hcju/isabelleps huggingface dataset:
abs_residual_pair_eqvt: fixes p :: "perm"   and \<alpha> :: "'a"   and P :: "'b" shows "p \<bullet> \<langle>\<alpha>,P\<rangle> = \<langle>p \<bullet> \<alpha>,p \<bullet> P\<rangle>"
mono: fixes c :: "'c multiset"   and a :: "'c multiset"   and d :: "'c multiset"   and b :: "'c multiset" assumes "c \<subseteq># a"   and "d \<subseteq># b" shows "c \<union># d \<subseteq># a \<union># b"
drop_emp_nel_2: fixes u :: "'b list"   and S :: "'b list set" assumes "u \<in> S\<^sub>+" shows "u \<in> S"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F F' \<Phi>\<^sub>F' \<tau>\<^sub>0 \<tau>\<^sub>1"
lhd_cons_def: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lhd_cons g s = lhd (lunstream g s)"
admits_diagonal_reduction_def: fixes A :: "'b mat" shows "admits_diagonal_reduction A = (\<exists>P Q. P \<in> carrier_mat (dim_row A) (dim_row A) \<and> Q \<in> carrier_mat (dim_col A) (dim_col A) \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q))"
semiring_simprules_3: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
tab_correct: fixes h :: "nat"   and n :: "nat" assumes "h < length bal_tab" shows "bal_i n h = (bal_tab ! h \<le> n)"
psi_init: fixes i :: "nat"   and x :: "nat" shows "\<psi> i \<triangleright> x = Partial_Recursive.list_encode (prefixes i x)"
alpha_set: fixes bs :: "atom set"   and x :: "'e"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'e" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
MapOps_def: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and d :: "'f set"   and ops :: "('g, 'e, 'h) MapOps" shows "MapOps \<alpha> d ops \<equiv> (\<forall>k. \<alpha> k \<in> d \<longrightarrow> lookup ops (MapOps.empty ops) k = None) \<and> (\<forall>e k k' M. \<alpha> k \<in> d \<and> \<alpha> k' \<in> d \<longrightarrow> lookup ops (update ops k e M) k' = (if \<alpha> k' = \<alpha> k then Some e else lookup ops M k'))"
map_add_upd_left: fixes m :: "'c"   and e2 :: "'c \<Rightarrow> 'd option"   and e1 :: "'c \<Rightarrow> 'd option"   and u1 :: "'d" assumes "m \<notin> dom e2" shows "e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
fv_ik\<^sub>s\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>s\<^sub>t: fixes x :: "'c"   and A :: "('d, 'c) stateful_strand_step list" assumes "x \<in> fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>s\<^sub>t A)" shows "x \<in> fv\<^sub>s\<^sub>s\<^sub>t A"
span_basis: shows "local.span basis = U\<^sub>V\<^sub>S"
hyp_port_strict_prefix: fixes f :: "'form"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list" assumes "f \<in> hyps_along t is" shows "strict_prefix (hyp_port_path_for t is f) is"
less_quasinorm_def: fixes N1 :: "'a quasinorm"   and N2 :: "'a quasinorm" shows "(N1 < N2) = (N1 \<le> N2 \<and> \<not> N2 \<le> N1)"
rev_flip_altsublist_chain: fixes xss :: "'w list list" assumes "flip_altsublist_chain xss" shows "flip_altsublist_chain (rev xss)"
nodes_sim: fixes a :: "'b"   and b :: "'e" assumes "a \<in> s1.b.V"   and "(a, b) \<in> R2" shows "b \<in> s2.b.V"
pickT: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'b \<Rightarrow> 'd list \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "M.Struct wtFsym wtPsym arOf resOf intT intF intP" shows "intT \<sigma> (Struct.pickT intT \<sigma>)"
wf_well_formed: fixes prog :: "cmd"   and procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list" assumes "WellFormProgs.wf prog procs" shows "well_formed procs"
