hcju/isabelleps huggingface dataset:
sets_select_sets: fixes M :: "'c measure measure"   and N :: "'c measure" assumes "sets M = sets (subprob_algebra N)" shows "sets (select_sets M) = sets N"
1: shows "(w \<sqinter> - d) * d\<^sup>T * top = bot"
secure_refinement_simpler_def: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and abs_steps :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val) \<Rightarrow> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val) \<Rightarrow> nat" shows "secure_refinement_simpler \<R>\<^sub>A \<R> P abs_steps \<equiv> closed_others \<R> \<and> preserves_modes_mem \<R> \<and> new_vars_private \<R> \<and> simpler_refinement_safe \<R>\<^sub>A \<R> P abs_steps \<and> conc.closed_glob_consistent P \<and> (\<forall>c\<^sub>1\<^sub>A mds\<^sub>A mem\<^sub>1\<^sub>A c\<^sub>1\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C. (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>c\<^sub>1\<^sub>C' mds\<^sub>C' mem\<^sub>1\<^sub>C'. \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<exists>c\<^sub>1\<^sub>A' mds\<^sub>A' mem\<^sub>1\<^sub>A'. abs.neval \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A (abs_steps \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<langle>c\<^sub>1\<^sub>A', mds\<^sub>A', mem\<^sub>1\<^sub>A'\<rangle>\<^sub>A \<and> (\<langle>c\<^sub>1\<^sub>A', mds\<^sub>A', mem\<^sub>1\<^sub>A'\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C) \<in> \<R>)))"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'b" shows "Poly_Mapping.range x = Set.range (lookup x) - {0::'b}"
assms: shows "z \<cdot> y \<le> x \<cdot> z"
right_unique_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "right_unique R = (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
current_current_of_bipartite: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "('a + 'a) \<times> ('a + 'a) \<Rightarrow> ennreal" assumes "web \<Gamma>"   and "current (bipartite_web_of \<Gamma>) f"   and "wave (bipartite_web_of \<Gamma>) f" shows "current \<Gamma> (web.current_of_bipartite \<Gamma> f)"
powr_closure_def: fixes f :: "'b \<Rightarrow> real" shows "powr_closure f \<equiv> {\<lambda>x. f x powr p |p. True}"
open_real_greaterThanLessThan: fixes a :: "real"   and b :: "real" shows "open {a<..<b}"
ext_cf_def: fixes S :: "('d, 'e) Ring_scheme"   and n :: "nat"   and c :: "nat \<times> (nat \<Rightarrow> 'd)" shows "ext_cf S n c = (n + fst c, \<lambda>i. if n \<le> i then snd c (sliden n i) else \<zero>\<^bsub>S\<^esub>)"
not_succ_Limit: fixes i :: "V" shows "\<not> Limit (ZFC_in_HOL.succ i)"
Nml_Nmlize_2: fixes t :: "'a local.term" assumes "Arr t" shows "Src \<^bold>\<lfloor>t\<^bold>\<rfloor> = Src t"
remove_for_all_empty_children: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> forall_M remove children \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r []"
rep_eq: fixes x :: "'a"   and xa :: "('t, 'c) fmap" shows "fmlookup (shift_keys x xa) = (\<lambda>xb. if x adds\<^sub>p xb then fmlookup xa (xb \<ominus> x) else None)"
that: fixes m :: "real" assumes "b < m"   and "P m"   and "dist (flow0 x m) u < e" shows "thesis"
map_zip_map2: fixes f :: "'b \<times> 'c \<Rightarrow> 'a"   and xs :: "'b list"   and g :: "'d \<Rightarrow> 'c"   and ys :: "'d list" shows "map f (zip xs (map g ys)) = map2 (\<lambda>x y. f (x, g y)) xs ys"
functor_axioms: shows "functor B D (L \<circ> G)"
UNIV_eq_I: fixes A :: "'d set" assumes "\<And>x. x \<in> A" shows "UNIV = A"
