hcju/isabelleps huggingface dataset:
no_matching_Goto_tail: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and rs :: "'c rule list" assumes "no_matching_Goto \<gamma> p (r # rs)" shows "no_matching_Goto \<gamma> p rs"
assms_2: shows "p \<in> dgrad_p_set d m"
image_subset_iff: fixes f :: "'r \<Rightarrow> 'q"   and A :: "'r set"   and B :: "'q set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
upper_plus: fixes A :: "'b interval"   and B :: "'b interval" shows "upper (A + B) = upper A + upper B"
inverse_negative_iff_negative: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse a) zero = less a zero"
s_cf_expr0: fixes p :: "'a" assumes "p \<in> carrier R" shows "pol_coeff S (s_cf R S X p) \<and> p = polyn_expr R X (fst (s_cf R S X p)) (s_cf R S X p)"
matrix_vector_right_distrib: fixes A :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec"   and y :: "('a, 'c) vec" shows "A *v (x + y) = A *v x + A *v y"
A5_NotDSource_level0: shows "isNotDSource level0 sA5"
monom_mult_assoc: fixes c :: "'c"   and s :: "'a"   and d :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult c s (monom_mult d t p) = monom_mult (c * d) (s + t) p"
D_def: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "GAUSS.D p a = GAUSS.C p a \<inter> {..(int p - 1) div 2}"
ide_def: fixes a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" shows "S.VV.ide a \<equiv> S.VV.comp a a \<noteq> S.VV.null \<and> (\<forall>f. (S.VV.comp f a \<noteq> S.VV.null \<longrightarrow> S.VV.comp f a = f) \<and> (S.VV.comp a f \<noteq> S.VV.null \<longrightarrow> S.VV.comp a f = f))"
mgu_sound: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<sigma>" shows "is_imgu \<sigma> {(s, t)}"
graph_def: fixes G :: "('b, 'c) pre_digraph" shows "graph G \<equiv> digraph G \<and> pseudo_graph G"
InfModel_intTI_intFI_intPI: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "MonotProblem wtFsym wtPsym arOf resOf parOf \<Phi>"   and "CM.Model wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> (MonotProblem.intTI wtFsym wtPsym arOf resOf parOf \<Phi>) (MonotProblem.intFI wtFsym wtPsym arOf resOf parOf \<Phi>) (MonotProblem.intPI wtFsym wtPsym arOf resOf parOf \<Phi>)"
list_succ_singleton: fixes x :: "'b" shows "list_succ [x] = list_succ []"
dual_inf: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y) ^ o = x ^ o \<squnion> y ^ o"
A1: shows "univ_for_pr \<in> PrimRec1"
Y_def: shows "Y = {q \<in> set qs. poly_deg (fst q) = d \<and> Suc m < card (snd q)}"
hyps_3: assumes "\<forall>x\<in>F. i < IArray.length (f x)"   and "F \<noteq> {}" shows "sum f F !! i = (\<Sum>x\<in>F. f x !! i)"
is_singletonI: fixes x :: "'b" shows "is_singleton {x::'b}"
pt_supp_finite_pi: fixes x :: "'a"   and pi :: "('b \<times> 'b) list" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp x)" shows "finite (supp (pi \<bullet> x))"
code: fixes r :: "'a language" shows "Coinductive_Language.Not r = Lang (\<not> \<oo> r) (\<lambda>a. Coinductive_Language.Not (\<dd> r a))"
can_in_hom: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Ide t"   and "Ide u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "\<guillemotleft>can u t : \<lbrace>t\<rbrace> \<Rightarrow> \<lbrace>u\<rbrace>\<guillemotright>"
rel_compile_env_entry: fixes \<O> :: "'fun \<Rightarrow> nat option"   and \<A> :: "'k \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and f :: "'k"   and fd1 :: "('b, 'l, 'fun, 'd, 'e) Inca.instr fundef"   and fd2 :: "('b, 'l, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef" assumes "compile_env_entry \<O> \<A> (f, fd1) = Ok (f, fd2)" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"
Fun_mkArr: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr (mkArr A B F)" shows "Fun (mkArr A B F) = restrict F A"