associated_sym: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "Group.monoid G"   and "a \<sim>\<^bsub>G\<^esub> b" shows "b \<sim>\<^bsub>G\<^esub> a"
vsv_appI: fixes a :: "V"   and b :: "V"   and aa :: "V"   and ba :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> set {\<langle>aa, ba\<rangle>}" shows "set {\<langle>aa, ba\<rangle>}\<lparr>a\<rparr> = b"
assms: shows "R[x\<turnstile>n>y] = AndL2 v.N u"
r2: shows "\<rho>\<^sub>2 \<in> carrier_mat d d"
Domain_iff_totalize_act: fixes act :: "('b \<times> 'b) set" shows "(Domain act = UNIV) = (totalize_act act = act)"
arr_dom: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "A1xA2.arr (A1xA2.dom f)"
finter_commute: fixes A :: "'b fset"   and B :: "'b fset" shows "A |\<inter>| B = B |\<inter>| A"
assms_4: shows "c = to_fun (shift 2 (T\<^bsub>a\<^esub> f)) b"
pdevs_domain_prod_of_pdevs: fixes x :: "'b pdevs"   and y :: "'c pdevs" shows "pdevs_domain (prod_of_pdevs x y) = pdevs_domain x \<union> pdevs_domain y"
fun_dom_scene_indep: fixes A :: "'a set"   and B :: "'a set" shows "(fds A \<bowtie>\<^sub>S fds B) = (A \<inter> B = {})"
lspasl_iu: fixes Gamma :: "bool"   and h2 :: "'a"   and h1 :: "'a"   and Delta :: "bool" assumes "Gamma \<and> (0::'a,h2\<triangleright>0::'a) \<and> h1 = (0::'a) \<longrightarrow> Delta" shows "Gamma \<and> (h1,h2\<triangleright>0::'a) \<longrightarrow> Delta"
measurable_prod_emb: fixes J :: "'b set"   and L :: "'b set"   and X :: "('b \<Rightarrow> 'c) set"   and M :: "'b \<Rightarrow> 'c measure" assumes "J \<subseteq> L"   and "X \<in> sets (Pi\<^sub>M J M)" shows "prod_emb L M J X \<in> sets (Pi\<^sub>M L M)"
UNION_singleton_eq_range: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(\<Union>x\<in>A. {f x}) = f ` A"
collision_react_subset: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real"   and s :: "real set"   and t :: "real set" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"   and "safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> s"   and "s \<subseteq> t" shows "safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> t"
starsetNat_eq_hypnat_of_nat_image_finite: fixes A :: "nat set" assumes "*s* A = hypnat_of_nat ` A" shows "finite A"
right_unique_OO: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_unique A"   and "right_unique B" shows "right_unique (A OO B)"
wf_rexp_valid_ENC: fixes n :: "nat"   and p :: "nat" assumes "n = 0 \<or> p < n" shows "local.wf n (valid_ENC n p)"
path_mirror: assumes "path (- g)" shows "path (- (- g))"
irreducible_altdef: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "comm_semiring_1.irreducible times one zero x = (x \<noteq> zero \<and> \<not> dvd.dvd times x one \<and> (\<forall>b. dvd.dvd times b x \<longrightarrow> dvd.dvd times x b \<or> dvd.dvd times b one))"
g: shows "finite {x. g x \<noteq> (0::'a)}"
of_nat_closed: fixes n :: "nat" shows "\<guillemotleft>n\<guillemotright>\<^sub>\<nat> \<in> carrier R"
exhaust: fixes y :: "sequent" obtains x1 :: "form multiset" and x2 :: "form multiset" where "y = ( x1 \<Rightarrow>* x2)"
assms_2: fixes xs :: "('a, 'b) tllist" assumes "\<not> is_TNil xs" shows "P (thd xs) xs"
cpoly_of_dist_right: fixes pR :: "real poly"   and g :: "real poly"   and pI :: "real poly" shows "cpoly_of (pR * g) (pI * g) = cpoly_of pR pI * map_poly complex_of_real g"
