hcju/isabelleps huggingface dataset:
consistency_def: fixes C :: "('c, 'd) form set set" shows "consistency C = (\<forall>S. S \<in> C \<longrightarrow> (\<forall>p ts. \<not> (Pred p ts \<in> S \<and> Neg (Pred p ts) \<in> S)) \<and> FF \<notin> S \<and> Neg TT \<notin> S \<and> (\<forall>Z. Neg (Neg Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. And A B \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. Neg (Or A B) \<in> S \<longrightarrow> S \<union> {Neg A, Neg B} \<in> C) \<and> (\<forall>A B. Or A B \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (And A B) \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {Neg B} \<in> C) \<and> (\<forall>A B. Impl A B \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (Impl A B) \<in> S \<longrightarrow> S \<union> {A, Neg B} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Forall P \<in> S \<longrightarrow> S \<union> {P[t/0]} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Neg (Exists P) \<in> S \<longrightarrow> S \<union> {Neg (P[t/0])} \<in> C) \<and> (\<forall>P. Exists P \<in> S \<longrightarrow> (\<exists>x. S \<union> {P[App x []/0]} \<in> C)) \<and> (\<forall>P. Neg (Forall P) \<in> S \<longrightarrow> (\<exists>x. S \<union> {Neg (P[App x []/0])} \<in> C)))"
ladj_def: fixes g :: "'d \<Rightarrow> 'c" shows "ladj g = (\<lambda>x. \<Sqinter> {y. x \<le> g y})"
ltake_plus_conv_lappend: fixes n :: "enat"   and m :: "enat"   and xs :: "'a llist" shows "ltake (n + m) xs = ltake n xs @\<^sub>l ltake m (ldrop n xs)"
prems_2: shows "\<rho>\<^sub>2 \<in> carrier_mat d d"
1: shows "[\<^bold>\<A>\<psi> (x\<^sup>P) in v]"
sum_less_eqE: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'b"   and x :: "'c" assumes "sum f A \<le> y"   and "x \<in> A"   and "finite A" shows "f x \<le> y"
f2'_nonneg: fixes x :: "nat" assumes "gx0 \<le> x" shows "0 \<le> f2' x"
tree_recurse_iterate: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<epsilon> :: "'a"   and l :: "'a"   and r :: "'a" assumes "\<And>x y z. f (f x y) z = f x (f y z)"   and "\<And>x. f x \<epsilon> = x"   and "\<And>x. f \<epsilon> x = x" shows "tree_recurse (f l) (f r) \<epsilon> = tree_iterate (\<lambda>x. f x l) (\<lambda>x. f x r) \<epsilon>"
Qmpgsurjec: fixes H :: "'a set"   and N :: "'a set" assumes "G \<triangleright> H"   and "G \<triangleright> N"   and "H \<subseteq> N" shows "gsurj\<^bsub>(G / H),(G / N)\<^esub> (Qm\<^bsub>G H,N\<^esub>)"
nonmeets4: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and u :: "'a"   and r :: "'a" assumes "class.interval meets \<I>" shows "\<not> (meets u r \<and> (\<exists>p. meets r p \<and> meets p u))"
Rep_formula': fixes x :: "('a, 'b, 'c) Formula.formula" shows "Formula.hereditarily_fs (Formula.formula.Rep_formula x)"
split_path_at_vertex: fixes u :: "nat"   and s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "u \<in> set (pathVertices_fwd s p)"   and "isPath s p t" obtains p1 :: "(nat \<times> nat) list" and p2 :: "(nat \<times> nat) list" where "p = p1 @ p2" and "isPath s p1 u" and "isPath u p2 t"
sub_unit_closed: shows "\<one> \<in> R\<setminus>\<pp>"
IH: fixes ys :: "'a list" assumes "x \<in> set xs" shows "\<exists>xxs as. as @ concat (map ((#) x) xxs) @ remove_cycles xs x ys = xs \<and> x \<notin> set as"
monotone_nn_integral_spmf: fixes f :: "'a \<Rightarrow> ennreal" shows "monotone (ord_spmf (=)) (\<le>) (\<lambda>p. integral\<^sup>N (measure_spmf p) f)"
