hcju/isabelleps huggingface dataset:
set_map_3: fixes f1 :: "'d \<Rightarrow> 'b"   and f2 :: "'e \<Rightarrow> 'c"   and f3 :: "'f \<Rightarrow> 'a"   and v :: "('d, 'e, 'f) expr" shows "\<V>\<^sub>E (map_expr f1 f2 f3 v) = f3 ` \<V>\<^sub>E v"
map_inj_on: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and ys :: "'h list" assumes "map f xs = map f ys"   and "inj_on f (set xs \<union> set ys)" shows "xs = ys"
outside_in: fixes y :: "'a"   and t :: "real" assumes "y \<in> outer"   and "0 \<le> t"   and "t \<in> existence_ivl0 y"   and "flow0 y t \<in> closure inner" shows "\<exists>T>0. T \<le> t \<and> (\<forall>s\<in>{0..<T}. flow0 y s \<in> outer) \<and> flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
dsafe_Var: fixes i :: "'b" shows "dsafe (Var i)"
dfg: shows "degree g \<le> degree (Polynomial.smult a f)"
elims: fixes x :: "('b, 'c) RBT_Impl.rbt"   and y :: "'b" assumes "rbt_max_opt x = y" obtains c :: "color" and lt :: "('b, 'c) RBT_Impl.rbt" and k :: "'b" and v :: "'c" where "x = Branch c__ lt__ k__ v__ rbt.Empty" and "y = k__"   | c :: "color" and lt :: "('b, 'c) RBT_Impl.rbt" and k :: "'b" and v :: "'c" and rc :: "color" and rlc :: "('b, 'c) RBT_Impl.rbt" and rk :: "'b" and rv :: "'c" and rrt :: "('b, 'c) RBT_Impl.rbt" where "x = Branch c lt k v (Branch rc rlc rk rv rrt)" and "y = rbt_max_opt (Branch rc rlc rk rv rrt)"   | "x = rbt.Empty" and "y = undefined"
thetaZOParT_ZObis: shows "thetaZOParT \<subseteq> ZObis"
M12: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and B :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'd" shows "\<turnstile> (\<box>[A]_f \<and> \<box>[B]_g) = \<box>[_actrans A f \<and> _actrans B g]_(f, g)"
assms_2: shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd' Ch'"
push_effect_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "push_effect f \<equiv> \<lambda>(u, v). augment_edge f (u, v) (min (excess f u) (cf_of f (u, v)))"
l7: shows "0 < snd p ^ 3 \<and> coprime (fst p ^ 3) (snd p ^ 3)"
take_bit_nonnegative: fixes n :: "nat"   and k :: "int" shows "0 \<le> take_bit n k"
induced_modulus_modulus: shows "\<Phi>.principal.induced_modulus d"
same_level_path_valid_path: 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 as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.same_level_path kind get_return_edges as" shows "CFG.valid_path kind get_return_edges as"
bounded_subset: fixes T :: "'a set"   and S :: "'a set" assumes "bounded T"   and "S \<subseteq> T" shows "bounded S"
that: fixes l :: "nat" assumes "i < l"   and "l < j"   and "tts (s l) t' = HOLD k'" shows "thesis"
swap_neighbors: fixes i :: "nat"   and l :: "'d list" assumes "i + 1 < length l" shows "l[i := l ! (i + 1), i + 1 := l ! i] = take i l @ [l ! (i + 1), l ! i] @ drop (i + 2) l"
add_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d" shows "a \<squnion> c \<le> b \<squnion> d"
apply_def: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "apply f x = f x"
whileLoopE_cond_fail: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and s :: "'d"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> (('e + 'c) \<times> 'd) set \<times> bool" assumes "\<not> C x s" shows "whileLoopE C   B x s = returnOk x s"
continuous_on_linepath: fixes s :: "real set"   and a :: "'a"   and b :: "'a" shows "continuous_on s (linepath a b)"
