hcju/isabelleps huggingface dataset:
mult_left_le_one_le: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and x :: "'e"   and y :: "'e" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x"   and "less_eq zero y"   and "less_eq y one" shows "less_eq (times y x) x"
gen_subalgebra_sig_sets: fixes G :: "'b set set"   and M :: "'b measure" assumes "G \<subseteq> Pow (space M)" shows "sets (gen_subalgebra M G) = sigma_sets (space M) G"
boundedI: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> x \<le> a" shows "x \<le> Min A"
image_mset_add_mset: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e"   and M :: "'e multiset" shows "image_mset f (add_mset a M) = add_mset (f a) (image_mset f M)"
configuration_non_empty: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "configuration (token_run x n) n \<noteq> {}"
Nonce_supply_lemma: fixes evs :: "event list" shows "\<exists>N. \<forall>n\<ge>N. Nonce n \<notin> used evs"
mix_pmf_pref_iff_more_likely: fixes b :: "'a pmf"   and w :: "'a pmf"   and \<alpha> :: "real"   and \<beta> :: "real" assumes "b \<succeq>[\<R>] w \<and> (w, b) \<notin> \<R>"   and "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}" shows "(\<beta> < \<alpha>) = (mix_pmf \<alpha> b w \<succeq>[\<R>] mix_pmf \<beta> b w \<and> (mix_pmf \<beta> b w, mix_pmf \<alpha> b w) \<notin> \<R>)"
top_unclocked_ignore: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and tp :: "top0"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and cnt :: "nat" assumes "evaluate_top ck env s tp (s', r)"   and "r \<noteq> Rerr (Rabort Rtimeout_error)" shows "evaluate_top False env (update_clock (\<lambda>_. cnt) s) tp (update_clock (\<lambda>_. cnt) s', r)"
set_replicate_conv_if: fixes n :: "nat"   and x :: "'a" shows "set (replicate n x) = (if n = 0 then {} else {x})"
in_out_arcs_reflexive: fixes v :: "'a"   and e :: "'b" assumes "v \<in> verts G"   and "e \<in> in_arcs G v" shows "\<exists>e'. e' \<in> out_arcs G v \<and> head G e' = tail G e"
makestep: shows "\<forall>i. X (Suc i) = make_step selector (X i)"
snd_conv: fixes x1 :: "'g"   and x2 :: "'f" shows "snd (x1, x2) = x2"
Int_emptyI: fixes A :: "'x set"   and B :: "'x set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
llength_LCons: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "LList2.llength (a ## r) = Suc (LList2.llength r)"
function_mult_def: fixes S :: "'d set"   and R :: "('e, 'f) ring_scheme"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "function_mult S R f g = (\<lambda>x\<in>S. f x \<otimes>\<^bsub>R\<^esub> g x)"
assms_4: fixes y :: "nat" shows "0 \<le> g y"
div_mult1_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (times a b) c = plus (times a (divide b c)) (divide (times a (modulo b c)) c)"
simps_1: fixes n :: "nat"   and t :: "'b tree234" shows "full\<^sub>i n (T\<^sub>i t) = full n t"
add2_Some: fixes x :: "'a option"   and y :: "'a"   and z :: "'a" shows "(x ++ Some y = Some z) = (\<exists>x'. x = Some x' \<and> x' \<oplus> y = Some z)"
derived_set_of_empty: fixes X :: "'c topology" shows "X derived_set_of {} = {}"
neq_fund_solI: fixes y :: "nat" assumes "\<not> is_square (Suc (y\<^sup>2 * D))" shows "y \<noteq> snd fund_sol"
a_remove_child_locs_def: fixes ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" shows "Shadow_DOM.remove_child_locs ptr owner_document = Shadow_DOM.set_child_nodes_locs ptr \<union> set_disconnected_nodes_locs owner_document"