sub_left_strict_mono_on: fixes k :: "nat" shows "SetInterval2.strict_mono_on (\<lambda>x. x - k) {k..}"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>c e. P (Eq (CN 0 c e))"   and "\<And>c e. P (NEq (CN 0 c e))"   and "\<And>c e. P (Lt (CN 0 c e))"   and "\<And>c e. P (Le (CN 0 c e))"   and "\<And>c e. P (Gt (CN 0 c e))"   and "\<And>c e. P (Ge (CN 0 c e))"   and "P T"   and "P F"   and "\<And>va. P (Lt (C va))"   and "\<And>va. P (Lt (Bound va))"   and "\<And>vd vb vc. P (Lt (CN (Suc vd) vb vc))"   and "\<And>va. P (Lt (Neg va))"   and "\<And>va vb. P (Lt (Add va vb))"   and "\<And>va vb. P (Lt (Sub va vb))"   and "\<And>va vb. P (Lt (Mul va vb))"   and "\<And>va. P (Lt (Floor va))"   and "\<And>va vb vc. P (Lt (CF va vb vc))"   and "\<And>va. P (Le (C va))"   and "\<And>va. P (Le (Bound va))"   and "\<And>vd vb vc. P (Le (CN (Suc vd) vb vc))"   and "\<And>va. P (Le (Neg va))"   and "\<And>va vb. P (Le (Add va vb))"   and "\<And>va vb. P (Le (Sub va vb))"   and "\<And>va vb. P (Le (Mul va vb))"   and "\<And>va. P (Le (Floor va))"   and "\<And>va vb vc. P (Le (CF va vb vc))"   and "\<And>va. P (Gt (C va))"   and "\<And>va. P (Gt (Bound va))"   and "\<And>vd vb vc. P (Gt (CN (Suc vd) vb vc))"   and "\<And>va. P (Gt (Neg va))"   and "\<And>va vb. P (Gt (Add va vb))"   and "\<And>va vb. P (Gt (Sub va vb))"   and "\<And>va vb. P (Gt (Mul va vb))"   and "\<And>va. P (Gt (Floor va))"   and "\<And>va vb vc. P (Gt (CF va vb vc))"   and "\<And>va. P (Ge (C va))"   and "\<And>va. P (Ge (Bound va))"   and "\<And>vd vb vc. P (Ge (CN (Suc vd) vb vc))"   and "\<And>va. P (Ge (Neg va))"   and "\<And>va vb. P (Ge (Add va vb))"   and "\<And>va vb. P (Ge (Sub va vb))"   and "\<And>va vb. P (Ge (Mul va vb))"   and "\<And>va. P (Ge (Floor va))"   and "\<And>va vb vc. P (Ge (CF va vb vc))"   and "\<And>va. P (Eq (C va))"   and "\<And>va. P (Eq (Bound va))"   and "\<And>vd vb vc. P (Eq (CN (Suc vd) vb vc))"   and "\<And>va. P (Eq (Neg va))"   and "\<And>va vb. P (Eq (Add va vb))"   and "\<And>va vb. P (Eq (Sub va vb))"   and "\<And>va vb. P (Eq (Mul va vb))"   and "\<And>va. P (Eq (Floor va))"   and "\<And>va vb vc. P (Eq (CF va vb vc))"   and "\<And>va. P (NEq (C va))"   and "\<And>va. P (NEq (Bound va))"   and "\<And>vd vb vc. P (NEq (CN (Suc vd) vb vc))"   and "\<And>va. P (NEq (Neg va))"   and "\<And>va vb. P (NEq (Add va vb))"   and "\<And>va vb. P (NEq (Sub va vb))"   and "\<And>va vb. P (NEq (Mul va vb))"   and "\<And>va. P (NEq (Floor va))"   and "\<And>va vb vc. P (NEq (CF va vb vc))"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
l_get_shadow_root_safe_def: fixes get_shadow_root_safe :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog" shows "l_get_shadow_root_safe get_shadow_root_safe \<equiv> \<forall>element_ptr h. pure (get_shadow_root_safe element_ptr) h"
psubsetD: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "order.greater B A"   and "c \<in> A" shows "c \<in> B"
assms_1: shows "continuous_on (apply_chart c ` s) (f \<circ> inv_chart c)"
ereal_INF_uminus_eq: fixes f :: "'a \<Rightarrow> ereal"   and S :: "'a set" shows "(INF x\<in>S. - f x) = - Sup (f ` S)"
in_degree_app_iso_eq: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and u :: "'e" assumes "wf_digraph G"   and "pre_digraph.digraph_isomorphism G hom"   and "u \<in> verts G" shows "in_degree (app_iso hom G) (iso_verts hom u) = in_degree G u"
power_dvd_iff_le_multiplicity: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "dvd.dvd times (power.power one times p n) x = (n \<le> factorial_semiring.multiplicity times one p x)"
hb_consistent_append: fixes suffix :: "'a list"   and prefix :: "'a list" assumes "hb_consistent suffix"   and "hb_consistent prefix"   and "\<And>s p. \<lbrakk>s \<in> set suffix; p \<in> set prefix\<rbrakk> \<Longrightarrow> \<not> s \<prec> p" shows "hb_consistent (prefix @ suffix)"
one_index: fixes i :: "'c" shows "1 $ i = (1::'b)"