LI_preserves_wellformedness: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S\<^sub>2 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>2 :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S\<^sub>1, \<theta>\<^sub>1) \<leadsto>\<^sup>* (S\<^sub>2, \<theta>\<^sub>2)"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1" shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>2 \<theta>\<^sub>2"
fmrel_on_fset_fmrel_restrict: fixes S :: "'c fset"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and n :: "('c, 'e) fmap" shows "fmrel_on_fset S P m n = fmrel P (fmrestrict_fset S m) (fmrestrict_fset S n)"
llast_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "llast (lappend xs ys) = (if lnull ys then llast xs else if lfinite xs then llast ys else undefined)"
intros_4: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "xs <~~> ys"   and "ys <~~> zs" shows "xs <~~> zs"
sgn_finite_2_def: shows "sgn = (\<lambda>x. x)"
hyps: shows "gval (fold gOr (map (\<lambda>x. Eq (V v) (L x)) ls) (Eq (V v) (L l))) s = gval (fold gOr (map (\<lambda>x. Eq (V v) (L x)) (l # ls)) (Bc False)) s"
assms: shows "R[a\<turnstile>c>b] = AndR <c>.M <d>.N e"
abNotNull: shows "\<not> (a = 0 \<and> b = 0)"
mult_cancel_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times c a = times c b) = (c = zero \<or> a = b)"
simps_6: fixes f1 :: "fdecl \<Rightarrow> 'a"   and f2 :: "mdecl \<Rightarrow> 'a"   and x2 :: "mdecl" shows "(case mdecl x2 of fdecl x \<Rightarrow> f1 x | mdecl x \<Rightarrow> f2 x) = f2 x2"
independent_in_carrier: fixes K :: "'a set"   and Us :: "'a list" assumes "local.independent K Us" shows "set Us \<subseteq> carrier R"
process_prop_3_def: fixes P :: "('c list \<times> 'c set) set \<times> 'c list set" shows "process_prop_3 P \<equiv> \<forall>xs X Y. (xs, Y) \<in> fst P \<and> X \<subseteq> Y \<longrightarrow> (xs, X) \<in> fst P"
smallI: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>c. 0 < c \<Longrightarrow> \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)" shows "f \<in> o[F](g)"
\<alpha>sl_def: fixes m :: "(lit, lit list) mapping" shows "\<alpha>sl m \<equiv> (the_default [] \<circ>\<circ> Mapping.lookup) m"
FAss: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and fs' :: "Objects.fields"   and h\<^sub>2' :: "heap" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,NonStatic:t in D"   and "fs' = fs((F, D) \<mapsto> v)"   and "h\<^sub>2' = h\<^sub>2(a \<mapsto> (C, fs'))" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h\<^sub>2', l\<^sub>2, sh\<^sub>2)\<rangle>"
pivotal_exists: fixes F :: "(indi \<Rightarrow> alt \<Rightarrow> real) \<Rightarrow> alt \<Rightarrow> real"   and b :: "alt" assumes "arrow F" shows "\<exists>i. arrow.pivotal F i b"
S_intersection_coeff2_def: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection_coeff2 p q \<equiv> (- p \<bullet> (M *v q) - sqrt (quarter_discrim p q)) / (p \<bullet> (M *v p))"
try_spmf_return_None: fixes q :: "'c spmf" shows "TRY return_pmf None ELSE q = q"
dvd_mult_cancel_left: fixes c :: "'b"   and a :: "'b"   and b :: "'b" shows "(c * a dvd c * b) = (c = (0::'b) \<or> a dvd b)"
bn_comp_fst_ge: fixes n :: "nat"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "bn_valid n p q" shows "n \<le> fst (bn_comp n p q r)"
map_gpv_bind_gpv: shows "map_gpv f (\<lambda>x. x) (x \<bind> Generative_Probabilistic_Value.Done) = map_gpv id (\<lambda>x. x) x \<bind> (\<lambda>x. map_gpv f (\<lambda>x. x) (Generative_Probabilistic_Value.Done x))"
set_zip_tr: fixes s :: "s"   and s' :: "s"   and ss :: "s list"   and T :: "x \<Rightarrow> x option" shows "(s, s') \<in> set (zip ss (tr_ss_f T ss)) \<longrightarrow> s' = tr_s_f T s"
float_power_bnds: fixes l1 :: "float"   and u1 :: "float"   and prec :: "nat"   and n :: "nat"   and l :: "float"   and u :: "float"   and x :: "real" assumes "(l1, u1) = float_power_bnds prec n l u"   and "x \<in> {real_of_float l..real_of_float u}" shows "x ^ n \<in> {real_of_float l1..real_of_float u1}"
