hcju/isabelleps huggingface dataset:
canonical_retraction_simplex_retraction1: fixes A :: "'a set set"   and C :: "'a set"   and a :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "C \<in> A"   and "a \<in> A" shows "fixespointwise (canonical_retraction A C) a"
assms_1: shows "spmf.admissible (\<lambda>fast_dice_roll. P (curry fast_dice_roll))"
hyps_1: shows "Q \<longmapsto> \<alpha> \<prec> Q'"
i_less_nrows: shows "i < ncols_iarray (matrix_to_iarray A)"
pmdl_closed_monom_mult: fixes p :: "'t \<Rightarrow>\<^sub>0 'd"   and B :: "('t \<Rightarrow>\<^sub>0 'd) set"   and c :: "'d"   and t :: "'a" assumes "p \<in> pmdl B" shows "monom_mult c t p \<in> pmdl B"
geom_rand_walk_pseudo_proj_False: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and n :: "nat" assumes "prob_grw p M geom_proc u d init" shows "geom_proc n = geom_proc n \<circ> pseudo_proj_False n"
eq: shows "x\<^sup>2 + y\<^sup>2 = 2"
hom_0: fixes x :: "'a poly" assumes "poly_x_minus_y x = 0" shows "x = 0"
lookup_Some_in\<delta>: fixes \<pi> :: "char list"   and op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "lookup_operator \<pi> = Some op" shows "op \<in> set ast\<delta>"
pmf_of_set_impl_def: fixes A :: "'b set" shows "pmf_of_set_impl A = mapping_of_pmf (pmf_of_set A)"
pinf_4: fixes t :: "'a" shows "\<exists>z. \<forall>x>z. (x \<noteq> t) = True"
CauchysMeanTheorem_Eq: fixes z :: "real list" assumes "pos z" shows "(gmean z = mean z) = (het z = 0)"
setops\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('d, 'e) stateful_strand_step list"   and B :: "('d, 'e) stateful_strand_step list" shows "setops\<^sub>s\<^sub>s\<^sub>t (A @ B) = setops\<^sub>s\<^sub>s\<^sub>t A \<union> setops\<^sub>s\<^sub>s\<^sub>t B"
sets_into_space: fixes x :: "'d set"   and M :: "'d measure" assumes "x \<in> sets M" shows "x \<subseteq> space M"
transitive_asymmetric_irreflexive_inf: fixes x :: "'a" assumes "transitive x" shows "asymmetric x = irreflexive_inf x"
hyps_11: shows "x = (SOME x'. x' \<in> U - V)"
SReal_Infinitesimal_zero: fixes x :: "real star" assumes "x \<in> \<real>"   and "x \<in> Infinitesimal" shows "x = 0"
faces: fixes h :: "'a set" assumes "h \<in> F" shows "\<exists>a b. a \<noteq> (0::'a) \<and> h = {x. a \<bullet> x \<le> b}"
conjug_pow: fixes x :: "'b list"   and z :: "'b list"   and y :: "'b list"   and k :: "nat" assumes "x \<cdot> z = z \<cdot> y" shows "x \<^sup>@ k \<cdot> z = z \<cdot> y \<^sup>@ k"
gt_ubound_def: fixes c :: "'a"   and b :: "'a option" shows "c >\<^sub>u\<^sub>b b = \<rhd>\<^sub>u\<^sub>b (<) c b"
set: fixes x :: "'b list" shows "set_iarray (IArray x) = set x"
backwards_res_act: fixes ts :: "traffic"   and c :: "cars"   and ts' :: "traffic"   and v :: "view" shows "(ts \<^bold>\<midarrow>r( c ) \<^bold>\<rightarrow> ts') \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts' c = ext v \<and> restrict v (res ts') c = lan v \<and> |lan v| = 1 \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (clm ts) c = lan v \<and> |lan v| = 1"
mat_row_def: fixes k :: "'a"   and i :: "'b" shows "mat_row k i = (\<chi>j. if i = j then k else (0::'a))"
Rep_formula_diff: fixes x :: "'a formula"   and y :: "'a formula" shows "Rep_formula (x - y) = Rep_formula x - Rep_formula y"
zero_eq_1_divide_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(zero = divide one a) = (a = zero)"
inter_sgs: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> K " shows "G \<guillemotright> H \<inter> K "
ar_coset_same4: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> I" shows "a \<uplus>\<^bsub>R\<^esub> I = I"
snd_def: fixes prod :: "'h \<times> 'g" shows "snd prod = (case prod of (x1, x2) \<Rightarrow> x2)"