not_equal_def: shows "Uprod.not_equal \<equiv> Abs_commute (\<noteq>)"
lunit_char_2: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and a :: "'d" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "T (monoidal_category.unity C \<iota>, monoidal_category.lunit C T \<alpha> \<iota> a) = C (T (\<iota>, a)) (category.inv C (\<alpha> (monoidal_category.unity C \<iota>, monoidal_category.unity C \<iota>, a)))"
assms: shows "SubRel.subpath g v1 es v2 subs"
try_gpv_bind_gpv: fixes gpv :: "('d, 'b, 'c) gpv"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) gpv"   and gpv' :: "('a, 'b, 'c) gpv" shows "TRY bind_gpv gpv f ELSE gpv' = bind_gpv (TRY map_gpv Some id gpv ELSE Done None) (\<lambda>x. case x of None \<Rightarrow> gpv' | Some x' \<Rightarrow> TRY f x' ELSE gpv')"
before_def: fixes ev :: "event"   and evs :: "event list" shows "(before ev on evs) = takeWhile (\<lambda>z. z \<noteq> ev) (rev evs)"
meet_comm: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
comp_eq_dest_lhs: fixes a :: "'c \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'c"   and c :: "'a \<Rightarrow> 'b"   and v :: "'a" assumes "a \<circ> b = c" shows "a (b v) = c v"
cas_append_iff: fixes u :: "'a"   and p :: "'b list"   and q :: "'b list"   and v :: "'a" shows "cas u (p @ q) v = (cas u p (awlast u p) \<and> cas (awlast u p) q v)"
nonneg_mod_div: fixes a :: "int"   and b :: "int" assumes "0 \<le> a"   and "0 \<le> b" shows "0 \<le> a mod b \<and> 0 \<le> a div b"
dim_span: fixes S :: "('a, 'b) vec set" shows "vec.dim (vec.span S) = vec.dim S"
of_nat_monom: shows "of_nat = MPoly_Type.monom 0 \<circ> of_nat"
asymp_equiv_compose: fixes f :: "'b \<Rightarrow> 'c"   and G :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "f \<sim>[G] g"   and "filterlim h G F" shows "f \<circ> h \<sim>[F] g \<circ> h"
add: fixes x :: "'a"   and y :: "'a" shows "f (x + y) = f x + f y"
group_l_invI: assumes "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y\<in>carrier R. y \<oplus> x = \<zero>" shows "Group.group (add_monoid R)"
wf_eq_minimal: fixes r :: "('c \<times> 'c) set" shows "wf r = (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q))"
trss_c'_split_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> trss fg" obtains csp :: "'n list multiset" where "c' = csp + c" and "mon_c fg csp = {}"
RSadj_cancel: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec" shows "RSadj x y (RSadj x y \<nu>) = \<nu>"
aux7: fixes a :: "'b set set"   and b :: "'b set set"   and p :: "('b set set, 'c) Combinators list"   and c :: "'b set set"   and d :: "'b set set" assumes "DenyAllFromTo a b \<in> set p"   and "allNetsDistinct (DenyAllFromTo c d # p)"   and "a \<noteq> c \<or> b \<noteq> d" shows "twoNetsDistinct a b c d"
fid_on_vempty: shows "fid_on []\<^sub>\<circ> = []\<^sub>\<circ>"
has_bochner_integral_implies_finite_norm: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "has_bochner_integral M f x" shows "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>"
finite_EqButGH: fixes p :: "pvar"   and p' :: "pvar" shows "finite {EqOnUC p p' u c |u c. (u, c) \<in> (UNIV - GH) \<times> UNIV}"
pair_measureI: fixes x :: "'d set"   and A :: "'d measure"   and y :: "'e set"   and B :: "'e measure" assumes "x \<in> sets A"   and "y \<in> sets B" shows "x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