adds_term_adds_pp: fixes u :: "'t"   and v :: "'t" shows "(u adds\<^sub>t v) = (component_of_term u = component_of_term v \<and> pp_of_term u adds\<^sub>p v)"
induct: fixes P :: "(('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> nat \<Rightarrow> nat \<times> nat list \<times> 'c list \<Rightarrow> bool"   and a0 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and a1 :: "'c \<Rightarrow> 'd"   and a2 :: "nat"   and a3 :: "nat \<times> nat list \<times> 'c list" assumes "\<And>index key p u ns xs. (find ((<) (Suc 0)) ns \<noteq> None \<Longrightarrow> P index key p (gcsort_round index key p ns xs)) \<Longrightarrow> P index key p (u, ns, xs)" shows "P a0 a1 a2 a3"
vreal_plus_closed: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" shows "x +\<^sub>\<real> y \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
simps_56: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "Stop \<noteq> Stmt.If x51 x52 x53"
linear_diff: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "vector_space_pair s1 s2"   and "linear s1 s2 f" shows "f (x - y) = f x - f y"
hyps: shows "satisfiable F' = satisfiable (filter (\<lambda>c. \<not> clauseTautology c) F')"
splitTree_def: fixes p :: "'c \<Rightarrow> bool"   and i :: "'c"   and t :: "('b, 'c) FingerTreeStruc" shows "splitTree p i t = (let (l, x, r) = nsplitTree p i t in (l, n_unwrap x, r))"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj" assumes "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter" shows "ptr |\<in>| shadow_root_ptr_kinds h"
lsl_Sup_closed: shows "Sup_closed_set (range \<nu>)"
column_add_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and n :: "'e"   and m :: "'e"   and q :: "'d" shows "column_add A n m q = (\<chi>i j. if j = n then A $ i $ n + A $ i $ m * q else A $ i $ j)"
jumpNestingOk_var_simp: fixes jmps :: "jump set"   and v :: "Term.var" shows "jumpNestingOk jmps (In2 v) = True"
removeAllListDiff: fixes x :: "'a"   and a :: "'a list"   and b :: "'a list" shows "removeAll x (list_diff a b) = list_diff (removeAll x a) b"
comparison_principle_le: fixes X :: "real set"   and Y :: "real set"   and f :: "real \<Rightarrow> real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and \<phi> :: "real \<Rightarrow> real"   and \<phi>' :: "real \<Rightarrow> real"   and \<psi> :: "real \<Rightarrow> real"   and \<psi>' :: "real \<Rightarrow> real" assumes "local_lipschitz X Y f"   and "\<And>x. x \<in> Y \<Longrightarrow> continuous_on {a..b} (\<lambda>t. f t x)"   and "{a..b} \<subseteq> X"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (\<phi> has_real_derivative \<phi>' x) (at x)"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (\<psi> has_real_derivative \<psi>' x) (at x)"   and "\<phi> ` {a..b} \<subseteq> Y"   and "\<psi> ` {a..b} \<subseteq> Y"   and "\<phi> a \<le> \<psi> a"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> \<phi>' x - f x (\<phi> x) \<le> \<psi>' x - f x (\<psi> x)" shows "\<forall>x\<in>{a..b}. \<phi> x \<le> \<psi> x"
compact_Int_closed: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "topological_space.closed open T" shows "topological_space.compact open (S \<inter> T)"
IH: assumes "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>" shows "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) (dbproj i D))\<rbrakk>\<^sub>d \<I>"
cases: fixes x :: "int \<times> int" obtains d :: "int" and i :: "int" where "x = (d, i)"
inner_prod_of_unit_vec: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "\<langle>unit_vec n i|unit_vec n i\<rangle> = 1"
emeasure_eq_measure: fixes A :: "'s stream set" shows "emeasure (T' I) A = ennreal (T'.prob A)"
default_def: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" shows "Mapping.default k v m = (if k \<in> Mapping.keys m then m else Mapping.update k v m)"