normalize_poly_old_def: fixes f :: "'c poly" shows "normalize f = Polynomial.smult (inverse (unit_factor (lead_coeff f))) f"
assms_7: shows "l \<circ> h \<circ> g \<le>\<le> g \<circ> h \<circ> l"
per_lemma_case: assumes "\<^bold>|rev z\<^bold>| + \<^bold>|rev y\<^bold>| \<le> \<^bold>|rev y \<^sup>@ b\<^bold>|" shows "rev y \<cdot> rev x = rev x \<cdot> rev y"
is_sublist_id: fixes vs :: "'a list" shows "is_sublist vs vs"
process_mru: shows "process_mru (votes sa) p = Some (r, v)"
hi_up_eq_un: shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) neg up = un [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> Brouwer_degree2 (n - Suc 0) neg"
One_fract_def: shows "1 = Fraction_Field.Fract (1::'c) (1::'c)"
simps_7: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and i :: "nat" shows "\<lbrakk> K\<^sub>1 strictly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> i\<^esup> = {\<rho>. \<forall>n\<ge>i. #\<^sub>\<le> \<rho> K\<^sub>2 n \<le> #\<^sub>< \<rho> K\<^sub>1 n}"
atLeastatMost_empty_iff2: fixes a :: "'b"   and b :: "'b" shows "({} = {a..b}) = (\<not> a \<le> b)"
surjective_pairing: fixes t :: "'b \<times> 'c" shows "t = (fst t, snd t)"
sheaf_morphisms_sheaf_spec: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "s \<in> comm_ring.sheaf_spec R addition multiplication zero unit U" shows "comm_ring.sheaf_spec_morphisms R addition multiplication zero unit U U s = s"
subst_bv1_bind_fv2: fixes t :: "Core.term"   and lev :: "nat"   and v :: "variable"   and T :: "typ" assumes "\<not> loose_bvar t lev" shows "subst_bv1 (bind_fv2 (v, T) lev t) lev (Fv v T) = t"
inner_same_Basis: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and u :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "inner u u = 1"
mult_isol_var: fixes u :: "'b \<times> 'c"   and x :: "'b \<times> 'c"   and v :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "less_eq_prod u x"   and "less_eq_prod v y" shows "less_eq_prod (times_prod' u v) (times_prod' x y)"
IH: fixes x :: "'a"   and ys :: "'a list" assumes "x \<noteq> y" shows "cnt y (remove_cycles xs x ys) \<le> cnt y ys + cnt y xs"
compact_LConsD: fixes x :: "'c"   and xs :: "'c llist" assumes "ccpo.compact lSup (\<sqsubseteq>) (LCons x xs)" shows "ccpo.compact lSup (\<sqsubseteq>) xs"
leg1_is_map: shows "is_left_adjoint (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)"
h0: fixes i :: "nat" assumes "i \<le> k" shows "h i \<noteq> 0"
infinite_le_lepoll: fixes A :: "'b set" shows "infinite A = (UNIV \<lesssim> A)"
orderE: fixes a :: "'b list"   and b :: "'b list" assumes "prefix a b" obtains "a = a \<sqinter> b"
WbisT_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> WretrT (theta \<union> WbisT)" shows "theta \<subseteq> WbisT"
unit_disc_fix_decompose_blaschke_rotation: fixes M :: "moebius" assumes "unit_disc_fix M" shows "\<exists>k \<phi>. cmod k < 1 \<and> M = moebius_rotation \<phi> + blaschke k"
idx_correct: fixes xs :: "'a list"   and y :: "'a"   and n :: "nat" assumes "distinct xs" shows "(idx xs y = Some n) = (n < length xs \<and> xs ! n = y)"
block_set_nempty_imp_points: assumes "\<B> \<noteq> {#}" shows "\<V> \<noteq> {}"
run_return_writer: fixes return :: "'c \<times> 'b list \<Rightarrow> 'a"   and x :: "'c" shows "run_writer (return_writer return x) = return (x, [])"