polyn_add_n: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "pol_coeff S (n, f)"   and "pol_coeff S (n, g)" shows "polyn_expr R X n (n, f) \<plusminus> polyn_expr R X n (n, g) = \<Sigma>\<^sub>e R (\<lambda>j. (f j \<plusminus>\<^bsub>S\<^esub> g j) \<cdot>\<^sub>r X^\<^bsup>R j\<^esup>) n"
entt_disjI1': fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<Longrightarrow>\<^sub>t B" shows "A \<Longrightarrow>\<^sub>t B \<or>\<^sub>A C"
mult_scaleR_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "real"   and x :: "'a"   and y :: "'a" assumes "class.real_algebra plus zero minus uminus scaleR times" shows "times (scaleR a x) y = scaleR a (times x y)"
assms_2: shows "\<not> (\<exists>n''<n'. \<exists>nid''. (n'', nid'') \<in> his t n nid)"
continuous_map_into_fulltopology: fixes X :: "'a topology"   and X' :: "'b topology"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_map X (subtopology X' T) f" shows "continuous_map X X' f"
ACI_ACI_nPlus: fixes r1 :: "'b rexp"   and s1 :: "'b rexp"   and r2 :: "'b rexp"   and s2 :: "'b rexp" assumes "ACI r1 s1"   and "ACI r2 s2" shows "ACI (ACI_nPlus r1 r2) (Plus s1 s2)"
normal_oDeriv: fixes F :: "ordinal \<Rightarrow> ordinal" shows "normal (oDeriv F)"
Abs_ainj_all: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and X' :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "good X"   and "good X'" shows "(Abs xs x X = Abs xs' x' X') = (xs = xs' \<and> (\<forall>y. (y = x \<or> fresh xs y X) \<and> (y = x' \<or> fresh xs y X') \<longrightarrow> X #[y \<and> x]_xs = X' #[y \<and> x']_xs))"
find_exist: fixes a :: "'c"   and xs :: "('c, 'd) bintree option list"   and t :: "('c, 'd) bintree" assumes "Binomial_Queue.find a xs = Some t" shows "Some t \<in> set xs"
iia_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and S :: "'d set"   and Is :: "'c set" shows "iia scf S Is \<equiv> \<forall>P P' x y. profile S Is P \<and> profile S Is P' \<and> x \<in> S \<and> y \<in> S \<and> (\<forall>i\<in>Is. (x \<^bsub>(P i)\<^esub>\<preceq> y) = (x \<^bsub>(P' i)\<^esub>\<preceq> y) \<and> (y \<^bsub>(P i)\<^esub>\<preceq> x) = (y \<^bsub>(P' i)\<^esub>\<preceq> x)) \<longrightarrow> (x \<^bsub>(scf P)\<^esub>\<preceq> y) = (x \<^bsub>(scf P')\<^esub>\<preceq> y)"
assms_1: shows "\<forall>i\<in>net_ips s1 \<union> net_ips s2. the ((netmap s1 ++ netmap s2) i) = \<sigma> i"
sample_vars_empty: fixes X :: "'v \<Rightarrow> int pmf"   and m :: "'m" shows "sample_vars X {} m = m"
WCFG_WhileBody: fixes c' :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node"   and b :: "expr" assumes "c' \<turnstile> n -et\<rightarrow> n'"   and "n \<noteq> (_Entry_)"   and "n' \<noteq> (_Exit_)" shows "while (b) c' \<turnstile> n \<oplus> 2 -et\<rightarrow> n' \<oplus> 2"
B: shows "z = \<lbrakk>\<guillemotleft>B\<guillemotright>\<rbrakk>e"
inv\<^sub>3E_2: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V" shows "B\<^sub>2 \<subseteq> S"
freeInQuadraticSub: shows "\<forall>At. eval (quadratic_sub var a b c d At) ((xs' @ x # xs)[var := sqrt C]) = eval (quadratic_sub var a b c d At) (xs' @ x # xs)"
map_carrier_vec: fixes h :: "'b \<Rightarrow> 'a"   and v :: "'b Matrix.vec"   and n :: "nat" shows "(map_vec h v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