unique_connecting_path_set_1: fixes v :: "'c"   and w :: "'c" assumes "v \<in> V\<^bsub>T\<^esub>"   and "w \<in> V\<^bsub>T\<^esub>" shows "v \<in> set (v \<leadsto>\<^bsub>T\<^esub> w)"
LIMSEQ_iff_nz: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
star_rd_le_sq: fixes x :: "'a" shows "star_rd x \<le> star_sq x"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "hindrance \<Gamma> f" obtains a :: "'a" where "a \<in> A \<Gamma>" and "a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" and "d_OUT f a < weight \<Gamma> a"
inject_2: fixes x21 :: "'b"   and x22 :: "'c \<Rightarrow> ('a, 'b, 'c) env option"   and y21 :: "'b"   and y22 :: "'c \<Rightarrow> ('a, 'b, 'c) env option" shows "(Env x21 x22 = Env y21 y22) = (x21 = y21 \<and> x22 = y22)"
subset_ccNeighbors: fixes S :: "Vars.var set"   and x :: "Vars.var"   and G :: "CoCalls" shows "(S \<subseteq> ccNeighbors x G) = ({x} G\<times> S \<sqsubseteq> G)"
case3: assumes "0 \<le> x"   and "y \<le> 0" shows "- 4611686018427387904 \<le> x * y"
conv_finite: fixes x :: "'a" shows "relation_algebra_rtc_class.finite x = relation_algebra_rtc_class.finite (x\<^sup>T)"
GFP_upperbound: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "x \<in> carrier L"   and "x \<sqsubseteq> f x" shows "x \<sqsubseteq> GFP f"
FWdelay_bisimulation_diverge_flip: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" assumes "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_diverge final2 r2 final1 r1 (\<lambda>t. flip (bisim t)) (flip bisim_wait) \<tau>move2 \<tau>move1"
dist_0_norm: fixes x :: "'b" shows "dist (0::'b) x = norm x"
le_INF_iff: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and u :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (u::'b::type) ((Inf::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set))) = (\<forall>i::'c::type\<in>A. less_eq u (f i))"
m_rcancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> \<zero>"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(b \<otimes> a = c \<otimes> a) = (b = c)"
gen_in: fixes w :: "'b list"   and G :: "'b list set" assumes "w \<in> G" shows "w \<in> \<langle>G\<rangle>"
spmf_eqI: fixes p :: "'h spmf"   and q :: "'h spmf" assumes "\<And>i. spmf p i = spmf q i" shows "p = q"
dynamic_to_static_overriding: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides old"   and "accmodi old \<noteq> Package"   and "wf_prog G" shows "G \<turnstile> new overrides\<^sub>S old"
map_aform_err_def: fixes I :: "real \<times> real pdevs \<Rightarrow> ((real \<times> real pdevs) \<times> real) option"   and p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "map_aform_err I p X = (let X0 = aform_err_to_aform X (degree_aform_err X) in I X0 \<bind> (\<lambda>(X1, e1). Some (acc_err p (aform_to_aform_err X1 (degree_aform_err X)) e1)))"
vertex_in_separator_2: fixes s :: "'c"   and t :: "'c"   and v :: "'a" assumes "s \<rightarrow>\<^bsub>T\<^esub> t"   and "v \<in> left_part s t"   and "v \<in> left_part t s" shows "v \<in> bag t"
Diff_insert2: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" shows "A - insert a B = A - {a} - B"
2: shows "punit.lt (rep_list a) \<oplus> lt g = punit.lt (rep_list g) \<oplus> lt a"
vsubst_ident: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "good X" shows "X #[z // z]_zs = X"
simps_4: fixes t :: "sterm"   and u :: "sterm" shows "frees (t $\<^sub>s u) = frees t |\<union>| frees u"