the_coext_unique: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' x f"   and "is_coext x' f g" shows "g = the_coext x' f"
multiplicity_eq_Max: shows "multiplicity p x = Max {n. p ^ n dvd x}"
valid_Act_strong: fixes X :: "'b"   and P :: "'state"   and \<alpha> :: "'act"   and x :: "('c, 'pred, 'act) formula" assumes "finite (supp X)" shows "P \<Turnstile> Act \<alpha> x = (\<exists>\<alpha>'' x' P'. Act \<alpha> x = Act \<alpha>'' x' \<and> P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x' \<and> bn \<alpha>'' \<sharp>* X)"
\<I>_trivial_def: fixes \<I> :: "('a, 'b) \<I>" shows "\<I>_trivial \<I> = (outs_\<I> \<I> = UNIV)"
prefix_code_1: fixes xs :: "'e list" shows "prefix [] xs = True"
assms: shows "fun_evaluate_match s env v pes err_v = (s', Rval vs)"
vsv_vrrestriction: fixes r :: "V"   and A :: "V" assumes "vsv r" shows "vsv (r \<restriction>\<^sup>r\<^sub>\<circ> A)"
simps_5: fixes f1 :: "'b"   and f2 :: "'c tree \<Rightarrow> 'c \<Rightarrow> 'c tree \<Rightarrow> 'b"   and x21 :: "'c tree"   and x22 :: "'c"   and x23 :: "'c tree" shows "(case \<langle>x21, x22, x23\<rangle> of \<langle>\<rangle> \<Rightarrow> f1 | \<langle>x, xa, xb\<rangle> \<Rightarrow> f2 x xa xb) = f2 x21 x22 x23"
simps_2: fixes d :: "bool"   and ds :: "bool list"   and l :: "tree"   and r :: "tree" shows "branch (d # ds) (Branching l r) = (if d then branch ds l else branch ds r)"
retractionI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C (C g f)" shows "category.retraction C g"
axioms: fixes m :: "int" assumes "poly_mod_prime_type TYPE('a) m" shows "poly_mod_type TYPE('a) m"
lim_const: fixes a :: "'a" shows "lim (\<lambda>m. a) = a"
comp_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's"   and C :: "'s set"   and G :: "'s \<Rightarrow> 's" assumes "S.arr (S.mkArr A B F)"   and "S.arr (S.mkArr B C G)" shows "S.mkArr B C G \<cdot>\<^sub>S S.mkArr A B F = S.mkArr A C (G \<circ> F)"
select_convs_6: fixes Pol :: "('c \<times> 'c) set"   and Map :: "'b \<Rightarrow> 'c"   and Doms :: "'c set"   and List :: "'b list"   and ListOp :: "'b list option"   and Set :: "'b set"   and more :: "'d" shows "Set \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = Set"
indRelRSTPO_to_SRel_and_TRel_4: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q" shows "\<forall>TP TQ. TP \<in>T P \<and> TQ \<in>T Q \<longrightarrow> (TP, TQ) \<in> TRel\<^sup>+"
mod_pos_geq: fixes l :: "int"   and k :: "int" assumes "0 < l"   and "l \<le> k" shows "k mod l = (k - l) mod l"
sconf_def: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and s :: "heap \<times> sfields \<times> sheap" shows "P,E \<turnstile> s \<surd> \<equiv> let (h, l, sh) = s in P \<turnstile> h \<surd> \<and> P,h \<turnstile> l (:\<le>) E \<and> P,h \<turnstile>\<^sub>s sh \<surd>"
insert_not_empty: fixes a :: "'x"   and A :: "'x set" shows "insert a A \<noteq> {}"
vector_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> top = x"   and "y \<sqinter> top = y" shows "(x \<squnion> y) \<sqinter> top = x \<squnion> y"
typ_of1_incr_bv_lev0: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ"   and Ts' :: "typ list" assumes "typ_of1 Ts t = Some ty" shows "typ_of1 (Ts' @ Ts) (incr_bv (length Ts') 0 t) = Some ty"
adjoint_preds_match: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and p :: "char"   and r :: "fml"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real" assumes "SPreds \<sigma> p = Afml r" shows "Preds (USubst.adjoint \<sigma> I \<omega>) p = (\<lambda>d. \<omega> \<in> fml_sem (repc I dotid d) r)"
induct: fixes P :: "'b negation_type list \<Rightarrow> bool"   and a0 :: "'b negation_type list" assumes "P []"   and "\<And>e. P [Pos e]"   and "\<And>e. P [Neg e]"   and "\<And>e v va. P (v # va) \<Longrightarrow> P (Pos e # v # va)"   and "\<And>e v va. P (v # va) \<Longrightarrow> P (Neg e # v # va)" shows "P a0"