finfx: shows "finite {i \<in> I. f i (x i) \<noteq> \<one>}"
reducible: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V" shows "\<exists>Us. (\<forall>U\<in>set Us. IrrFinGroupRepresentation G smult U) \<and> 0 \<notin> set Us \<and> V = (\<Oplus>U\<leftarrow>Us. U)"
Exit_path: fixes n :: "'node" assumes "valid_node n" shows "\<exists>as. n -as\<rightarrow>\<^sub>\<surd>* (_Exit_)"
AbstTermP_Zero: fixes t :: "tm"   and v :: "name" shows "{OrdP t} \<turnstile> AbstTermP \<guillemotleft>Var v\<guillemotright> t Zero Zero"
simps_3: fixes f :: "'b \<Rightarrow> bool"   and a :: "'b"   and as :: "'b negation_type list" shows "cnf_to_bool f (Neg a # as) = (\<not> f a \<and> cnf_to_bool f as)"
blinfun_apply_blinfun_compose: fixes a :: "'c \<Rightarrow>\<^sub>L 'a"   and b :: "'b \<Rightarrow>\<^sub>L 'c"   and c :: "'b" shows "blinfun_apply (a o\<^sub>L b) c = blinfun_apply a (blinfun_apply b c)"
unfold: shows "matches = (\<Lambda> pat. smap\<cdot>slength oo sfilter\<cdot>(endswith\<cdot>pat) oo sinits)"
B: shows "\<forall>\<^sub>F x in F. 0 \<le> f x"
numeral_tree_simps_1: fixes n :: "num" shows "root (numeral n) = numeral n"
convex_hull_finite_union_cones: fixes I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}" shows "convex hull \<Union> (S ` I) = sum S I"
ln_exp_cf2_upper_bound_neg: fixes x :: "real" assumes "x \<le> 0" shows "x \<le> ln (exp_cf2 x)"
inv_one_finalD': fixes g :: "graph"   and f :: "face"   and a :: "nat"   and b :: "nat" assumes "Invariants.inv g"   and "f \<in> \<F> g"   and "\<not> final f"   and "(a, b) \<in> \<E> f" shows "\<exists>f'\<in>\<F> g. final f' \<and> f' \<noteq> f \<and> (b, a) \<in> \<E> f'"
sup_mono: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (sup a b) (sup c d)"
subst_comp_all: fixes \<delta> :: "'e \<Rightarrow> ('c, 'f) Term.term"   and \<theta> :: "'f \<Rightarrow> ('c, 'd) Term.term"   and M :: "('c, 'e) Term.term set" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta> \<circ>\<^sub>s \<theta> = M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta> \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
strip_all_vars_subst_bv_Fv: fixes x :: "variable"   and \<tau> :: "typ"   and B :: "Core.term" shows "strip_all_vars (subst_bv (Fv x \<tau>) B) = strip_all_vars B"
compact_Inter: fixes \<F> :: "'b set set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S"   and "\<F> \<noteq> {}" shows "compact (\<Inter> \<F>)"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg (f' \<star> f) (g \<star> g') f'fgg'.\<eta> f'fgg'.\<epsilon>"
antisym_conv3: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
SXAlloc_def2: fixes G :: "prog"   and P :: "'b assn"   and Y :: "vals"   and s :: "state"   and Z :: "'b" shows "SXAlloc G P Y s Z = (\<forall>s'. G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s' \<longrightarrow> P Y s' Z)"
interpret_floatarith_divide: fixes a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Mult a (Inverse b)) vs = interpret_floatarith a vs / interpret_floatarith b vs"
distinct_1: fixes x2 :: "'b" shows "None \<noteq> Some x2"
c5: assumes "ip \<in> kD rt"   and "\<pi>\<^sub>3 r = unk" shows "P (rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r)))"
slo_defs_4: fixes r :: "('a \<times> 'a) set" shows "irrefl r = (\<forall>a. (a, a) \<notin> r)"
nf_ACI_norm_ACI: fixes \<phi> :: "('a, 'b) aformula" shows "nf_ACI \<langle>\<phi>\<rangle>"