prime_dvd_mult_eq_int: fixes p :: "int"   and a :: "int"   and b :: "int" assumes "prime p" shows "(p dvd a * b) = (p dvd a \<or> p dvd b)"
scaleRat_less2: fixes a :: "'a"   and b :: "'a"   and k :: "rat" assumes "a < b"   and "k < 0" shows "k *R b < k *R a"
lQuot_star: fixes c :: "'b"   and A :: "'b list set" shows "lQuot c (star A) = lQuot c A @@ star A"
EphK_EphK: fixes X :: "ephkey"   and X' :: "ephkey" shows "(EphK X = EphK X') = (X = X')"
lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'e"   and n :: "nat" shows "prod g {..<Suc n} = g 0 * (\<Prod>i<n. g (Suc i))"
ext_inject: fixes cfields :: "fdecl list"   and methods :: "mdecl list"   and init :: "stmt"   and more :: "'b"   and cfields' :: "fdecl list"   and methods' :: "mdecl list"   and init' :: "stmt"   and more' :: "'b" shows "(\<lparr>cfields = cfields, methods = methods, init = init, \<dots> = more\<rparr> = \<lparr>cfields = cfields', methods = methods', init = init', \<dots> = more'\<rparr>) = (cfields = cfields' \<and> methods = methods' \<and> init = init' \<and> more = more')"
assms_4: shows "A, n \<turnstile> (p # ps, a) # branch"
utos_type: fixes w :: "bit list" shows "utos (norm_unsigned w) = utos w"
mem_samp_primes_excl_prime: fixes x :: "nat"   and y :: "nat" assumes "x \<in> set_spmf (sample_set_excl prime_set {y})" shows "prime x"
iso_lower_hemisphere_reduced_homology_group: fixes p :: "int"   and n :: "nat" shows "hom_boundary (1 + p) (subtopology (nsphere (Suc n)) {x. x (Suc n) \<le> 0}) {x. x (Suc n) = 0} \<in> Group.iso (relative_homology_group (1 + p) (subtopology (nsphere (Suc n)) {x. x (Suc n) \<le> 0}) {x. x (Suc n) = 0}) (reduced_homology_group p (nsphere n))"
terminates_elim_cases_1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>Skip \<down> s" obtains s :: "'e" where "sb__ = Normal sc__"   | f :: "'f" where "sb__ = Fault f"   | "sb__ = Stuck"   | s :: "'e" where "sb__ = Abrupt s"
sup2: fixes B :: "'d"   and k :: "'d"   and b :: "'d"   and a :: "'d" assumes "B \<equiv> sup k b" shows "sup a B \<equiv> sup k (sup a b)"
dvd_eq_mod_eq_0: 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 a :: "'a"   and b :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "dvd.dvd times a b = (modulo b a = zero)"
af_congruent_axioms_def: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" shows "af_congruent_axioms eq \<equiv> \<forall>\<phi> \<psi> \<nu>. eq \<phi> \<psi> \<longrightarrow> eq (af_letter \<phi> \<nu>) (af_letter \<psi> \<nu>)"
l_set_disconnected_nodes_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs \<equiv> l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs \<and> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs"
path_AppendE: fixes l1 :: "'vtx"   and l3 :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and l2 :: "'vtx"   and s :: "'lbl"   and l2' :: "'vtx" assumes "path l1 l3 (xs @ [(l2, s, l2')])" obtains "l3 = l2'" and "path l1 l2 xs" and "s \<in>\<^sub>A weights l2 l2'"
C_nonneg: fixes off :: "nat"   and k :: "nat" shows "0 \<le> C off k"
Refl_under_Under: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "Refl r"   and "A \<noteq> {}" shows "Under r A = \<Inter> (under r ` A)"
fair_clss_seq_def: fixes Ns :: "'a literal multiset set llist" shows "fair_clss_seq Ns = (let N' = Liminf_llist Ns - Rf (Liminf_llist Ns) in concls_of (inferences_from N' - Ri N') \<subseteq> Sup_llist Ns \<union> Rf (Sup_llist Ns))"
lfp_chain_induct: fixes f :: "'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "mono f"   and "\<And>x. P x \<Longrightarrow> P (f x)"   and "\<And>C. \<lbrakk>set_chain C; \<forall>x\<in>C. P x\<rbrakk> \<Longrightarrow> P (Sup C)" shows "P (lfp f)"
IH: fixes S :: "'node SDG_node set" assumes "n'' \<in> set ns"   and "n\<^sub>1 \<in> S" shows "n'' \<in> HRB_slice S"
