hcju/isabelleps huggingface dataset:
map\<^sub>0_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>D (F.map\<^sub>0 a) = F.map\<^sub>0 a"
mpi_less_Arg: fixes z :: "complex" shows "- pi < Arg z"
a_def: shows "a = r \<circ> s \<circ> t"
w\<^sub>g_def: shows "w\<^sub>g = f\<^sub>0g\<^sub>1.p\<^sub>0 \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1"
Max_V_viterbi: fixes \<O>\<^sub>s :: "'a set"   and \<K> :: "'b \<Rightarrow> 'b pmf"   and \<O> :: "'b \<Rightarrow> 'a pmf"   and \<S> :: "'b set"   and s :: "'b"   and os :: "'a list"   and o :: "'a" assumes "HMM2 \<O>\<^sub>s \<K> \<O> \<S>"   and "s \<in> \<S>" shows "(MAX t\<in>\<S>. HMM2.viterbi_prob \<K> \<O> \<S> s t os) = Max {emeasure (MC_syntax.T (HMM_defs.K \<K> \<O>) (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'} |as. length as = length os \<and> set as \<subseteq> \<S>}"
simps_2: fixes s :: "('c, 'd) fmap"   and pfx_a :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "rem_condless_act s pfx_a (a # as) = (if fst a \<subseteq>\<^sub>f exec_plan s pfx_a then rem_condless_act s (pfx_a @ [a]) as else rem_condless_act s pfx_a as)"
four_points_on_circline_iff_cross_ratio_real: fixes z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "distinct [z, u, v, w]" shows "is_real (to_complex (cross_ratio z u v w)) = (\<exists>H. {z, u, v, w} \<subseteq> circline_set H)"
that: fixes c :: "('a, 'b) chart" assumes "c \<in> atlas"   and "x \<in> domain c"   and "domain c \<subseteq> X" shows "thesis"
unsat_wSts0: shows "\<not> satisfiable (grounding_of_clss (wrp.N_of_wstate (lhd wSts) \<union> wrp.P_of_wstate (lhd wSts) \<union> wrp.Q_of_wstate (lhd wSts)))"
not_one_less_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one zero"
simps_5: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and x2 :: "'g" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
assms_1: shows "finite {t. f t = 0 \<and> a < t \<and> t < b}"
card_older_senior_senior_states: fixes x :: "nat"   and q :: "'b"   and n :: "nat" assumes "x \<in> configuration q n" shows "card (older_seniors x n) = card (senior_states q n)"
intros_2: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
separatedin_def: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "separatedin X S T \<equiv> S \<subseteq> topspace X \<and> T \<subseteq> topspace X \<and> S \<inter> X closure_of T = {} \<and> T \<inter> X closure_of S = {}"
join_comm: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
equiv_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "Equiv_Relations.equiv A r = (refl_on A r \<and> sym r \<and> trans r)"
Det_Sim: fixes P :: "('c, 'd) action"   and S :: "'e \<Rightarrow> 'd"   and Q :: "('c, 'e) action"   and P' :: "('c, 'd) action"   and Q' :: "('c, 'e) action" assumes "P \<preceq>S Q"   and "P' \<preceq>S Q'" shows "P \<box> P' \<preceq>S Q \<box> Q'"
T_subset: shows "T \<subseteq> {butterfly a b |a b. a \<in> basisA \<and> b \<in> basisB}"
bijective_def: fixes f :: "'a \<Rightarrow> 'b" shows "bijective f \<equiv> injective f \<and> surjective f"
incoming_outside: fixes x :: "nat" assumes "x \<notin> cf.V" shows "cf.incoming x = {}"
servK_notin_authKeysD: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and A :: "agent"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace>\<rbrace> \<in> parts (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV" shows "servK \<notin> authKeys evs"