pos_recurrent_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a" shows "MC_syntax.pos_recurrent K x = (MC_syntax.recurrent K x \<and> MC_syntax.U' K x x \<noteq> \<infinity>)"
A: fixes a :: "'a" shows "((1::'a) l\<rightarrow> a) l\<rightarrow> (1::'a) = (1::'a)"
MDSV: shows "min_dist s v = length p1 + 1"
nonPrincipalID: fixes r :: "'a sequent list \<times> 'a sequent"   and A :: "'a form" assumes "r \<in> Ax" shows "\<not> rightPrincipal r A \<and> \<not> leftPrincipal r A"
snd_safe_distance_def: shows "snd_safe_distance = (v\<^sub>o - v\<^sub>e)\<^sup>2 / (2 * (a\<^sub>o - a\<^sub>e))"
simple_integral_mono: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal" assumes "simple_function M f"   and "simple_function M g"   and "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x" shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
l_coset_def: fixes G :: "('e, 'f) monoid_scheme"   and a :: "'e"   and H :: "'e set" shows "a <#\<^bsub>G\<^esub> H = (\<Union>h\<in>H. {a \<otimes>\<^bsub>G\<^esub> h})"
chain_iterates: fixes f :: "'a set \<Rightarrow> 'a set" assumes "mono f" shows "chain (\<lambda>n. (f ^^ n) {})"
bij_imp_bij_inv: fixes f :: "'a \<Rightarrow> 'b" assumes "bij f" shows "bij (inv f)"
Bignorabimus_equiv: fixes \<alpha> :: "game" shows "Bignorabimus \<alpha> = {M. \<forall>I \<omega> X. \<omega> \<in> game_sem I \<alpha> X \<longrightarrow> \<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> M)}"
X: shows "entropy b S X = - (LINT x|(S \<Otimes>\<^sub>M T). Pxy x * log b (Px (fst x)))"
numeral_power_int_eq_of_complex_cancel_iff: fixes x :: "num"   and n :: "int"   and y :: "complex" shows "(numeral x powi n = of_complex y) = (numeral x powi n = y)"
cond_bind_spmf_fst_map_spmf_fst: fixes p :: "'h spmf"   and f :: "'h \<Rightarrow> ('i \<times> 'j) spmf"   and x :: "'i" shows "cond_bind_spmf_fst p (map_spmf fst \<circ> f) x = cond_bind_spmf p f ({x} \<times> UNIV)"
max_y_impl_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and j :: "nat" shows "max_y_impl a b x j = (if j < length b \<and> big_e a b x j \<noteq> [] then Min (set (big_e a b x j)) else Max (set a))"
of_fun_def: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "of_fun f n = IArray (map f [0..<n])"
nil_iter: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice" assumes "finite_or_infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "finite_or_infinite_iteration.iter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) \<top> = nil"
suffix_ConsI: fixes xs :: "'a list"   and ys :: "'a list"   and y :: "'a" assumes "suffix xs ys" shows "suffix xs (y # ys)"
mult_relative_homology_group: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "(\<otimes>\<^bsub>relative_homology_group (int p) X S\<^esub>) = (\<lambda>R S. \<Union>r\<in>R. \<Union>s\<in>S. {r + s})"
Un_insert_right: fixes A :: "'k set"   and a :: "'k"   and B :: "'k set" shows "A \<union> insert a B = insert a (A \<union> B)"
face_of_imp_subset: fixes T :: "'a set"   and S :: "'a set" assumes "T face_of S" shows "T \<subseteq> S"
multi_member_this: fixes x :: "'f"   and XS :: "'f multiset" shows "x \<in># {#x#} + XS"
disc_1: fixes x1 :: "'a" shows "isl (Inl x1)"
induct: fixes P :: "('c \<Rightarrow> 'd) \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> 'd"   and a1 :: "'d \<Rightarrow> bool"   and a2 :: "'c list" assumes "\<And>f p. P f p []"   and "\<And>f p a as. (\<And>x. \<lbrakk>x = f a; \<not> p x\<rbrakk> \<Longrightarrow> P f p as) \<Longrightarrow> P f p (a # as)" shows "P a0 a1 a2"
inject: fixes x :: "'d"   and ya :: "'d" shows "(node_ptr.Ext x = node_ptr.Ext ya) = (x = ya)"
int_simps_1: fixes x :: "'b \<Rightarrow> 'c" shows "\<turnstile> x = x = #True"