simps_1: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" shows "pre_digraph.cas G u [] v = (u = v)"
Lambert_W'_antimono: fixes x :: "real"   and y :: "real" assumes "- exp (- 1) \<le> x"   and "x \<le> y"   and "y < 0" shows "Lambert_W' y \<le> Lambert_W' x"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<R>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm T2"
structure_is_natural_isomorphism: shows "natural_isomorphism C.CC.comp (\<cdot>\<^sub>D) T\<^sub>DoFF.map FoT\<^sub>C.map T\<^sub>DoFF.map"
assms_2: shows "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"
strong_cases: fixes a1 :: "(name \<times> ty) list"   and a2 :: "trm"   and a3 :: "ty"   and x :: "name" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "(name \<times> ty) list" and x :: "name" and T :: "ty" where "a1 = \<Gamma>__" and "a2 = Var x" and "a3 = T" and "valid \<Gamma>__" and "(x, T) \<in> set \<Gamma>__"   | \<Gamma> :: "(name \<times> ty) list" and e\<^sub>1 :: "trm" and T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" and e\<^sub>2 :: "trm" where "a1 = \<Gamma>'__" and "a2 = App e\<^sub>1 e\<^sub>2" and "a3 = T\<^sub>2__" and "\<Gamma>'__ \<turnstile> e\<^sub>1 : T\<^sub>1__\<rightarrow>T\<^sub>2__" and "\<Gamma>'__ \<turnstile> e\<^sub>2 : T\<^sub>1__"   | \<Gamma> :: "(name \<times> ty) list" and T\<^sub>1 :: "ty" and e :: "trm" and T\<^sub>2 :: "ty" where "\<lbrakk>x__ \<sharp> a1; x__ \<sharp> a2; x__ \<sharp> a3\<rbrakk> \<Longrightarrow> a1 = \<Gamma> \<and> a2 = Lam [x__].e \<and> a3 = T\<^sub>1\<rightarrow>T\<^sub>2 \<and> x__ \<sharp> \<Gamma> \<and> (x__, T\<^sub>1) # \<Gamma> \<turnstile> e : T\<^sub>2"
select_convs_2: fixes states :: "'a \<Rightarrow> 'c"   and msgs :: "('a, 'b) message \<Rightarrow> nat"   and more :: "'d" shows "msgs \<lparr>states = states, msgs = msgs, \<dots> = more\<rparr> = msgs"
mpoly_map_vars_monom: fixes f :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and c :: "'a" assumes "bij f" shows "mpoly_map_vars f (MPoly_Type.monom m c) = MPoly_Type.monom (permutep (inv f) m) c"
elims: fixes x :: "'a list"   and y :: "'a tree" assumes "bst_of_list x = y" obtains "x = []" and "y = \<langle>\<rangle>"   | xa :: "'a" and xs :: "'a list" where "x = xa # xs" and "y = \<langle>bst_of_list (filter (\<lambda>y. y < xa) xs), xa, bst_of_list (filter ((<) xa) xs)\<rangle>"
prj_sym_2: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "pr1 a1 a0 \<cdot> local.sym a0 a1 = pr0 a0 a1"
cases: fixes \<Gamma> :: "('b, 'd) \<I>"   and a :: "('e, 'b, 'd) gpv" assumes "\<Gamma> \<turnstile>g a \<surd>" obtains gpv :: "('e, 'b, ('e, 'b, 'd) rpv) generat spmf" where "a = GPV gpv" and "\<And>out c. IO out c \<in> set_spmf gpv \<Longrightarrow> out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. \<Gamma> \<turnstile>g c input \<surd>)"
zero_least3: fixes x :: "'a" shows "n x \<cdot> o = o"
none_case: fixes l :: "'a" assumes "\<sigma> l = None" shows "\<sigma>' (\<beta> l) = None"
f_def: shows "f \<equiv> \<lambda>Y. vars b \<union> X \<union> L c Y"
induct: fixes P :: "signs \<Rightarrow> bool"   and signs :: "signs" assumes "P Pos"   and "P Neg" shows "P signs"
Cons_5: shows "j_to_jb_index ((m, b) # n_as) j = (bj, lj)"
linorder_rank_def: fixes R :: "('a \<times> 'a) set"   and A :: "'a set"   and x :: "'a" shows "linorder_rank R A x = card {y \<in> A - {x}. (y, x) \<in> R}"
abs_pos: shows "0 < \<bar>c * q n - d * p n\<bar>"