return_time_pos: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x"   and "closed P" shows "0 < auto_ll_on_open.return_time f X P x"
id_upt_nrows_mat_1: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "id_upt_k A (nrows A)" shows "A = mat (1::'b)"
a1: shows "supported_instruction (fst i) = True \<and> fst i \<noteq> ctrl_type RETT"
mat_col_eqI: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list" assumes "mat nr nc m1"   and "mat nr nc m2"   and "\<And>i. i < nc \<Longrightarrow> col m1 i = col m2 i" shows "m1 = m2"
distinct_1: fixes x1 :: "name"   and x21 :: "Lambda_Z.term"   and x22 :: "Lambda_Z.term" shows "Var x1 \<noteq> App x21 x22"
lemDirectionCollinear: fixes u :: "'a Point"   and v :: "'a Point"   and d :: "'a Vector" shows "collinear u v (v \<leadsto> d) = (\<exists>\<beta>. from u to v = (- \<beta>) ** d)"
Re_2: shows "(Im (Arccos w))\<^sup>2 \<le> (cmod w)\<^sup>2"
pre_discover_def: fixes u :: "'v"   and v :: "'v"   and s0 :: "'s"   and s :: "'s" shows "pre_discover u v s0 s \<equiv> post_get_pending u (Some v) s0 s \<and> \<not> gds_is_discovered gds v s"
r_auxhyp_prim: shows "prim_recfn 1 r_auxhyp"
case_unit_unfold: fixes x :: "'a"   and u :: "unit" shows "case u of () \<Rightarrow> x \<equiv> x"
gij: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "g i \<subseteq> g j"
assms: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>"
assms: shows "s' = snd (fst (shift_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
le_\<I>_def: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('a, 'b) \<I>" shows "(\<I> \<le> \<I>') = (outs_\<I> \<I> \<subseteq> outs_\<I> \<I>' \<and> (\<forall>x\<in>outs_\<I> \<I>. responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x))"
neq: fixes x :: "real" assumes "0 \<le> x" shows "s + complex_of_real x \<noteq> 0"
bij_enumerate: fixes S :: "nat set" assumes "infinite S" shows "bij_betw (enum S) UNIV S"
simps_2: fixes \<sigma> :: "nat \<Rightarrow> val"   and x :: "nat" shows "cexpr_sem \<sigma> (CVar x) = \<sigma> x"
restrict_extensional_sub: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "A \<subseteq> B" shows "restrict f A \<in> extensional B"
assms: fixes F :: "('a, 'b) labeled_graph"   and f :: "('b \<times> 'c) set" assumes "finite_graph F"   and "subgraph F (fst R)"   and "extensible (F, fst R) G f"   and "graph_homomorphism F G f" shows "extensible (F, snd R) G f"
simps_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(Nom x2 = Nom y2) = (x2 = y2)"
s': shows "S \<subseteq> {x. a' \<bullet> x \<le> b'}"
linorder_neqE_linordered_idom: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "x \<noteq> y" obtains "less x y"   | "less y x"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> B" shows "x \<bullet> y = 0"
flattening_lemma_i: fixes \<pi> :: "('c, 'd) sas_plus_operator list list"   and \<Psi> :: "('c, 'd) sas_plus_problem" assumes "\<forall>ops\<in>set \<pi>. \<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)" shows "\<forall>op\<in>set (concat \<pi>). op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
INFM_EX: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>\<^sub>\<infinity>x. P x" shows "\<exists>x. P x"
is_irred_map_scale: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'c" assumes "\<not> is_red F p" shows "\<not> is_red F (c \<cdot> p)"