assms: shows "\<forall>x. S1 x \<le> S2 x"
array_shr_def: fixes a :: "'b array"   and i :: "nat"   and k :: "nat" shows "array_shr a i k \<equiv> Array.len a \<bind> (\<lambda>l. safe_sblit a i (i + k) (l - (i + k)))"
simps: fixes a1 :: "sequent list \<times> sequent"   and a2 :: "form" shows "leftPrincipal a1 a2 = (\<exists>C A Ps. a1 = (Ps, C) \<and> a2 = A \<and> C = ( \<LM> A  \<RM> \<Rightarrow>* \<Empt>) \<and> A \<noteq> ff)"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b + (a + c) = a + (b + c)"
assms: shows "s1 \<subseteq>\<^sub>f s2"
nat_filtration_not_borel_info': fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b measure" assumes "f \<in> nat_filtration n \<rightarrow>\<^sub>M N"   and "set_discriminating n f N" shows "f \<circ> pseudo_proj_False n = f"
bindU_returnU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'d \<rightarrow> udom\<cdot>'d"   and returnU :: "udom \<rightarrow> udom\<cdot>'d"   and bindU :: "udom\<cdot>'d \<rightarrow> (udom \<rightarrow> udom\<cdot>'d) \<rightarrow> udom\<cdot>'d"   and x :: "udom"   and f :: "udom \<rightarrow> udom\<cdot>'d" assumes "class.monad fmapU returnU bindU" shows "bindU\<cdot>(returnU\<cdot>x)\<cdot>f = f\<cdot>x"
poly_pos_between_leq_leq: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> 0 < poly p x) = ((b < a \<or> 0 < poly p a) \<and> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> poly p x \<noteq> 0))"
pseudo_proj_True_img: fixes n :: "nat"   and w :: "bool stream" assumes "pseudo_proj_True n w = w" shows "w \<in> range (pseudo_proj_True n)"
LeadsTo_UN_UN_noindex: fixes F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set" assumes "\<And>i. F \<in> A i \<longmapsto>w A' i" shows "F \<in> \<Union> (range A) \<longmapsto>w \<Union> (range A')"
exp_inv_limit_0_right: shows "((\<lambda>t. exp (- inverse t)) \<longlongrightarrow> 0) (at_right 0)"
FreeUltrafilterNat_HNatInfinite: fixes X :: "nat \<Rightarrow> nat" assumes "\<forall>u. \<forall>\<^sub>F n in \<U>. u < X n" shows "star_n X \<in> HNatInfinite"
12384: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<sqinter> - y = - (x \<squnion> y)"
Re_winding_number_poly_linepth: fixes pp :: "complex poly" assumes "lead_coeff pp = 1"   and "\<forall>x\<in>proots pp. Im x \<noteq> 0" shows "((\<lambda>r. 2 * Re (winding_number (poly pp \<circ> (\<lambda>x. complex_of_real (linepath (- r) (of_real r) x))) 0) + cindex_pathE (poly pp \<circ> (\<lambda>x. complex_of_real (linepath (- r) (of_real r) x))) 0) \<longlongrightarrow> 0) at_top"
subringE_1: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "subring H R" shows "H \<subseteq> carrier R"
comparable: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and sgn :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.nice_ordered_field plus minus less_eq less zero times abs uminus divide inverse one sgn"   and "less_eq a c \<or> less_eq c a"   and "less_eq b c \<or> less_eq c b" shows "less_eq a b \<or> less_eq b a"
Mojmir_rabin_smallest_accepting_rank: shows "\<MM>.smallest_accepting_rank = \<MM>.smallest_accepting_rank\<^sub>\<R>"
simps: fixes a1 :: "trm"   and a2 :: "trm" shows "a1 \<Down> a2 = ((\<exists>x e. a1 = Lam [x].e \<and> a2 = Lam [x].e) \<or> (\<exists>x e\<^sub>1 e\<^sub>2 e' e e\<^sub>2'. a1 = App e\<^sub>1 e\<^sub>2 \<and> a2 = e' \<and> x \<sharp> (e\<^sub>1, e\<^sub>2, e') \<and> e\<^sub>1 \<Down> Lam [x].e \<and> e\<^sub>2 \<Down> e\<^sub>2' \<and> e[x::=e\<^sub>2'] \<Down> e'))"