emb_coerce: fixes x :: "'e" assumes "DEFL('e) \<sqsubseteq> DEFL('f)" shows "EMB('f)\<cdot>(COERCE('e, 'f)\<cdot>x) = EMB('e)\<cdot>x"
PowD: fixes A :: "'c set"   and B :: "'c set" assumes "A \<in> Pow B" shows "A \<subseteq> B"
mset_concat_map_cons: fixes F :: "'c \<Rightarrow> 'b"   and G :: "'c \<Rightarrow> 'b list"   and S :: "'c list" shows "mset (concat (map (\<lambda>c. F c # G c) S)) = mset (map F S) + mset (concat (map G S))"
presFreshAll_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresFreshAll h hA (fromMOD MOD)" shows "presFreshAll (fromIMor h) (fromIMorAbs hA) MOD"
InvokeElim: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and C :: "Class"   and m :: "Method" assumes "(M, l, s, n, ll, t) \<in> Step"   and "get_ins M l = Some (invokeS C m)" shows "\<exists>code l0 v ops hh u k R par. mbody_is C m (par, code, l0) \<and> (fst s, par, R, ops) \<in> Frame \<and> ((par, code, l0), l0, ([], R, snd (snd s)), k, hh, v) \<in> Exec \<and> n = Suc k \<and> t = (v # ops, fst (snd s), hh) \<and> ll = l + 1"
bind_def: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'a list" shows "List.bind xs f = concat (map f xs)"
explode_union: fixes a :: "ZF"   and b :: "ZF" shows "explode (HOLZF.union a b) = explode a \<union> explode b"
a: shows "a = HMA_Connect.spectral_radius (\<chi>i j. complex_of_real (st_mat A $h i $h j))"
rep_Bonk_Schramm_extension_def: shows "rep_Bonk_Schramm_extension \<equiv> quot_type.rep Rep_Bonk_Schramm_extension"
Un_Diff_Int: fixes A :: "'h set"   and B :: "'h set" shows "A - B \<union> A \<inter> B = A"
silent_move_into_silent_move2: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "\<tau>mredT s s'" shows "\<exists>tl. mthr.silent_move2 s tl s'"
sor: shows "(\<sigma>, s) \<in> oreachable (opnet onp p\<^sub>1) (\<lambda>a b. oarrivemsg I a) (other U (net_tree_ips p\<^sub>1))"
ltl_const_equiv_implies_prop_equiv_1: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>C true\<^sub>n = \<phi> \<sim>\<^sub>P true\<^sub>n"
sum_power2_eq_zero_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(plus (power.power one times x 2) (power.power one times y 2) = zero) = (x = zero \<and> y = zero)"
Or_def: fixes P :: "('d, 'e, 'f) formula"   and Q :: "('d, 'e, 'f) formula" shows "(P || Q) = ! (! P && ! Q)"
inv_ordsetmap: fixes a :: "'b"   and b :: "'b" assumes "a \<in> f ` P"   and "b \<in> f ` P"   and "codomain.greater_eq b a" shows "the_inv_into P f b \<succeq> the_inv_into P f a"
interp_ins_def: fixes ops :: "('b \<times> 'b option) list" shows "interp_ins ops \<equiv> foldl insert_spec [] ops"
le_take_same: fixes i :: "nat"   and j :: "nat"   and a :: "'a list"   and b :: "'a list" assumes "i \<le> j"   and "take j a = take j b" shows "take i a = take i b"
not_dbm_lt_le_impl: fixes a :: "'d"   and b :: "'d" assumes "\<not> Lt a \<prec> Le b" shows "b < a"
isPath_bwd_cases: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "isPath s p t" obtains "p = []" and "t = s"   | p' :: "(nat \<times> nat) list" and u :: "nat" where "p = p' @ [(u, t)]" and "isPath s p' u" and "(u, t) \<in> E"
if_is_pr: fixes p :: "nat \<Rightarrow> nat"   and q1 :: "nat \<Rightarrow> nat"   and q2 :: "nat \<Rightarrow> nat" assumes "p \<in> PrimRec1"   and "q1 \<in> PrimRec1"   and "q2 \<in> PrimRec1" shows "(\<lambda>x. if p x = 0 then q1 x else q2 x) \<in> PrimRec1"
T_init_observation_eq: fixes s :: "'a"   and o :: "'b"   and os :: "'b list"   and o' :: "'b" shows "emeasure (T (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>xs \<omega>'. \<omega> = xs @- \<omega>' \<and> map snd xs = os} = emeasure (T (s, o')) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>xs \<omega>'. \<omega> = xs @- \<omega>' \<and> map snd xs = os}"