odd_one: 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" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) one"
case_3: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b" shows "(case Protected of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f3"
fls_lr_inverse_X_inv_power_1: fixes n :: "nat"   and x :: "'a" shows "fls_left_inverse (fls_X_inv ^ n) x = fls_shift (- int n) (fls_const x)"
Scons_Scons_eq: fixes M :: "('b, 'c) Old_Datatype.node set"   and N :: "('b, 'c) Old_Datatype.node set"   and M' :: "('b, 'c) Old_Datatype.node set"   and N' :: "('b, 'c) Old_Datatype.node set" shows "(Scons M N = Scons M' N') = (M = M' \<and> N = N')"
root_int_main_2: fixes n :: "int"   and p :: "nat"   and y :: "int"   and b :: "bool" assumes "0 \<le> n"   and "root_int_main p n = (y, b)" shows "b = (y ^ p = n)"
parts_trans: fixes X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "X \<in> parts G"   and "G \<subseteq> parts H" shows "X \<in> parts H"
resolve_action_schema_def: fixes n :: "char list" shows "resolve_action_schema n = index_by ast_action_schema.name (actions D) n"
strand_vars_split_3: fixes S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list" shows "fv\<^sub>s\<^sub>t (S @ S') = fv\<^sub>s\<^sub>t S \<union> fv\<^sub>s\<^sub>t S'"
ucast_down_wi: fixes x :: "int" assumes "is_down ucast" shows "ucast (word_of_int x) = word_of_int x"
simps: fixes s :: "('a, 'b option) insert_message list"   and l :: "nat"   and u :: "nat"   and w :: "('a, 'b option) insert_message" shows "concurrent s l u w = idx s (P w) \<bind> (\<lambda>p_pos. idx s (S w) \<bind> (\<lambda>s_pos. Inr (if p_pos \<le> l \<and> u \<le> s_pos then [\<lbrakk>I w\<rbrakk>] else [])))"
learn_bc_extra_wrt_goedel: fixes \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and extra :: "(nat \<Rightarrow> nat option) set \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> bool" assumes "goedel_numbering \<chi>"   and "learn_bc \<psi> U s"   and "\<And>f n. f \<in> U \<Longrightarrow> extra U f n (\<psi> (the (s (f \<triangleright> n))))" shows "\<exists>t. learn_bc \<chi> U t \<and> (\<forall>f\<in>U. \<forall>n. extra U f n (\<chi> (the (t (f \<triangleright> n)))))"
assms_4: fixes x :: "Vars.var"   and e :: "exp"   and exp :: "exp" assumes "x \<notin> fv e"   and "P e"   and "P exp" shows "P (let x be e in exp )"
maintained_holds_iff: fixes G :: "('b, 'c) labeled_graph"   and e\<^sub>L :: "'b allegorical_term"   and e\<^sub>R :: "'b allegorical_term" assumes "graph G" shows "maintained (translation e\<^sub>L, translation (A_Int e\<^sub>L e\<^sub>R)) G = G \<Turnstile> e\<^sub>L \<sqsubseteq> e\<^sub>R"
Rstep_CohideR: fixes j :: "nat"   and A :: "('sf, 'sc, 'sz) formula list"   and S :: "('sf, 'sc, 'sz) formula list" shows "Rrule_result CohideR j (A, S) = [(A, [S ! j])]"
substitutes_on_bilateral_substitutes_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "substitutes_on A f" shows "bilateral_substitutes_on A f"
lex_pp_refl: fixes s :: "('a, 'b) pp" shows "lex_pp s s"
segment_not_inc_iff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq> b = (b \<notin> segment D a)"
group_by'_def: fixes f :: "'f \<Rightarrow> 'd \<times> 'e"   and s :: "'f set" shows "group_by' f s = (\<lambda>a. let (fa, uu_) = f a in (fa, (snd \<circ> f) ` Set.filter (\<lambda>a'. fa = fst (f a')) s)) ` s"