mk_lossless_parametric: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (rel_spmf A) (rel_spmf A) mk_lossless mk_lossless"
orthogonal_ket: fixes i :: "'c"   and j :: "'c" shows "is_orthogonal (ket i) (ket j) = (i \<noteq> j)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) F G \<tau>"
IH_2: fixes d :: "nat"   and t' :: "'a tree" assumes "ins n d x l = Unbal t'" shows "height l \<le> height t' \<and> height t' \<le> height l + 1"
word_of_int_numeral: fixes bin :: "num" shows "word_of_int (numeral bin) = numeral bin"
butterfly_scaleR_left: fixes r :: "real"   and \<psi> :: "'b"   and \<phi> :: "'a" shows "butterfly (r *\<^sub>R \<psi>) \<phi> = complex_of_real r *\<^sub>C butterfly \<psi> \<phi>"
union_inter: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) + sum g (A \<inter> B) = sum g A + sum g B"
chine_eq_apex: shows "\<mu>.chine = \<mu>.apex"
A: shows "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)"
simps_2: fixes a :: "'b set"   and b :: "'b set"   and n :: "nat" shows "trivial_series a b (Suc n) = b"
Par1F: fixes P :: "pi"   and \<alpha> :: "Early_Semantics.freeRes"   and P' :: "pi"   and Q :: "pi" assumes "P \<longmapsto>\<^sub>e \<alpha> \<prec> P'" shows "P \<parallel> Q \<longmapsto>\<^sub>e \<alpha> \<prec> P' \<parallel> Q"
fdemodalisation1: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(ad (ad z) \<cdot>  |x\<rangle> y = (zero_class.zero::'a)) = (ad (ad z) \<cdot> x \<cdot> ad (ad y) = (zero_class.zero::'a))"
empty: fixes g :: "'c \<Rightarrow> 'a" shows "sum g {} = (0::'a)"
Zero_fract_def: shows "0 = Fract (0::'a) (1::'a)"
simps_27: fixes v :: "'b rexp"   and va :: "'b rexp"   and vf :: "'b rexp" shows "le_rexp (Times v va) (Star vf) = False"
p_imp_false: fixes P :: "'a upred" shows "(P \<Rightarrow> false) = (\<not> P)"
nms: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and t :: "state"   and a :: "action" assumes "\<not> matches \<gamma> m p"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>Rule m a # rs, Undecided\<rangle> \<Rightarrow>\<^sub>s t"
function_mult_comp_is_seq_mult: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and g :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "g \<in> carrier (function_ring (carrier Zp) Zp)" shows "f \<otimes>\<^bsub>function_ring (carrier Zp) Zp\<^esub> g \<circ> s = (f \<circ> s) \<otimes>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> (g \<circ> s)"
Max_by_Union_distrib: fixes A :: "'c set"   and S :: "'c set set"   and cmp_f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "A = \<Union> S"   and "S \<noteq> {}"   and "{} \<notin> S"   and "inj_on cmp_f A" shows "Max_by cmp_f A = Max_by cmp_f (Max_by cmp_f ` S)"
squarefree_factorial_semiring'': fixes n :: "'b" assumes "n \<noteq> (0::'b)" shows "squarefree n = (\<forall>p. prime p \<longrightarrow> multiplicity p n \<le> 1)"
that: fixes v' :: "value" assumes "\<Gamma>' \<turnstile>\<^sub>v t \<down> v'"   and "v' \<approx>\<^sub>e v" shows "thesis"
nat_pow_one: fixes G :: "('c, 'd) monoid_scheme"   and n :: "nat" assumes "Group.monoid G" shows "\<one>\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>"
NTCatCod_def: fixes \<eta> :: "('j, 'g, 'k, 'h, 'l, 'i, 'm) NatTrans_scheme" shows "NTCatCod \<eta> \<equiv> CatCod (NTCod \<eta>)"
cod_simp: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "VVV.cod f = VxVxV.cod f"
mod_mult_self3: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and c :: "'d"   and b :: "'d"   and a :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (times c b) a) b = modulo a b"
IH: assumes "finite V"   and "finite (inputs M2)" shows "finite (TS M2 M1 \<Omega> V m n)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_singleton_axioms ext" shows "ext_singleton ext"