preserves_comp_2: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'a \<times> 'd \<Rightarrow> 'e"   and f2' :: "'d"   and f2 :: "'d" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A2 (A2 f2' f2)" shows "(\<lambda>f1. F (f1, A2 f2' f2)) = vertical_composite.map A1 B (\<lambda>f1. F (f1, f2)) (\<lambda>f1. F (f1, f2'))"
closed_imp_locally_compact: fixes S :: "'a set" assumes "closed S" shows "locally compact S"
OrdP_sf: fixes t :: "tm" shows "Sigma_fm (OrdP t)"
reciprocal_degree: fixes P :: "'b poly" shows "reciprocal_poly (degree P) P = reflect_poly P"
instance_of\<^sub>t\<^sub>s_def: fixes ts\<^sub>1 :: "fterm list"   and ts\<^sub>2 :: "fterm list" shows "instance_of\<^sub>t\<^sub>s ts\<^sub>1 ts\<^sub>2 = (\<exists>\<sigma>. ts\<^sub>1 = ts\<^sub>2 \<cdot>\<^sub>t\<^sub>s \<sigma>)"
assms_2: shows "\<not> is_Prim\<^sub>0 \<^bold>\<lfloor>u\<^bold>\<rfloor>"
monoI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<sqsubseteq> f y" shows "order.mono less_eq f"
WT_gpvD_2: fixes \<Gamma> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and out :: "'a"   and c :: "('c, 'a, 'b) rpv"   and input :: "'b" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out" shows "\<Gamma> \<turnstile>g c input \<surd>"
to_hma\<^sub>m_row_def: fixes A :: "'b mat"   and i :: "'d" shows "to_hma\<^sub>m_row A i = (\<chi>j. A $$ (mod_type_class.to_nat i, mod_type_class.to_nat j))"
invfun_mem: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and b :: "'f" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B"   and "b \<in> B" shows "(f\<inverse>\<^bsub>B,A\<^esub>) b \<in> A"
disjunct_set_mset_diff: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "M \<inter># N = \<Empt>" shows "set_mset (M - N) = set_mset M"
prod_mset_image: fixes X :: "int poly multiset" shows "\<Prod>\<^sub># (image_mset of_int_poly X) = of_int_poly (\<Prod>\<^sub># X)"
wf_step: fixes S :: "'a set" assumes "finite S" shows "wf {((ts', ws', r'), ts, ws, r). step (ts, ws, r) (ts', ws', r') \<and> set ts' \<subseteq> S \<and> dom ws \<subseteq> {0..<n} \<and> ran ws \<subseteq> S}"
no_head_genE: fixes u :: "'b list"   and C :: "'b list set"   and b :: "'b list" assumes "u \<in> no_head_gen C b" obtains z :: "'b list" and k :: "nat" where "z \<in> C" and "z \<noteq> b" and "u = z \<cdot> b \<^sup>@ k"
length_Casts_vs: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Ts :: "ty list"   and vs :: "val list"   and vs' :: "val list" assumes "P \<turnstile> Ts Casts vs to vs' " shows "length Ts = length vs"
real_lepoll_natnat: shows "UNIV \<lesssim> Pow UNIV"
Hadjicostas_Chapman_formula_aux: fixes z :: "complex" assumes "- 2 < Re z" shows "Hadjicostas_integral z = Gamma (z + 2) * pre_zeta 1 (z + 2)"
step_11: shows "\<alpha> = Cantor_sum \<alpha>s ms"
rec_nat_0_imp: fixes f :: "nat \<Rightarrow> 'c"   and f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c \<Rightarrow> 'c" assumes "f = rec_nat f1 f2" shows "f 0 = f1"
semiring_norm_68: fixes n :: "num" shows "(num.One \<le> n) = True"
inject_1: fixes x11 :: "e list"   and x12 :: "e list"   and y11 :: "e list"   and y12 :: "e list" shows "(LBase x11 x12 = LBase y11 y12) = (x11 = y11 \<and> x12 = y12)"
simps_5: fixes m :: "'b match_expr" shows "matcheq_matchNone (MatchNot (MatchNot m)) = matcheq_matchNone m"
le_Integer_numeral_simps_3: shows "le\<cdot>1\<cdot>0 = FF"
dual_all: fixes P :: "'b dual \<Rightarrow> bool" shows "(\<forall>x. P (dual x)) = (\<forall>x'. P x')"
reachable_step: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a" shows "greachable A w V p = insert p (\<Union> (greachable A w V ` gsuccessors A w V p))"
a_consistent_var\<^sub>2: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, e, Upd x # S)" shows "a_consistent (ae, a, as) ((x, e) # \<Gamma>, e, S)"