assms_4: fixes n :: "nat" assumes "1 \<le> n" shows "cmod (f n) \<le> C"
simps_15: fixes \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula" shows "Formula.future_bounded (formula.Until \<phi> I \<psi>) = (Formula.future_bounded \<phi> \<and> Formula.future_bounded \<psi> \<and> right I \<noteq> \<infinity>)"
lin_indpt_cone_def: shows "lin_indpt_cone \<equiv> \<Union> {finite_cone B |B. B \<subseteq> A \<and> card B = dim_span A \<and> lin_indpt B}"
degree_lagrange_interpolation_poly: fixes xs_ys :: "('a \<times> 'a) list" shows "degree (lagrange_interpolation_poly xs_ys) \<le> length xs_ys - 1"
osum_def: fixes r :: "('e \<times> 'e) set"   and r' :: "('f \<times> 'f) set" shows "r +o r' = map_prod Inl Inl ` r \<union> map_prod Inr Inr ` r' \<union> {(Inl a, Inr a') |a a'. a \<in> Field r \<and> a' \<in> Field r'}"
0: fixes x :: "complex" assumes "dist \<xi> x < r" shows "((\<lambda>x. exp (h x) / g x) has_field_derivative 0) (at x)"
l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs"
span_eq_real: fixes S :: "'c set" assumes "real_vector_space_on S" shows "module_on.span (*\<^sub>R) = span"
le_add_diff_inverse: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
unique_moebius_pt: fixes M1 :: "moebius"   and M2 :: "moebius" assumes "moebius_pt M1 = moebius_pt M2" shows "M1 = M2"
toplevel_summands_lang_UN: fixes n :: "nat"   and s :: "'b rexp" shows "lang n s = \<Union> (lang n ` toplevel_summands s)"
Rep_cases: fixes y :: "atom" assumes "y \<in> {a. sort_of a = s}" obtains x :: "'a" where "y = Rep x"
list_all2_eq_fun_conv_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'b list"   and ys :: "'c list" shows "list_all2 (\<lambda>x y. x = f y) xs ys = (xs = map f ys)"
prod: fixes g :: "'a \<Rightarrow> nat"   and A :: "'a set" shows "\<Phi> (prod g A) = (\<Prod>x\<in>A. \<Phi> (g x))"
le_of_nat_term_order_zero_min: fixes to :: "'a nat_term_order"   and t :: "'a" shows "le_of_nat_term_order to (0::'a) t"
"2_3": shows "v \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
dim_row_of_one_mat_less_pow: fixes m :: "nat"   and G1 :: "complex Matrix.mat"   and n :: "nat"   and G2 :: "complex Matrix.mat"   and i :: "nat" assumes "gate m G1"   and "gate n G2"   and "i < dim_row (1\<^sub>m (dim_col G1 * dim_col G2))" shows "i < 2 ^ (m + n)"
is_SNF_PSQ: shows "is_SNF A\<^sup>T (P, S, Q)"
simps: fixes e :: "'a"   and a :: "'b"   and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "BsSkewBinomialHeapStruc.elem_to_mset (BsSkewBinomialHeapStruc.Element e a q) = {#(e, a)#} + \<Sum>\<^sub># (image_mset BsSkewBinomialHeapStruc.elem_to_mset (image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q))))"
phi_1: shows "\<forall>x\<in>{0..1}. \<gamma>1 x = (\<gamma>2 \<circ> \<phi>) x"
bisimParAssoc: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> Q \<parallel> R \<sim> P \<parallel> (Q \<parallel> R)"
induct: fixes x :: "ind"   and P :: "ind \<Rightarrow> bool" assumes "Nat.Nat x"   and "P Zero_Rep"   and "\<And>i. \<lbrakk>Nat.Nat i; P i\<rbrakk> \<Longrightarrow> P (Suc_Rep i)" shows "P x"
inf_compl_bot_left2: fixes x :: "'b upred"   and y :: "'b upred" shows "(x \<and> \<not> x \<and> y) = false"
igFreshIGAbs1_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igFreshIGAbs1 MOD \<equiv> \<forall>ys y s X. isInBar (ys, s) \<and> igWls MOD s X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD ys y X)"