nonint_def: shows "nonint \<equiv> \<forall>ucl. \<forall>u\<in>GL. out (doo st0 ucl) u = out (doo st0 (purge GH ucl)) u"
no_free_return: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Return], Undecided\<rangle> \<Rightarrow> t"   and "matches \<gamma> m p" shows "False"
j_max: shows "Max {i. suffix i w \<Turnstile>\<^sub>n \<psi>} < j"
simps: fixes P :: "(char list list \<times> char list exp) prog"   and a1 :: "char list exp list"   and a2 :: "heap \<times> sfields \<times> sheap"   and a3 :: "char list exp list"   and a4 :: "heap \<times> sfields \<times> sheap" shows "P \<turnstile> \<langle>a1,a2\<rangle> [\<Rightarrow>] \<langle>a3,a4\<rangle> = ((\<exists>s. a1 = [] \<and> a2 = s \<and> a3 = [] \<and> a4 = s) \<or> (\<exists>e s\<^sub>0 v s\<^sub>1 es es' s\<^sub>2. a1 = e # es \<and> a2 = s\<^sub>0 \<and> a3 = Val v # es' \<and> a4 = s\<^sub>2 \<and> P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle> \<and> P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>) \<or> (\<exists>e s\<^sub>0 e' s\<^sub>1 es. a1 = e # es \<and> a2 = s\<^sub>0 \<and> a3 = throw e' # es \<and> a4 = s\<^sub>1 \<and> P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>))"
fold_coeffs_0_eq: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "fold_coeffs f 0 = id"
lubH_le_flubH: fixes H :: "'a set" assumes "H = {x. (x, f x) \<in> r \<and> x \<in> A}" shows "(lub H cl, f (lub H cl)) \<in> r"
bd_def: shows "bd = fst (subset_dfs A (nfa_startnode A))"
Mertens_lemma: fixes n :: "nat" assumes "n \<noteq> 0" shows "\<bar>(\<Sum>d = 1..n. mangoldt d / real d) - ln (real n)\<bar> \<le> 4"
has_white_path_to_eq_imp': fixes x :: "'a"   and y :: "'a"   and s' :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and s :: "'g process_name \<Rightarrow> ('e, 'g, 'f, 'a) Model.local_state" assumes "(x has_white_path_to y) s'"   and "\<forall>r'. \<Union> (ran ` obj_fields ` set_option (sys_heap s' r')) = \<Union> (ran ` obj_fields ` set_option (sys_heap s r'))"   and "\<forall>r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r')"   and "sys_fM s' = sys_fM s" shows "(x has_white_path_to y) s"
pair_measure_null_product: fixes M :: "'b measure"   and N :: "'c measure" assumes "emeasure M (space M) = 0" shows "emeasure (M \<Otimes>\<^sub>M N) (space (M \<Otimes>\<^sub>M N)) = 0"
Rep_reg_lan: fixes x :: "'a reg_lan" shows "Rep_reg_lan x \<in> range lang"
prod_frame_match: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a \<times> 'c"   and y :: "'b \<times> 'd"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and B' :: "'c \<Rightarrow> 'd \<Rightarrow> assn" assumes "hn_ctxt A (fst x) (fst y) \<Longrightarrow>\<^sub>t hn_ctxt A' (fst x) (fst y)"   and "hn_ctxt B (snd x) (snd y) \<Longrightarrow>\<^sub>t hn_ctxt B' (snd x) (snd y)" shows "hn_ctxt (A \<times>\<^sub>a B) x y \<Longrightarrow>\<^sub>t hn_ctxt (A' \<times>\<^sub>a B') x y"
a_host_shadow_root_rel_host: fixes h :: "(_) heap"   and shadow_root :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<in> local.a_host_shadow_root_rel h"
f: shows "mcont lub ord lSup (\<sqsubseteq>) f"
closure_eq: fixes S :: "'b set" shows "(closure S = S) = closed S"