Dag_unique_all_impl_simp: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag"   and P :: "dag \<Rightarrow> bool" assumes "Dag p l r t" shows "(\<forall>t. Dag p l r t \<longrightarrow> P t) = P t"
count_list_append: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a" shows "count_list (xs @ ys) x = count_list xs x + count_list ys x"
Cons_both: fixes x :: "'b"   and xs :: "'b list"   and t' :: "'b ttree"   and t :: "'b ttree" shows "(x # xs \<in> paths (t' \<otimes>\<otimes> t)) = (if possible t' x \<and> possible t x then xs \<in> paths (nxt t' x \<otimes>\<otimes> t) \<or> xs \<in> paths (t' \<otimes>\<otimes> nxt t x) else if possible t' x then xs \<in> paths (nxt t' x \<otimes>\<otimes> t) else if possible t x then xs \<in> paths (t' \<otimes>\<otimes> nxt t x) else False)"
simps_1: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" shows "is_path_undir G v [] v' = (v = v' \<and> v' \<in> nodes G)"
flat_ord_def: fixes b :: "'a"   and x :: "'a"   and y :: "'a" shows "flat_ord b x y = (x = b \<or> x = y)"
less_eq_vec_def: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(x \<le> y) = (\<forall>i. x $h i \<le> y $h i)"
intros_3: fixes v :: "'a"   and w :: "'a"   and vs :: "'a list" assumes "v \<rightarrow> w"   and "walk (w # vs)" shows "walk (v # w # vs)"
ereal_infty_mult: fixes a :: "ereal" shows "\<infinity> * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
sorted_spvec_add_spvec: fixes a :: "(nat \<times> 'a) list"   and b :: "(nat \<times> 'a) list" assumes "sorted_spvec a"   and "sorted_spvec b" shows "sorted_spvec (add_spvec a b)"
other_stepsE: fixes U :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and J :: "'a set"   and \<sigma> :: "'a \<Rightarrow> 'b"   and \<sigma>' :: "'a \<Rightarrow> 'b" assumes "other_steps U J"   and "U \<sigma> \<sigma>'" shows "\<forall>j\<in>J. \<sigma>' j = \<sigma> j"
le_cong_l: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x .= y"   and "y \<sqsubseteq> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq> z"
foundation: shows "wf {(x, y). x \<in> elts y}"
Atm_Sbis: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and atm :: "'b"   and tval :: "'c \<Rightarrow> 'a \<Rightarrow> bool" assumes "PL_Indis indis"   and "PL_Indis.compatAtm aval indis atm" shows "(Atm atm, Atm atm) \<in> PL_Indis.Sbis tval aval indis"
l4: shows "\<not> (abscissa A = abscissa D \<and> ordinate A = ordinate D)"
at_right_to_top: shows "at_right 0 = filtermap inverse at_top"
tl_lem1: fixes l :: "'a list" assumes "l \<noteq> []"   and "tl l = []" shows "length l = 1"
Op2_eq_b_least2: shows "loc_Op2 = b_least2"
IH_2: fixes i :: "nat" assumes "i < n" shows "\<bar>v $ i\<bar>\<^sup>2 \<le> \<parallel>v\<parallel>\<^sup>2"
seqI: fixes g1 :: "'a1"   and f1 :: "'a1"   and g2 :: "'a2"   and f2 :: "'a2" assumes "A1.seq g1 f1 \<and> A2.seq g2 f2" shows "A1xA2.seq (g1, g2) (f1, f2)"
functional_converse_relation: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "left_unique R"   and "right_total R" shows "\<forall>y. \<exists>!x. R x y"
i_is_x_axis: fixes i :: "real \<times> real"   and j :: "real \<times> real" assumes "R2 i j" shows "i = (1, 0)"
case_2: fixes f1 :: "name \<Rightarrow> 'a"   and f2 :: "name \<Rightarrow> 'a"   and x2 :: "name" shows "(case BoundOutputS x2 of InputS x \<Rightarrow> f1 x | BoundOutputS x \<Rightarrow> f2 x) = f2 x2"
accp_induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "Wellfounded.accp r a"   and "\<And>x. \<lbrakk>Wellfounded.accp r x; \<forall>y. r y x \<longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