inv_from_def: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and m :: "nat" shows "inv_from n p A m \<equiv> \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> m < j \<longrightarrow> p A i j"
left_unique_conversep: fixes A :: "'g \<Rightarrow> 'f \<Rightarrow> bool" shows "left_unique A\<inverse>\<inverse> = right_unique A"
cterms_includes_sterms_of_seq_reachable: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('f \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and I :: "'g seq_action \<Rightarrow> bool" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "automaton.trans A = seqp_sos \<Gamma>" shows "\<Union> (sterms \<Gamma> ` snd ` reachable A I) \<subseteq> cterms \<Gamma>"
double_compl: fixes x :: "'a" shows "- (- x) = x"
rec_1: fixes f1 :: "brick list \<Rightarrow> 'a"   and f2 :: "brick list \<Rightarrow> wall \<Rightarrow> 'a \<Rightarrow> 'a"   and x1 :: "brick list" shows "rec_wall f1 f2 (basic x1) = f1 x1"
insert_subset_eq_iff: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" shows "(add_mset a A \<subseteq># B) = (a \<in># B \<and> A \<subseteq># B - {#a#})"
\<theta>_def: fixes xb :: "('f, 'g) F \<times> 'h" shows "\<theta> xb = F id \<langle>id, \<lambda>a. snd xb\<rangle> (fst xb)"
lift_Pareto_SD_strict_iff: fixes p :: "'b pmf"   and q :: "'b pmf" assumes "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts" shows "p \<prec>[Pareto (SD \<circ> R')] q = p \<prec>[Pareto (SD \<circ> R)] q"
prj_chine_assoc_3: shows "doms.Prj\<^sub>0\<^sub>0 \<cdot> doms.chine_assoc = doms.Prj\<^sub>0"
permute_prefs_from_table: fixes \<sigma> :: "'c \<Rightarrow> 'c"   and xs :: "('c \<times> 'd set list) list" assumes "\<sigma> permutes fst ` set xs" shows "prefs_from_table xs \<circ> \<sigma> = prefs_from_table (map (\<lambda>(x, y). (inv \<sigma> x, y)) xs)"
groups_disjoint_pairwise: fixes G1 :: "'a set"   and G2 :: "'a set" assumes "G1 \<in> \<G>"   and "G2 \<in> \<G>"   and "G1 \<noteq> G2" shows "disjnt G1 G2"
degen_path_sound: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and u :: "'Q \<times> nat"   and p :: "('Q \<times> nat) list"   and v :: "'Q \<times> nat" assumes "path (g_E (degeneralize_ext m)) u p v" shows "path E (fst u) (map fst p) (fst v)"
sum_representation_eq: fixes basis :: "('a, 'b) vec set"   and v :: "('a, 'b) vec"   and B :: "('a, 'b) vec set" assumes "vec.independent basis"   and "v \<in> vec.span basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. vec.representation basis v b *s b) = v"
assoc'_simps_4: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "C.dom \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] = C.dom f \<star>\<^sub>C C.dom g \<star>\<^sub>C C.dom h"
euclid_ext2_def: 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 gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "euclidean_ring_gcd.euclid_ext2 divide minus zero times one modulo unit_factor normalize gcd uminus a b = (fst (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b), snd (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b), divide (uminus b) (gcd a b), divide a (gcd a b), gcd a b)"
eucl_sup: fixes x :: "'c"   and y :: "'c" shows "sup x y = (\<Sum>i\<in>Basis. sup (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)"
norm_in_cf: fixes x :: "'a itrm" shows "norm x \<in> CF"
a_add_comm: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> y = y \<oplus> x"