cases: fixes x :: "'b list" obtains "xa__ = []"   | x :: "'b" where "xa__ = [xb__]"   | x :: "'b" and y :: "'b" and xs :: "'b list" where "xa__ = x # y # xs"
in_set_repl_setFin: fixes f :: "face"   and fs :: "face list"   and f' :: "face" assumes "f \<in> set fs"   and "final f" shows "f \<in> set (replace f' [setFinal f'] fs)"
inject_2: fixes x2 :: "bool"   and y2 :: "bool" shows "(<\<bottom>, outM x2> = <\<bottom>, outM y2>) = (x2 = y2)"
invar_del_min: fixes h :: "'b hp option" assumes "invar h" shows "invar (del_min h)"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "C.arr (trg\<^sub>C f)"
post_get_pending_def: fixes u :: "'v"   and Vs :: "'v option"   and s0 :: "'s"   and s :: "'s" shows "post_get_pending u Vs s0 s \<equiv> pre_get_pending s0 \<and> inres (gds_get_pending gds s0) (u, Vs, s)"
LLs_LLq: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "'b"   and t2 :: "'b" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "t1 \<in> Syntax_Arith.atrm var Var num suc pls tms"   and "t2 \<in> Syntax_Arith.atrm var Var num suc pls tms" shows "Syntax_Arith.LLs var Var FvarsT Fvars subst eql cnj imp exi fls pls t1 t2 = cnj (Syntax_Arith.LLq var Var FvarsT Fvars subst eql exi pls t1 t2) (Syntax_with_Connectives_False.neg imp fls (eql t1 t2))"
assms_2: shows "b \<notin> ptrm_fvs X"
bigstepT_the_cost: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "nat"   and s' :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> t \<Down> s'" shows "\<down>\<^sub>t (c, s) = t"
invariant_def: fixes r :: "'s"   and s :: "'s" shows "invariant r s = (\<lambda>(ws, ps, N). (r, s) \<in> snd ` set ws \<union> set ps \<and> distinct (map snd ws @ ps) \<and> bij_betw (map_prod post post) (set (map snd ws @ ps)) N \<and> (\<forall>(w, r', s')\<in>set ws. fold delta (rev w) r = r' \<and> fold delta (rev w) s = s' \<and> wf_word n (rev w) \<and> wf_state r' \<and> wf_state s') \<and> (\<forall>(r', s')\<in>set ps. (\<exists>w. fold delta w r = r' \<and> fold delta w s = s') \<and> wf_state r' \<and> wf_state s' \<and> final r' = final s' \<and> (\<forall>a\<in>set (\<sigma> n). (post (delta a r'), post (delta a s')) \<in> N)))"
sup_to_inf: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = - (- x \<sqinter> - y)"
valid_weight_repr_def: fixes l :: "(('e \<times> 'e) \<times> 'f) list" shows "valid_weight_repr l \<equiv> distinct (map (uedge \<circ> fst) l)"
set_tree_pathD: fixes x :: "'a"   and t :: "'a tree" assumes "x \<in> set_tree t" shows "\<exists>p. x = root (traverse_tree p t)"
split_volatile_Write\<^sub>s\<^sub>b_in_outstanding_refs: fixes a :: "nat"   and xs :: "'b memref list" assumes "a \<in> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs" shows "\<exists>sop v ys zs A L R W. xs = ys @ Write\<^sub>s\<^sub>b True a sop v A L R W # zs"
subspace_dim_equal: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set"   and S :: "'e set"   and T :: "'e set" assumes "finite_dimensional_vector_space scale Basis"   and "module.subspace scale S"   and "module.subspace scale T"   and "S \<subseteq> T"   and "vector_space.dim scale T \<le> vector_space.dim scale S" shows "S = T"
irrefl: fixes a :: "'c list" shows "\<not> strict_prefix a a"
proots_count_reciprocal: fixes P :: "'a poly"   and p :: "nat"   and A :: "'a set" assumes "degree P \<le> p"   and "P \<noteq> 0"   and "(0::'a) \<notin> A" shows "proots_count (reciprocal_poly p P) A = proots_count P {x. inverse x \<in> A}"
