((ts ! xa) x) \<le> f ((ts ! xa) x)"
elims_2: fixes x :: "Runs.var set option"   and xa :: "Runs.var set" assumes "in_progressS x xa" obtains S :: "Runs.var set" and S' :: "Runs.var set" where "x = Some S" and "xa = S'" and "S' \<subseteq> S"
assms_1: shows "\<forall>p\<in>ps. (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<in> cbox (x, y) (x + \<delta>, y + \<delta>)"
fps_X_nth: fixes n :: "nat" shows "fps_X $ n = (if n = 1 then 1::'b else (0::'b))"
scene_override_overshadow_right: fixes S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a"   and S\<^sub>3 :: "'a"   and X :: "'a scene" shows "S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 \<oplus>\<^sub>S S\<^sub>3 on X on X = S\<^sub>1 \<oplus>\<^sub>S S\<^sub>3 on X"
bl_bin_bl': fixes n :: "nat"   and bs :: "bool list"   and w :: "int" shows "bin_to_bl (n + length bs) (bl_to_bin_aux bs w) = bin_to_bl_aux n w bs"
simps: fixes a1 :: "('c, 'd) tm"   and a2 :: "('c, 'd) tm" shows "(a1 \<unrhd>\<^sub>e\<^sub>m\<^sub>b a2) = ((\<exists>t. a1 = t \<and> a2 = t) \<or> (\<exists>t u s. a1 = t \<and> a2 = s \<and> t \<unrhd>\<^sub>e\<^sub>m\<^sub>b u \<and> u \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s))"
ncoplanar_perm_18: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet D A B C"
cod_cod: fixes f :: "'a" assumes "C.arr f" shows "C.cod (C.cod f) = C.cod f"
rel_sum_simps_1: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'b"   and b1 :: "'c" shows "rel_sum R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
fps_unit_factor_shift: fixes n :: "nat"   and f :: "'a fps" assumes "n \<le> subdegree f" shows "unit_factor (fps_shift n f) = unit_factor f"
\<xi>': shows "adjoint_equivalence_in_bicategory (\<cdot>\<^sub>S) (\<star>\<^sub>S) S.\<a> S.\<i> S.src S.trg (S.UP f) (S.UP g) (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f)) \<xi>'"
limI: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c" assumes "X \<longlonglongrightarrow> L" shows "lim X = L"
C_wf: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s C"
g_def: shows "g \<equiv> \<lambda>i. SOME p. p \<in> f (Suc i) \<and> (\<exists>q. is_red {p} q \<and> \<not> is_red (f i) q)"
norm_EM_remainder_le_strong_nat: fixes Y :: "real set"   and n :: "nat"   and D :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and C :: "real" assumes "finite Y"   and "\<forall>x. \<bar>pbernpoly n x\<bar> \<le> D"   and "continuous_on {a..} f"   and "continuous_on {a..} g"   and "continuous_on {a..} g'"   and "(g \<longlongrightarrow> C) at_top"   and "\<And>x. a \<le> x \<Longrightarrow> norm (f x) \<le> g' x"   and "\<And>x. x \<in> {a..} - Y \<Longrightarrow> (g has_real_derivative g' x) (at x)" shows "\<forall>x. a \<le> real x \<longrightarrow> norm (EM_remainder n f (int x)) \<le> D / fact n * (C - g (real x))"
add_less_imp_less_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a + c < b + c" shows "a < b"
ereal_le_epsilon: fixes x :: "ereal"   and y :: "ereal" assumes "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e" shows "x \<le> y"
add_closed_converse_right: fixes g :: "'m"   and x :: "'m" assumes "g + x \<in> M"   and "g \<in> M" shows "x \<in> M"
axioms_3: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F"
pref_profile_wf_def: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "pref_profile_wf agents alts R \<equiv> (agents \<noteq> {} \<and> alts \<noteq> {}) \<and> (\<forall>i. i \<in> agents \<longrightarrow> finite_total_preorder_on alts (R i)) \<and> (\<forall>i x y. i \<notin> agents \<longrightarrow> \<not> R i x y)"
sup_pnat: fixes x :: "pnat"   and y :: "pnat" shows "x \<squnion> y \<equiv> pnat_min x y"
in_Chains_finite: fixes M :: "'a set" assumes "M \<in> Chains leq"   and "M \<noteq> {}"   and "finite M" shows "lub M \<in> M"
antisym: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a"   and "a \<le> b" shows "a = b"