phi_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and phis :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option"   and g :: "'d"   and v :: "'c" shows "CFG_SSA_wf_base.phi \<alpha>n defs phis g v = phis g (CFG_SSA_wf_base.defNode \<alpha>n defs phis g v, v)"
Dynkin_def: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "Dynkin \<Omega> M = \<Inter> {D. Dynkin_system \<Omega> D \<and> M \<subseteq> D}"
nth_mask: fixes n :: "nat"   and i :: "nat" shows "mask n !! i = (i < n \<and> i < size (mask n))"
bin_code: fixes u\<^sub>0 :: "'b list"   and u\<^sub>1 :: "'b list"   and us :: "'b list list"   and vs :: "'b list list" assumes "binary_code u\<^sub>0 u\<^sub>1"   and "us \<in> lists {u\<^sub>0, u\<^sub>1}"   and "vs \<in> lists {u\<^sub>0, u\<^sub>1}"   and "concat us = concat vs" shows "us = vs"
assms_2: fixes x :: "'a" shows "h x \<le> 1"
open_Collect_positive: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "continuous_on s f" shows "\<exists>A. open A \<and> A \<inter> s = {x \<in> s. 0 < f x}"
when_def: fixes zero :: "'q"   and a :: "'q"   and P :: "bool" shows "zero.when zero a P = (if P then a else zero)"
Delta_null_same_emeasure: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure" assumes "A \<Delta> B \<in> null_sets M"   and "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M A = emeasure M B"
ensuresI: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A - B co A \<union> B"   and "F \<in> transient (A - B)" shows "F \<in> A ensures B"
IO: fixes out :: "'b"   and c :: "('a, 'b, 'c) rpv"   and gpv :: "('a, 'b, 'c) gpv"   and input :: "'c"   and \<Gamma> :: "('b, 'c) \<I>"   and x :: "'a" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out"   and "x \<in> results_gpv \<Gamma> (c input)" shows "x \<in> results_gpv \<Gamma> gpv"
eqv: fixes p :: "perm" assumes "p \<bullet> T = S"   and "p \<bullet> ba xs = ba ys"   and "supp p \<subseteq> set (ba xs) \<union> set (ba ys)" shows "p \<bullet> f xs T = f ys S"
assms: shows "(l, a) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)"
diff_self: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus a a = zero"
that_3: assumes "prime (gauss_int_norm x)"   and "[gauss_int_norm x = 1] (mod 4)"   and "0 < ReZ x"   and "0 < ImZ x"   and "ReZ x \<noteq> ImZ x" shows "thesis"
d_delta_lnexp_cf7: fixes x :: "real" assumes "0 < numer_cf7 x"   and "0 < numer_cf7 (- x)" shows "((\<lambda>x. ln (exp_cf7 x) - x) has_real_derivative diff_delta_lnexp_cf7 x) (at x)"
X: shows "cop ds \<union> X'' \<subseteq> fp_cop_F ds"
field_differentiable_imp_differentiable: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a filter" assumes "f field_differentiable F" shows "f differentiable F"
on_circline_unit_circle: fixes z :: "complex" shows "on_circline unit_circle (of_complex z) = (cmod z = 1)"
cindexE_ubd_def: fixes f :: "real \<Rightarrow> real" shows "cindexE_ubd f = (\<Sum>x | jumpF f (at_right x) \<noteq> 0. jumpF f (at_right x)) - (\<Sum>x | jumpF f (at_left x) \<noteq> 0. jumpF f (at_left x))"
induct: fixes E :: "('a \<times> 'a) set"   and x1 :: "'a"   and x2 :: "'a list"   and x3 :: "'a"   and P :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a \<Rightarrow> bool" assumes "path E x1 x2 x3"   and "\<And>u. P u [] u"   and "\<And>u v l w. \<lbrakk>(u, v) \<in> E; path E v l w; P v l w\<rbrakk> \<Longrightarrow> P u (u # l) w" shows "P x1 x2 x3"