2094: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> y \<sqinter> (z \<squnion> top)) \<sqinter> (x \<squnion> y) = bot"
R27_d: shows "pmf (sds R27) d = 1 - pmf (sds R27) a"
SymSystem_rel: fixes init :: "(nat \<Rightarrow> 'c) set"   and p :: "'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" shows "SymSystem init p r = {.x. \<forall>u. u \<in> init \<longrightarrow> prec_st p r u x.} \<circ> [: x \<leadsto> y . \<exists>u. u \<in> init \<and> (\<box> lft_rel_st r) u x y :]"
return_def: shows "return = COERCE(udom \<rightarrow> udom\<cdot>'e, 'd \<rightarrow> 'd\<cdot>'e)\<cdot>returnU"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.dom (g \<cdot> f) = C.dom f"
arev_neq: fixes a :: "'b" assumes "a \<in> arcs G" shows "rev_G a \<noteq> a"
cexpr_typing_cong: fixes e :: "cexpr"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<Gamma>' :: "nat \<Rightarrow> pdf_type"   and t :: "pdf_type" assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<Gamma> x = \<Gamma>' x" shows "(\<Gamma> \<turnstile>\<^sub>c e : t) = (\<Gamma>' \<turnstile>\<^sub>c e : t)"
get_action_case_simp: fixes m :: "'c match_expr"   and r :: "'c rule" shows "get_action (case r of Rule m' x \<Rightarrow> Rule (MatchAnd m m') x) = get_action r"
insertion_one: fixes f :: "nat \<Rightarrow> 'a" shows "insertion f 1 = (1::'a)"
wf_trm_subst_rm_vars': fixes \<delta> :: "'a \<Rightarrow> ('fun, 'a) Term.term"   and v :: "'a"   and X :: "'a set" assumes "wf\<^sub>t\<^sub>r\<^sub>m (\<delta> v)" shows "wf\<^sub>t\<^sub>r\<^sub>m (rm_vars X \<delta> v)"
independent_explicit_module: fixes s :: "'c set" shows "m2.independent s = (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. u v *b v) = (0::'c) \<longrightarrow> v \<in> t \<longrightarrow> u v = (0::'a))"
discrete_imp_closed: fixes e :: "real"   and S :: "'c set" assumes "0 < e"   and "\<forall>x\<in>S. \<forall>y\<in>S. dist y x < e \<longrightarrow> y = x" shows "closed S"
stake_gener_bool_list: fixes n :: "nat" shows "stake n ` streams UNIV = gener_bool_list n"
assms: shows "mustT (if tst then c1 else c2) s"
FF_dim_3: shows "mat_of_rows n (map of_int_hom.vec_hom fs) \<in> carrier_mat (length (map of_int_hom.vec_hom fs)) n"
abelian_group_def: fixes G :: "('e, 'f) ring_scheme" shows "abelian_group G \<equiv> abelian_monoid G \<and> abelian_group_axioms G"
subcls1I1: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and D :: "qtname" assumes "class G C = Some c"   and "C \<noteq> Object"   and "D = super c" shows "G\<turnstile>C\<prec>\<^sub>C1D"
simps_12: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (Imp p q) = (Ifm bs p \<longrightarrow> Ifm bs q)"
delta_pos_sol: shows "delta_pos = F (delta_neg, delta_pos)"
image_constant_conv: fixes c :: "'c"   and A :: "'d set" shows "(\<lambda>x. c) ` A = (if A = {} then {} else {c})"
similar_mat_witD_6: fixes n :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "P \<in> carrier_mat n n"
continuous_within_avoid: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b" assumes "continuous (at x within s) f"   and "f x \<noteq> a" shows "\<exists>e>0. \<forall>y\<in>s. dist x y < e \<longrightarrow> f y \<noteq> a"
Abs_dlist_inverse: fixes y :: "'c list" assumes "y \<in> {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None}" shows "list_of_dlist (Abs_dlist y) = y"
prod: shows "prod_dfs A B (0, 0) = (tab, ps)"
partition_from_equivalence: fixes S :: "('d, 'e) eq_object_scheme" assumes "equivalence S" shows "Congruence.partition (carrier S) classes\<^bsub>S\<^esub>"