lead_coeff_rebase_poly: fixes p :: "'p mod_ring poly" shows "lead_coeff (#p) = @lead_coeff p"
infinite_growing: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a set" assumes "class.linorder less_eq less"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. less x y" shows "infinite X"
typ_of_fun: fixes t :: "Core.term"   and u :: "Core.term"   and R :: "typ"   and U :: "typ" assumes "typ_of (t $ u) = Some R"   and "typ_of t = Some (U \<rightarrow> R)" shows "typ_of u = Some U"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a' \<Rightarrow>\<^sub>S a\<guillemotright>"   and "S.iso (R f)" shows "S.iso f"
Lim_def: fixes "open" :: "'a set \<Rightarrow> bool"   and A :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "class.t2_space open" shows "t2_space.Lim open A f = (THE l. filterlim f (topological_space.nhds open l) A)"
subst_of_simps_1: shows "subst_of [] = Var"
scaleR_sum_left: fixes f :: "'d \<Rightarrow> real"   and A :: "'d set"   and x :: "'c" shows "sum f A *\<^sub>R x = (\<Sum>a\<in>A. f a *\<^sub>R x)"
pathstart_def: fixes g :: "real \<Rightarrow> 'b" shows "pathstart g = g 0"
not_less_iff_gr_or_eq: fixes x :: "complex"   and y :: "complex" shows "(\<not> x <\<^sub>\<complex> y) = (y <\<^sub>\<complex> x \<or> x = y)"
integral_norm_bound_integral: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f integrable_on S"   and "g integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x" shows "norm (integral S f) \<le> integral S g"
Liminf_state_fin: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist"   and Sts' :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "lfinite Sts"   and "lfinite Sts'"   and "\<not> lnull Sts"   and "\<not> lnull Sts'"   and "llast Sts' = llast Sts" shows "Liminf_state Sts' = Liminf_state Sts"
fold_matches_first_step_not_None: fixes x :: "typ"   and xs :: "typ list"   and y :: "typ"   and ys :: "typ list"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip (x # xs) (y # ys)) (Some subs) = Some subs'" obtains point :: "variable \<times> String.literal set \<Rightarrow> typ option" where "raw_match' x y subs = Some point" and "fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip xs ys) (Some point) = Some subs'"
cr_pmf_def: shows "cr_pmf \<equiv> \<lambda>x y. x = pmf y"
adv: assumes "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g absorb \<A> cnv \<surd>" shows "advantage (absorb \<A> cnv) (obsf_resource (sim |\<^sub>= 1\<^sub>C \<rhd> ideal_resource)) (obsf_resource real_resource) \<le> adv"
simps: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "NoCardinalityAnalysis Aexp Aheap" shows "NoCardinalityAnalysis.prognosis Aexp ae as a (\<Gamma>, e, S) = (\<lambda>_. many) f|` (edom (ArityAnalysis.ABinds Aexp \<Gamma>\<cdot>ae) \<union> edom (Aexp e\<cdot>a) \<union> edom (ArityAnalysis.AEstack Aexp as S))"
simps_2: fixes e :: "(name, hf) finfun"   and t :: "tm"   and u :: "tm" shows "eval_fm e (t EQ u) = (\<lbrakk>t\<rbrakk>e = \<lbrakk>u\<rbrakk>e)"
star_induct: fixes w :: "'a list"   and A :: "'a list set"   and P :: "'a list \<Rightarrow> bool" assumes "w \<in> A\<star>"   and "P []"   and "\<And>u v. \<lbrakk>u \<in> A; v \<in> A\<star>; P v\<rbrakk> \<Longrightarrow> P (u @ v)" shows "P w"
real_arch_simple: fixes x :: "'c" shows "\<exists>n. x \<le> of_nat n"
summable_sums_iff: fixes f :: "nat \<Rightarrow> 'a" shows "summable f = f sums suminf f"
reflect: shows "ChamberComplexApartmentSystemTriangle X \<A> A B' B D C E z"