right_exact_surjec: fixes M :: "('f, 'a) Module"   and N :: "('g, 'a) Module"   and f :: "'f \<Rightarrow> 'g"   and p :: "'g \<Rightarrow> 'h"   and e :: "'h" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "p \<in> mHom R N (Zm R e)"   and "exact3 R M f N p (Zm R e)" shows "surjec\<^bsub>M,N\<^esub> f"
continuous_on_minus: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. - f x)"
elims_2: fixes x :: "'b nres"   and xa :: "'b nres" assumes "x \<le> xa" obtains uu_ :: "'b nres" where "x = uu_" and "xa = FAILi"   | a :: "'b set" and b :: "'b set" where "x = RES a" and "xa = RES b" and "a \<subseteq> b"
less: fixes n :: "nat" shows "emeasure lebesgue (S - C n) < ennreal ((1 / 2) ^ n)"
ord_fam_lifted_q: fixes q :: "'q" assumes "q \<in> Q" shows "tiebreaker_lifting Bot_FL Inf_FL Bot_G (entails_q q) (Inf_G_q q) (Red_I_q q) (Red_F_q q) (\<G>_F_L_q q) (\<G>_I_L_q q) (\<lambda>g Cl Cl'. False)"
simps_1: shows "hom heap.Empty = \<langle>\<rangle>"
similar_matrix_def: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'c) vec, 'c) vec" shows "similar_matrix A B = (\<exists>P. invertible P \<and> A = matrix_inv P ** B ** P)"
tab_simps_4: shows "local.dom tab = p\<^sub>1"
power_Suc_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero a"   and "less a one" shows "less (times a (power.power one times a n)) (power.power one times a n)"
permutation_fix: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < Suc n"   and "j < Suc n" shows "{p. p permutes {0..<Suc n} \<and> p i = j} = permutation_insert i j ` {q. q permutes {0..<n}}"
prems_4: shows "\<mu> R\<^sub>r \<eta> \<in> old (\<sigma> 0)"
assms_3: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and f' :: "Label \<Rightarrow> sterm option"   and u' :: "sterm" assumes "Obj f T = s"   and "Obj f T => Obj f' T"   and "l \<in> dom f'"   and "u => u'"   and "z = (the (f' l)\<^bsup>[Obj f' T,u']\<^esup>)" shows "Q"
prems_2: shows "takeWhile ((\<noteq>) v) (rev (u # l)) = c"
not_Ici_eq_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b" assumes "class.preorder less_eq less" shows "ord.atLeast less_eq l \<noteq> {}"
def_of_hd: fixes y :: "'a list"   and a :: "'a"   and x :: "'a list" shows "y = a # x \<longrightarrow> hd y = a"
stirling_integral_bound_aux: fixes n :: "nat" assumes "1 < n" obtains c :: "real" where "\<And>s. 0 < Re s \<Longrightarrow> cmod (stirling_integral n s) \<le> c / Re s ^ (n - 1)"
inv2: shows "local.inv_part local.diff_ev A 0 0"
simple_match_port_UNIV: fixes s :: "16 word"   and e :: "16 word" shows "({p. simple_match_port (s, e) p} = UNIV) = (s = 0 \<and> e = - 1)"
lenlex_irreflexive: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "\<And>x. (x, x) \<notin> r" shows "(xs, xs) \<notin> lenlex r"
homeomorphic_maps_involution: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'a" assumes "continuous_map X X f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f (f x) = x" shows "homeomorphic_maps X X f f"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> [] \<Rightarrow>\<^sub>z t"
H_choice: fixes p :: "'a"   and x :: "'a"   and q :: "'a"   and y :: "'a" assumes "H p x q"   and "H p y q" shows "H p (x + y) q"
Times_assoc: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Times (Times r s) t = Times r (Times s t)"