set_bset_eqvt: fixes p :: "perm"   and B :: "'a set['b]" shows "p \<bullet> set_bset B = set_bset (p \<bullet> B)"
L3_5v: fixes b :: "'b set"   and i :: "'b set"   and c :: "'b com" shows "SEM (While b i c) = (\<lambda>x. \<Union>k. SEM (fwhile b c k) x)"
preserves_reflects_arr: fixes f :: "'b \<times> 'b" shows "B.hseq (fst f) (snd f) = B.VV.arr f"
Abs_run_cases: fixes x :: "'b run" obtains y :: "nat \<Rightarrow> clock \<Rightarrow> bool \<times> 'b tag_const" where "x = Abs_run y" and "y \<in> {\<rho>. \<forall>c. mono (\<lambda>n. time (\<rho> n c))}"
vinsert_set_insert_eq: fixes B :: "V set"   and a :: "V"   and b :: "V" assumes "small B" shows "set (insert a (insert b B)) = vinsert a (set (insert b B))"
lc_upd: fixes t :: "sterm"   and l :: "Label"   and s :: "sterm" shows "lc (Upd t l s) = (lc t \<and> body s)"
norm_le: fixes x :: "'a"   and y :: "'b" shows "(norm x \<le> norm y) = (x \<bullet> x \<le> y \<bullet> y)"
simps_4: fixes z :: "octo" shows "Im3 (Octonions.cnj z) = - Im3 z"
trace_parallel_plan_strips_length_gt_one_if: fixes k :: "nat"   and I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>) - 1" shows "1 < length (trace_parallel_plan_strips I \<pi>)"
chop_always_possible: shows "\<forall>i. \<exists>j k. N_Chop(i,j,k)"
minus_mod_self2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus a b) b = modulo a b"
monoidal_category_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d" shows "monoidal_category C T \<alpha> \<iota> \<equiv> ((category C \<and> product_category C C) \<and> product_category C (product_category.comp C C) \<and> binary_endofunctor C T) \<and> (natural_isomorphism (product_category.comp C (product_category.comp C C)) C (binary_endofunctor.ToTC C T) (binary_endofunctor.ToCT C T) \<alpha> \<and> equivalence_functor C C (\<lambda>f. T (partial_magma.cod C \<iota>, f))) \<and> equivalence_functor C C (\<lambda>f. T (f, partial_magma.cod C \<iota>)) \<and> monoidal_category_axioms C T \<alpha> \<iota>"
Times_def: fixes r :: "'b language"   and s :: "'b language" shows "Times r s = Times_Plus (TimesLR r s)"
continuous_on_compose2: fixes t :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "continuous_on t g"   and "continuous_on s f"   and "f ` s \<subseteq> t" shows "continuous_on s (\<lambda>x. g (f x))"
fps_lr_inverse_one_one_1: shows "fps_left_inverse 1 (1::'a) = 1"
imp_dfa_def: shows "imp_dfa = binop_dfa (\<longrightarrow>)"
Fun_in_terms_of_comp: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f" shows "set_category_given_img.Fun S (set_category.some_img S \<AA>) f = restrict (set_category.img S \<AA> \<circ> S f \<circ> set_category.mkPoint S \<AA> (partial_magma.dom S f)) (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f))"
gAnd_commute: fixes a :: "'a gexp"   and b :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (gAnd a b) s = gval (gAnd b a) s"
mem_diff: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc"   and "\<forall>x. var_asm_not_written mds' x \<longrightarrow> mem' x = mem'' x \<and> dma mem' x = dma mem'' x" shows "\<langle>c', mds', mem''\<rangle> \<in> loc_reach lc"
ltl_models_equiv_prop_entailment: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> = {\<psi>. w \<Turnstile>\<^sub>n \<psi>} \<Turnstile>\<^sub>P \<phi>"