tiny_digraphI': fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>"   and "\<CC>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<CC>\<lparr>Arr\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>" shows "tiny_digraph \<alpha> \<CC>"
FIXvar: fixes b :: "BExpr"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<beta> :: "(Location \<times> Location) set" assumes "PhiWhile b \<Phi> (s, t, \<beta>)" shows "(b, \<Phi>, \<beta>, s, t) \<in> var"
p: fixes s :: "'s" assumes "guard s"   and "I s" shows "p \<le> weight_spmf (loop_spmf.while guard body s)"
a_Prj1: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, eP, eV : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2" shows "\<Gamma> \<turnstile> Prj1 e, Prj1 eP, Prj1 eV : \<tau>\<^sub>1"
is_total_def: fixes x :: "'a" shows "is_total x \<equiv> reflexive (x ; x\<^sup>T)"
meld_def: fixes q1 :: "('c, 'd) BinomialHeap"   and q2 :: "('c, 'd) BinomialHeap" shows "meld q1 q2 \<equiv> Abs_BinomialHeap (BinomialHeapStruc.meld (Rep_BinomialHeap q1) (Rep_BinomialHeap q2))"
path_valid_node_1: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit_wf.path n as n'" shows "CFGExit_wf.valid_node n"
eq_iff_4: fixes var :: "Vars.var"   and exp :: "exp"   and vara :: "Vars.var"   and expa :: "exp" shows "(Lam [var]. exp = Lam [vara]. expa) = ([[atom var]]lst. exp = [[atom vara]]lst. expa)"
mult_powr_eq_mult_powr_iff: fixes m1 :: "int"   and m2 :: "int"   and e1 :: "int"   and e2 :: "int" assumes "odd m1"   and "odd m2" shows "(real_of_int m1 * 2 powr real_of_int e1 = real_of_int m2 * 2 powr real_of_int e2) = (m1 = m2 \<and> e1 = e2)"
assms_1: shows "vars_term s \<inter> vars_term t = {}"
no_nonpos_Int_in_ball': fixes z :: "'a" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" obtains d :: "real" where "0 < d" and "\<And>t. t \<in> ball z d \<Longrightarrow> t \<notin> \<int>\<^sub>\<le>\<^sub>0"
nle__lt: 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.Le Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Lt Bet Cong C D A B"
injects: fixes C :: "C"   and C' :: "C" shows "(C\<cdot>C = C\<cdot>C') = (C = C')"
le_supI: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x"   and "b \<le> x" shows "sup a b \<le> x"
tagged_partial_division_of_def: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" shows "(s tagged_partial_division_of i) = (finite s \<and> (\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and> (\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow> interior K1 \<inter> interior K2 = {}))"
aadd_minus_r: fixes x :: "ant" shows "x + - x = 0"
index_image: fixes xs :: "'a list"   and X :: "'a set" assumes "distinct xs"   and "set xs = X" shows "index xs ` X = {0..<length xs}"
continuous_within_subset: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and t :: "'a set" assumes "continuous (at x within s) f"   and "t \<subseteq> s" shows "continuous (at x within t) f"
simps_2: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and as :: "'a list" assumes "U_function_ring R" shows "U_function_ring.polynomial R (a # as) = (\<lambda>x\<in>carrier R. a \<oplus>\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> U_function_ring.polynomial R as x)"
scope_cases: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and v :: "'b \<times> nat list"   and i :: "('b, 'a) antecedent"   and v' :: "'b \<times> nat list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "Solved_Task.scope' freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions v i v'" obtains c :: "'b" and is' :: "nat list" and "is" :: "nat list" and ia :: "nat" where "v = (c, is)" and "i = Solved_Task.in_port_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions (c, is) ia" and "v' = (c, is')" and "c \<in> set conclusions" and "is' \<in> (#) 0 ` it_paths (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) c, 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c)))" and "prefix (is @ [ia]) is'"