length_listsetD: fixes vs :: "'c list"   and VS :: "'c set list" assumes "vs \<in> listset VS" shows "length vs = length VS"
intros_2: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat"   and n :: "nat" assumes "((\<sigma>, k), \<sigma>', k') \<in> cp"   and "n cd\<^bsup>path \<sigma>\<^esup>\<rightarrow> k"   and "path \<sigma> (Suc k) \<noteq> path \<sigma>' (Suc k')"   and "path \<sigma> n \<in> dom local.att" shows "((\<sigma>, n), \<sigma>', k') \<in> cop"
openin_subset: fixes U :: "'d topology"   and S :: "'d set" assumes "openin U S" shows "S \<subseteq> topspace U"
get_M_document_put_M_shadow_root: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and setter :: "('aj \<Rightarrow> 'aj) \<Rightarrow> ('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> ('z, 'x, 'y, 'ag option) RShadowRoot_scheme"   and v :: "'aj"   and h' :: "(_) heap"   and getter :: "('z, 'x, 'y, (\<lparr>RShadowRoot.nothing :: unit, mode :: shadow_root_mode, child_nodes :: (_) node_ptr list, \<dots> :: 'ag option\<rparr> + 'ah) option) RDocument_scheme \<Rightarrow> 'ak"   and document_ptr :: "(_) document_ptr" assumes "h \<turnstile> put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr setter v \<rightarrow>\<^sub>h h'"   and "\<And>x. is_shadow_root_kind\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t x \<Longrightarrow> getter (cast\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (setter (\<lambda>_. v) (the (cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t x)))) = getter x" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr getter) h h'"
index_correct_2: fixes idx_s :: "('Q, ('Q, 'L) ta_rule dlist) HashMap.hashmap" assumes "hta_idx_s H = Some idx_s" shows "hll_idx.is_index lhs (ls.\<alpha> \<delta>) idx_s"
cur_def: shows "cur = (if ne = 0 then 0 else \<tau> \<sigma> (ne - 1))"
elimmeets: fixes p :: "'a"   and s :: "'a"   and r :: "'a"   and q :: "'a" shows "(p \<parallel> s \<and> (\<exists>t. p \<parallel> t \<and> t \<parallel> s) \<and> (\<exists>t. r \<parallel> t \<and> t \<parallel> q)) = False"
ext_extensional0: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "extensional0 S f"   and "extensional0 S g"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "f = g"
smalloI_tendsto: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "((\<lambda>x. f x / g x) \<longlongrightarrow> (0::'b)) F"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)" shows "f \<in> o[F](g)"
insert_extract_weights_cong_shared: fixes m :: "real mat convnet"   and f :: "nat \<Rightarrow> real" assumes "shared_weight_net m"   and "\<And>x. x < count_weights True (remove_weights m) \<Longrightarrow> f x = extract_weights True m x" shows "m = insert_weights True (remove_weights m) f"
remap_lang1: fixes t :: "'a tree"   and TA :: "('b, 'a, 'c) tree_automaton_rec_scheme"   and f :: "'b \<Rightarrow> 'd" assumes "t \<in> ta_lang TA" shows "t \<in> ta_lang (ta_remap f TA)"
mono_neutral_cong: fixes T :: "'b set"   and S :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite T"   and "finite S"   and "\<And>i. i \<in> T - S \<Longrightarrow> h i = (1::'c)"   and "\<And>i. i \<in> S - T \<Longrightarrow> g i = (1::'c)"   and "\<And>x. x \<in> S \<inter> T \<Longrightarrow> g x = h x" shows "prod g S = prod h T"
h3g: assumes "MPoly_Type.degree p2 var = 1" shows "p2 = A + B * Var var"