basic_trans_rules_21: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x \<le> y"   and "y < z" shows "x < z"
verit_comp_simplify1_1: fixes a :: "'d" shows "(a < a) = False"
jcdi: shows "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> i"
L_bisimilar_is_L_transform_bisimulation: shows "L_transform.is_bisimulation L_bisimilar"
add_mset_add_single: fixes a :: "'d"   and A :: "'d multiset" shows "add_mset a A = A + {#a#}"
implies_itself_same: fixes x :: "'a"   and y :: "'a" shows "x \<leadsto> x = y \<leadsto> y"
face_cycle_pred_succ: fixes G :: "('e, 'f) pre_digraph"   and M :: "'f pre_map"   and a :: "'f" assumes "digraph_map G M" shows "pre_digraph_map.face_cycle_pred M (pre_digraph_map.face_cycle_succ M a) = a"
length_filter_le: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "length (filter P xs) \<le> length xs"
CompDiag_Can_Inv: fixes t :: "'a local.term" assumes "Can t"   and "Diag t" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Inv t = Cod t"
prems_2: shows "sum ((^) 2) (insert x X) = sum ((^) 2) Y"
real_uminus: fixes a :: "'a" shows "real_of (- a) = - real_of a"
newQuery_equiv_def: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set"   and st :: "nat set"   and t :: "'b option list" assumes "getIJ getIJ" shows "New_max.newQuery V Q (st, t) = New_max.projectQuery V ((\<lambda>tab. New_max.semiJoin tab (st, t)) ` Q)"
cball_diff_eq_sphere: fixes a :: "'a"   and r :: "real" shows "cball a r - ball a r = sphere a r"
zip_eq_ConsE: fixes xs :: "'k list"   and ys :: "'l list"   and xy :: "'k \<times> 'l"   and xys :: "('k \<times> 'l) list" assumes "zip xs ys = xy # xys" obtains x :: "'k" and xs' :: "'k list" and y :: "'l" and ys' :: "'l list" where "xs = x # xs'" and "ys = y # ys'" and "xy = (x, y)" and "xys = zip xs' ys'"
sub: fixes n :: "nat" shows "A (Suc n) \<subseteq> A n"
applyUnitPropagateEffect: fixes state :: "State" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "\<not> getConflictFlag state"   and "getQ state \<noteq> []" shows "let uLiteral = hd (getQ state); state' = applyUnitPropagate state in \<exists>uClause. formulaEntailsClause (getF state) uClause \<and> isUnitClause uClause uLiteral (elements (getM state)) \<and> getM state' = getM state @ [(uLiteral, False)]"
sincos_total_pi_half: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y"   and "x\<^sup>2 + y\<^sup>2 = 1" shows "\<exists>t\<ge>0. t \<le> pi / 2 \<and> x = cos t \<and> y = sin t"
in_scc_of_self: fixes u :: "'a" assumes "u \<in> verts G" shows "u \<in> G.scc_of u"
simps_3: fixes x3 :: "dB"   and y3 :: "dB" shows "(Abs x3 = Abs y3) = (x3 = y3)"
Q_sub: shows "pp_of_term ` {v. v \<prec>\<^sub>t u \<and> d (pp_of_term v) \<le> dgrad_max d \<and> component_of_term v < length fs \<and> \<not> is_sig_GB_in d G v} \<subseteq> dgrad_set d (dgrad_max d)"
free_tv_le_new_tv: fixes n :: "nat"   and t :: "'a"   and m :: "nat" assumes "new_tv n t"   and "m \<in> free_tv t" shows "m < n"
BangUnfold: fixes P :: "pi" shows "!P \<equiv>\<^sub>s P \<parallel> !P"
gen_supp: fixes T :: "ty"   and Xs :: "W.tvar list" shows "supp (gen T Xs) = supp T - supp Xs"
Ssub: shows "S \<subseteq> topspace (Euclidean_space n)"
less_multiset_def: fixes M' :: "'b multiset"   and M :: "'b multiset" shows "(M' < M) = ((M', M) \<in> mult {(x', x). x' < x})"