inject_2: fixes x2 :: "t list"   and y2 :: "t list" shows "(Type x2 = Type y2) = (x2 = y2)"
total_conv_surjective: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.total x = matrix_bounded_idempotent_semiring.surjective (x\<^sup>t)"
max_next_pcs_not_empty: fixes pc :: "nat"   and bp :: "instr list"   and x :: "nat \<times> integer"   and s :: "integer" assumes "pc < length bp"   and "x \<in> set (exec (bp ! pc) (pc, s))" shows "max_next_pcs bp \<noteq> {}"
finite_entropy_integrable_transform: fixes S :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and Px :: "'b \<Rightarrow> real"   and T :: "'f measure"   and Y :: "'a \<Rightarrow> 'f"   and Py :: "'f \<Rightarrow> real"   and f :: "'f \<Rightarrow> 'b" assumes "finite_entropy S X Px"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>x. x \<in> space T \<Longrightarrow> 0 \<le> Py x"   and "X = (\<lambda>x. f (Y x))"   and "f \<in> T \<rightarrow>\<^sub>M S" shows "integrable T (\<lambda>x. Py x * log b (Px (f x)))"
permutation_permutes: fixes p :: "'b \<Rightarrow> 'b" shows "permutation p = (\<exists>S. finite S \<and> p permutes S)"
cases: fixes G :: "prog"   and a1 :: "ty"   and a2 :: "ty" assumes "G\<turnstile>a1\<preceq>? a2" obtains S :: "ty" and T :: "ty" where "a1 = Sa__" and "a2 = T__" and "G\<turnstile>Sa__\<preceq>T__"   | S :: "ty" and T :: "ty" where "a1 = S" and "a2 = T" and "G\<turnstile>S\<succ>T"
inject_2: fixes x2 :: "'d"   and y2 :: "'d" shows "(Lt x2 = Lt y2) = (x2 = y2)"
coinduct: fixes X :: "('g, 'h, 'i) gpv \<Rightarrow> ('j, 'k, 'l) gpv \<Rightarrow> bool"   and x1 :: "('g, 'h, 'i) gpv"   and x2 :: "('j, 'k, 'l) gpv"   and A :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'l \<Rightarrow> bool" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>gpv gpv'. x1 = gpv \<and> x2 = gpv' \<and> rel_spmf (rel_generat A C (R ===> (\<lambda>uu_ uua_. X uu_ uua_ \<or> rel_gpv'' A C R uu_ uua_))) (the_gpv gpv) (the_gpv gpv')" shows "rel_gpv'' A C R x1 x2"
simps_2: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and l :: "'a tree"   and y :: "'a"   and r :: "'a tree" shows "ins3_tm n d x \<langle>l, y, r\<rangle> =1 (case cmp x y of LT \<Rightarrow> ins3_tm n (d + 1) x l \<bind> up3_tm y r False | EQ \<Rightarrow> return Same2 | GT \<Rightarrow> ins3_tm n (d + 1) x r \<bind> up3_tm y l True)"
take_all_iff: fixes n :: "nat"   and xs :: "'a list" shows "(take n xs = xs) = (|xs| \<le> n)"
sum_defs_2: fixes a :: "V" shows "ZFC_Cardinals.Inl a \<equiv> \<langle>0, a\<rangle>"
simps_1: fixes t :: "'b tree" shows "root (mirror t) = root t"
compactE_image: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and C :: "'b set"   and f :: "'b \<Rightarrow> 'a set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "\<And>T. T \<in> C \<Longrightarrow> open (f T)"   and "S \<subseteq> \<Union> (f ` C)" obtains C' :: "'b set" where "C' \<subseteq> C" and "finite C'" and "S \<subseteq> \<Union> (f ` C')"
reads_singleton: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a" shows "reads {preserved f} f h h'"
natural_transformation_axioms: shows "natural_transformation B C H L \<tau>'\<tau>.map"
cong: fixes a :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "(a, a') \<in> E"   and "(b, b') \<in> E" shows "(a \<cdot> b, a' \<cdot> b') \<in> E"
eq: shows "gcd (m * k) (a * b) = gcd a m * gcd k b"
match_4: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d"   and f :: "'d"   and h :: "'d" assumes "category C"   and "partial_magma.arr C (C g f)"   and "partial_magma.arr C (C h g)" shows "partial_magma.arr C (C h (C g f))"