compare_expansions_LT: fixes F :: "'a"   and G :: "'a"   and c :: "real"   and c' :: "real"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and g :: "real \<Rightarrow> real" assumes "compare_expansions F G = (LT, c, c')"   and "trimmed G"   and "(f expands_to F) basis"   and "(g expands_to G) basis"   and "basis_wf basis" shows "f \<in> o(g)"
simps_1: fixes uu :: "iface \<Rightarrow> ('a word \<times> nat) list option" shows "iiface_constrain uu MatchAny = MatchAny"
res_add_eq: fixes x :: "int"   and y :: "int" shows "x \<oplus> y = (x + y) mod int p"
converse_rtranclp_induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "P b"   and "\<And>y z. \<lbrakk>r y z; r\<^sup>*\<^sup>* z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
is_dghm_axioms: shows "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
continuous_on_open_avoid: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and a :: "'b" assumes "continuous_on s f"   and "open s"   and "x \<in> s"   and "f x \<noteq> a" shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
done_only_from_recv_marker: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and cid :: "nat"   and p :: "'p"   and q :: "'p" assumes "c \<turnstile> ev \<mapsto> c'"   and "channel cid = Some (p, q)"   and "snd (cs c cid) \<noteq> Done"   and "snd (cs c' cid) = Done" shows "ev = RecvMarker cid q p"
prefixes_rev: fixes xs :: "'a list" shows "prefixes (rev xs) = map rev (suffixes xs)"
max_x_impl': fixes y :: "nat list"   and b :: "nat list"   and a :: "nat list"   and i :: "nat" assumes "length y \<le> length b" shows "max_x_impl' a b y i = hlde_ops.max_x' a b y i"
reachable_SKIP: shows "reachable \<bottom> = UNIV"
set_fst_sinter_with_subset1: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list" shows "set (map fst (sinter_with f xs ys)) \<subseteq> set (map fst xs)"
headR_correct: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t"   and "t \<noteq> Empty" shows "FingerTreeStruc.headR t = last (FingerTreeStruc.toList t)"
exponent_float: fixes m :: "int"   and e :: "int" assumes "odd m"   and "m \<noteq> 0" shows "exponent (float_of (real_of_int m * 2 powr real_of_int e)) = e"
Abs'_def: fixes x :: "int" shows "Abs' x = Abs (x mod int CARD('a))"
PiE_mono: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and C :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi\<^sub>E A B \<subseteq> Pi\<^sub>E A C"
interior_Int: fixes S :: "'a set"   and T :: "'a set" shows "interior (S \<inter> T) = interior S \<inter> interior T"
cd_impl_icd_cd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and l :: "nat"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob i \<pi> l"   and "IFC_def.is_icdi prob i \<pi> k"   and "\<not> IFC_def.is_icdi prob i \<pi> l" shows "IFC_def.is_cdi prob k \<pi> l"
power3_eq_cube: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.monoid_mult one times" shows "power.power one times a 3 = times (times a a) a"
L_def: shows "L = fst ` \<Union> ((edges \<circ> snd) ` Rs) \<union> {S_Bot, S_Top, S_Idt} \<union> S_Const ` C \<union> fst ` edges (fst R)"
exp_complex_eqI: fixes w :: "complex"   and z :: "complex" assumes "\<bar>Im w - Im z\<bar> < 2 * pi"   and "exp w = exp z" shows "w = z"
lm: fixes l :: "nat" assumes "j + 1 + l \<le> n" shows "matrix_sum d f (j + 1 + l) = f j + matrix_sum d f' (j + 1 + l)"
Class_widen: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and T :: "ty" assumes "P \<turnstile> Class C \<le> T" shows "\<exists>D. T = Class D"
noStuckI': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Stuck \<Longrightarrow> False" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Stuck}"
hrp_comp_def: fixes RR' :: "('j \<Rightarrow> 'h \<Rightarrow> assn) \<times> ('j \<Rightarrow> 'i \<Rightarrow> assn)"   and S :: "('j \<times> 'g) set" shows "hrp_comp RR' S \<equiv> (hr_comp (fst RR') S, hr_comp (snd RR') S)"