idt_eq: fixes x :: "'b"   and y :: "'b" assumes "(S_Idt, x, y) \<in> edges G" shows "Eps (P x) = Eps (P y)"
mat_adjoint_def: fixes A :: "'a mat" shows "adjoint A \<equiv> mat_of_rows (dim_row A) (map conjugate (cols A))"
bconf_BinOp: fixes P :: "'a prog"   and sh :: "sheap"   and e1 :: "'b exp"   and bop :: "bop"   and e2 :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (e1 \<guillemotleft>bop\<guillemotright> e2,b) \<surd>) = (case val_of e1 of None \<Rightarrow> P,sh \<turnstile>\<^sub>b (e1,b) \<surd> | \<lfloor>v\<rfloor> \<Rightarrow> P,sh \<turnstile>\<^sub>b (e2,b) \<surd>)"
partial_nth_Some_imp_elem: fixes l :: "'a list"   and n :: "nat"   and x :: "'a" assumes "partial_nth l n = Some x" shows "x \<in> set l"
wellformed_1: fixes e :: "'b" assumes "e \<in> arcs G" shows "tail G e \<in> verts G"
Markov_Decision_Process_Invariant_axioms: shows "Markov_Decision_Process_Invariant K S"
lpp_map_key_PP: fixes p :: "('x, nat) pp \<Rightarrow>\<^sub>0 'b" shows "pp_pm.lpp (Poly_Mapping.map_key PP p) = pp.mapping_of (lpp p)"
simps_1: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c" shows "[] \<Squnion>\<^bsub>f\<^esub> y = y"
inj_map_strong: fixes x :: "'a set['b]"   and xa :: "'a set['b]"   and f :: "'a \<Rightarrow> 'c"   and fa :: "'a \<Rightarrow> 'c" assumes "\<And>z za. \<lbrakk>z \<in> set_bset x; za \<in> set_bset xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_bset f x = map_bset fa xa" shows "x = xa"
norm_signed10: fixes xs :: "bit list" shows "norm_signed (\<one> # \<zero> # xs) = \<one> # \<zero> # xs"
f_aggregate_drop_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate (xs \<up> (n * k)) k ag = f_aggregate xs k ag \<up> n"
category_axioms_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "category_axioms C \<equiv> ((\<forall>g f. C g f \<noteq> partial_magma.null C \<longrightarrow> partial_magma.arr C (C g f)) \<and> (\<forall>f. (partial_magma.domains C f \<noteq> {}) = (partial_magma.codomains C f \<noteq> {})) \<and> (\<forall>h g f. partial_magma.arr C (C h g) \<longrightarrow> partial_magma.arr C (C (C h g) f) \<longrightarrow> partial_magma.arr C (C g f))) \<and> (\<forall>h g f. partial_magma.arr C (C h (C g f)) \<longrightarrow> partial_magma.arr C (C g f) \<longrightarrow> partial_magma.arr C (C h g)) \<and> (\<forall>g f h. partial_magma.arr C (C g f) \<longrightarrow> partial_magma.arr C (C h g) \<longrightarrow> partial_magma.arr C (C (C h g) f)) \<and> (\<forall>g f h. partial_magma.arr C (C g f) \<longrightarrow> partial_magma.arr C (C h g) \<longrightarrow> C (C h g) f = C h (C g f))"
intro: fixes D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'c" assumes "\<And>x. partial_magma.ide D x \<Longrightarrow> \<exists>y e. initial_arrow_to_functor C D G x y e" shows "right_adjoint_functor_axioms C D G"
measure_nonneg': fixes M :: "'a measure"   and A :: "'a set" shows "\<not> Sigma_Algebra.measure M A < 0"
add_increasing2: fixes c :: "'k multiset"   and b :: "'k multiset"   and a :: "'k multiset" assumes "{#} \<subseteq># c"   and "b \<subseteq># a" shows "b \<subseteq># a + c"
usphere_def: shows "usphere \<equiv> \<lambda>n. subtopology (nsphere n) (upper n)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. T (f, \<I>)) local.map (\<lambda>f. if arr f then f \<cdot> \<r>[local.dom f] else null)"
shift_def: fixes lab :: "'a list \<Rightarrow> 'b"   and k :: "'a" shows "BNF_Greatest_Fixpoint.shift lab k = (\<lambda>kl. lab (k # kl))"