hs_dfs_pref_dfs: fixes \<Sigma>i :: "('a, unit) HashMap.hashmap"   and post :: "'a \<Rightarrow> 'a dlist" shows "wa_precise_refine (det_wa_wa (hs_dfs_dwa \<Sigma>i post)) (dfs_algo (hs.\<alpha> \<Sigma>i) (hs_R post)) hs_dfs_\<alpha>"
add_pos_pos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less zero b" shows "less zero (plus a b)"
integral_zero: fixes M :: "'c measure" shows "LINT x|M. (0::'b) = (0::'b)"
hseq_char': fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "B.hseq \<nu> \<mu> = (\<nu> \<star>\<^sub>B \<mu> \<noteq> B.null)"
divide_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f / g = f * floatarith.Inverse g"
commute: fixes a :: "'e multiset"   and b :: "'e multiset" shows "a \<union># b = b \<union># a"
one: shows "\<parallel>fs ! i\<parallel>\<^sup>2 \<le> int (N * m) * 1"
gc_fair: fixes Ns :: "('f \<times> 'l) set llist" assumes "chain (\<leadsto>GC) Ns"   and "active_subset (lhd Ns) = {}"   and "passive_subset (Liminf_llist Ns) = {}" shows "empty_ord.fair Ns"
ide_composite: shows "ide (\<mu> \<star> \<nu> \<star> \<pi>)"
ins: shows "insertion e (map_poly real_of_float p) = insertion e (map_poly real_of_float pl) + insertion e (map_poly real_of_float pr)"
3: shows "ipd (\<pi> k) pd\<rightarrow> ipd (\<pi> j)"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> synth H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>agt. P (Agent agt)"   and "\<And>n. P (Number n)"   and "\<And>X. \<lbrakk>X \<in> synth H; P X\<rbrakk> \<Longrightarrow> P (Hash X)"   and "\<And>X Y. \<lbrakk>X \<in> synth H; P X; Y \<in> synth H; P Y\<rbrakk> \<Longrightarrow> P \<lbrace>X, Y\<rbrace>"   and "\<And>X K. \<lbrakk>X \<in> synth H; P X; Key K \<in> H\<rbrakk> \<Longrightarrow> P (Crypt K X)" shows "P x"
stateless_callee_def: fixes callee :: "'h \<Rightarrow> ('i, 'k, 'l) gpv"   and s :: "'j" shows "stateless_callee callee s = map_gpv (\<lambda>b. (b, s)) id \<circ> callee"
lprefix_antisym: fixes xs :: "'f llist"   and ys :: "'f llist" assumes "lprefix xs ys"   and "lprefix ys xs" shows "xs = ys"
option_rel_id: fixes R :: "('a \<times> 'a) set" assumes "R = Id" shows "\<langle>R\<rangle>option_rel = Id"
conf_ClassD: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and C :: "char list" assumes "P,h \<turnstile> v :\<le> Class C" shows "v = Null \<or> (\<exists>a obj T. v = Addr a \<and> h a = \<lfloor>obj\<rfloor> \<and> obj_ty obj = T \<and> subtype P T (Class C))"
is_terminal: fixes x' :: "'c"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map x' (J_C.MkIde D) f" shows "\<exists>!g. x.is_coext x' f g"
pow_generator_eq_iff_cong: fixes x :: "nat"   and y :: "nat" assumes "finite (carrier G)" shows "(\<^bold>g [^] x = \<^bold>g [^] y) = [x = y] (mod order G)"
that: shows "(state_dp_consistency.crel_vs R OO local.rel_state (=)) a b"
elims_3: fixes x :: "bool \<times> (bool \<times> 'b) rexp" assumes "\<not> final_a x" obtains b :: "bool" and r :: "(bool \<times> 'b) rexp" where "x = (b, r)" and "\<not> (final r \<or> b \<and> nullable r)"
rpd_match: fixes i :: "nat"   and j :: "nat"   and test :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and r :: "'a Regex.regex" assumes "i < j" shows "Regex.match test r i j = (\<Squnion> (Regex.match test ` Regex.rpd test j r)) i (j - 1)"
ex_unique_moebius_three_points_fun: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo"   and w1 :: "complex_homo"   and w2 :: "complex_homo"   and w3 :: "complex_homo" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3"   and "w1 \<noteq> w2"   and "w1 \<noteq> w3"   and "w2 \<noteq> w3" shows "\<exists>!f. is_moebius f \<and> f z1 = w1 \<and> f z2 = w2 \<and> f z3 = w3"