mod_eq_dvd_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "(modulo a c = modulo b c) = dvd.dvd times c (minus a b)"
hyps: shows "pred_of_seq (apply f xq_) = pred_of_seq xq_ \<bind> f"
cont_compose: fixes c :: "'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'b" assumes "cont c"   and "cont f" shows "cont (\<lambda>x. c (f x))"
lemma_4_3_i_a: fixes a :: "'a"   and b :: "'a" shows "a \<squnion>1 b = (a l\<rightarrow> b) r\<rightarrow> b"
pivot'_def: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and s :: "('c, 'a) state" shows "pivot' x\<^sub>i x\<^sub>j s \<equiv> \<T>_update (pivot_tableau' x\<^sub>i x\<^sub>j (\<T> s)) s"
elims_3: fixes x :: "'b \<times> 'c"   and xa :: "'b \<times> 'c" assumes "\<not> alpha_residual x xa" obtains \<alpha>1 :: "'b" and P1 :: "'c" and \<alpha>2 :: "'b" and P2 :: "'c" where "x = (\<alpha>1, P1)" and "xa = (\<alpha>2, P2)" and "[bn \<alpha>1]set. (\<alpha>1, P1) \<noteq> [bn \<alpha>2]set. (\<alpha>2, P2)"
ospecD2: fixes x :: "'e option"   and P :: "'e \<Rightarrow> bool" assumes "ospec x P" shows "\<exists>y. x = Some y \<and> P y"
Diff_partition: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "A \<union> (B\<setminus>A) = B"
prems_1: shows "set (a # ws) \<subseteq> carrier_vec n"
euclidean_size_normalize: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_euclidean_semiring divide plus minus zero times one modulo euclidean_size unit_factor normalize" shows "euclidean_size (normalize a) = euclidean_size a"
open_scaling: fixes c :: "real"   and s :: "'c set" assumes "c \<noteq> 0"   and "open s" shows "open ((*\<^sub>R) c ` s)"
tanh_real_nonneg_iff: fixes x :: "real" shows "(0 \<le> tanh x) = (0 \<le> x)"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) src\<^sub>B B.map \<phi>" shows "equivalence_functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) src\<^sub>B"
sterm_determines_frechet: fixes I :: "('a, 'b, 'c) interp"   and J :: "('d, 'e, 'c) interp"   and \<theta>1 :: "('a, 'c) trm"   and \<theta>2 :: "('d, 'c) trm"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec" assumes "is_interp I"   and "is_interp J"   and "dfree \<theta>1"   and "dfree \<theta>2"   and "sterm_sem I \<theta>1 = sterm_sem J \<theta>2" shows "frechet I \<theta>1 (fst \<nu>) (snd \<nu>) = frechet J \<theta>2 (fst \<nu>) (snd \<nu>)"
eq: shows "m\<^sup>2 = p * n\<^sup>2"
induced_automorphism_C0: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g ` C0 = g ` C0"
fair: shows "effective_redundancy_criterion.fair_clss_seq \<Gamma>' Rf (\<lambda>N. Ri N \<union> (\<Gamma>' - \<Gamma>)) Ns"
int_liminf: shows "\<integral>\<^sup>+ x. liminf (\<lambda>n. ennreal (G n x)) \<partial>M = 2 * \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M"
terminates_on_stop: fixes "next" :: "'a \<Rightarrow> 'b \<times> 'a" shows "terminates_on (\<lambda>_. False, next) = UNIV"
bridge'_nominals: fixes k :: "'b"   and j :: "'b"   and p :: "('c, 'b) fm" shows "nominals (bridge' k j p) \<union> {k, j} = nominals p \<union> {k, j}"
\<omega>_of_Cantor_Nil: fixes ms :: "nat list" shows "\<omega>_of_Cantor [] ms = []"
count_image_mset: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c multiset"   and x :: "'b" shows "count (image_mset f A) x = sum (count A) (f -` {x} \<inter> set_mset A)"
CollectD: fixes a :: "'f"   and P :: "'f \<Rightarrow> bool" assumes "a \<in> {x. P x}" shows "P a"