bigo_bounded: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x. (0::'b) \<le> f x"   and "\<forall>x. f x \<le> g x" shows "f \<in> O(g)"
igWlsAbsIsInBar_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igWlsAbsIsInBar MOD \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> isInBar (us, s)"
ctor1_def: shows "ctor1 = unfold1 (F1map id dtor1 dtor2) (F2map id dtor1 dtor2)"
UNF_onI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<And>a b c. \<lbrakk>a \<in> A; (a, b) \<in> r\<^sup>!; (a, c) \<in> r\<^sup>!\<rbrakk> \<Longrightarrow> b = c" shows "UNF_on r A"
prv_all_imp_cnj_rev: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (all x (imp (cnj \<phi> \<psi>) \<chi>))" shows "bprv (all x (imp \<phi> (imp \<psi> \<chi>)))"
simps: fixes r :: "nat"   and x :: "'b"   and ts :: "'b tree list" shows "invar_btree (Node r x ts) = (Ball (set ts) invar_btree \<and> map rank ts = rev [0..<r])"
Posix_simp: fixes s :: "'b list"   and r :: "'b rexp"   and v :: "'b val" assumes "s \<in> fst (simp r) \<rightarrow> v" shows "s \<in> r \<rightarrow> snd (simp r) v"
s_has_vector_derivative: fixes t :: "real" assumes "0 \<le> t"   and "v\<^sub>e / a\<^sub>e \<le> 0"   and "a\<^sub>e \<noteq> 0" shows "(ego.s has_vector_derivative ego2.s' t) (at t within {0..})"
list_update_append: fixes xs :: "'g list"   and ys :: "'g list"   and n :: "nat"   and x :: "'g" shows "(xs @ ys)[n := x] = (if n < length xs then xs[n := x] @ ys else xs @ ys[n - length xs := x])"
assms_1: shows "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
distinct_17: fixes x2 :: "'c" shows "Notify x2 \<noteq> WokenUp"
minus_formula_def: fixes x :: "'a formula"   and y :: "'a formula" shows "x - y = Abs_formula (Rep_formula x - Rep_formula y)"
bdd_belowI2: fixes A :: "'d set"   and m :: "'e multiset"   and f :: "'d \<Rightarrow> 'e multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "subset_mset.bdd_below (f ` A)"
chinese_remainder: fixes a :: "nat"   and b :: "nat"   and u :: "nat"   and v :: "nat" assumes "coprime a b"   and "a \<noteq> 0"   and "b \<noteq> 0" shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b"
\<pi>_def: shows "\<pi> \<equiv> \<lambda>k. (\<lambda>y. odiff y (\<omega>_sum (take k \<alpha>s))) \<circ> ordermap D VWF"
isomorphic_def: fixes a :: "'a \<times> 'a"   and a' :: "'a \<times> 'a" shows "S.VV.isomorphic a a' = (\<exists>f. S.VV.in_hom f a a' \<and> S.VV.iso f)"
alpha_Tree_permute_rep_commute: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('b, 'c, 'd) Tree\<^sub>\<alpha>" shows "p \<bullet> rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha> =\<^sub>\<alpha> rep_Tree\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
ball_empty: fixes P :: "'d \<Rightarrow> bool" shows "Ball {} P = True"
simps_3: fixes v :: "'a" shows "pnPlus (Atom v) Zero = Atom v"
assigns_def: fixes t :: "Term.term" shows "assigns t = (case t of In1 (Inl e) \<Rightarrow> assignsE e | In1 (Inr s) \<Rightarrow> {} | In2 v \<Rightarrow> assignsV v | In3 es \<Rightarrow> assignsEs es)"
eps_match: fixes test :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "nat"   and r :: "'a Regex.regex" shows "Regex.eps test i r = Regex.match test r i i"
map_fun_id2_in: fixes g :: "'d \<Rightarrow> 'f"   and h :: "'g \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'g" shows "map_fun g h f = map_fun g id (h \<circ> f)"