hcju/isabelleps huggingface dataset:
le_inf_iff: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and x :: "'a::type"   and y :: "'a::type"   and z :: "'a::type" assumes "class.semilattice_inf (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (x::'a::type) ((inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (y::'a::type) (z::'a::type)) = (less_eq x y \<and> less_eq x z)"
cong_red_in_ctx: fixes t :: "lam"   and t' :: "lam"   and E :: "ctx" assumes "t \<longrightarrow>o t'" shows "E\<lbrakk>t\<rbrakk> \<longrightarrow>o E\<lbrakk>t'\<rbrakk>"
normalize_denom_pos: fixes r :: "int \<times> int"   and p :: "int"   and q :: "int" assumes "Rat.normalize r = (p, q)" shows "0 < q"
assms: shows "local.etp.HCP_adv \<A> \<sigma> (if \<sigma> then b1 else b0) D \<le> HCP_ad"
LIMSEQ_SUP: fixes X :: "nat \<Rightarrow> 'd" assumes "incseq X" shows "X \<longlonglongrightarrow> \<Squnion> range X"
igFreshBinp_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) model_scheme"   and ys :: "'c"   and y :: "'f"   and binp :: "'j \<Rightarrow> 'h option" shows "igFreshBinp MOD ys y binp \<equiv> liftAll (igFreshAbs MOD ys y) binp"
nr'_def: shows "nr' \<equiv> (\<pi>\<^sub>2 s, \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r)"
simps_97: fixes x6 :: "char list"   and x4 :: "val" shows "Var x6 \<noteq> Val x4"
star_decompose_3: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> = x\<^sup>\<star> * (y * x\<^sup>\<star>)\<^sup>\<star>"
select_convs_3: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.tab \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = tab"
wlog_positive_x_axis: fixes u :: "complex_homo"   and v :: "complex_homo"   and P :: "complex_homo \<Rightarrow> complex_homo \<Rightarrow> bool" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "\<And>M u v. \<lbrakk>unit_disc_fix M; u \<in> unit_disc; v \<in> unit_disc; u \<noteq> v; P (moebius_pt M u) (moebius_pt M v)\<rbrakk> \<Longrightarrow> P u v"   and "\<And>x. \<lbrakk>is_real x; 0 < Re x; Re x < 1\<rbrakk> \<Longrightarrow> P 0\<^sub>h (of_complex x)" shows "P u v"
GreenThm_typeI_divisible_region_boundary: fixes \<gamma> :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "gen_division s (cubeImage ` two_chain)"   and "two_chain_horizontal_boundary two_chain \<subseteq> \<gamma>"   and "\<gamma> \<subseteq> two_chain_boundary two_chain" shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} \<gamma>"
assms_1: shows "- pi / 2 \<le> \<alpha> \<and> \<alpha> \<le> pi / 2"
MaybeBoolInt_not_num_1: fixes r :: "value option"   and f :: "int \<Rightarrow> int \<Rightarrow> bool"   and n :: "value option" assumes "\<forall>n. r \<noteq> Some (Num n)" shows "MaybeBoolInt f n r = invalid"
alphaXX'yx: shows "X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs"
fun_upd_comp: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'c"   and y :: "'e" shows "f \<circ> g(x := y) = (f \<circ> g)(x := f y)"
Min_insert2: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "\<And>b. b \<in> A \<Longrightarrow> a \<le> b" shows "Min (insert a A) = a"
SUP_empty: fixes f :: "'b \<Rightarrow> 'a" shows "\<Squnion> (f ` {}) = \<bottom>"
meet_def: fixes x :: "nat"   and y :: "nat" shows "nat_dvd.meet x y = (THE inf. nat_dvd.is_inf x y inf)"
da_elim_cases_34: fixes Env :: "env"   and B :: "lname set"   and unop :: "unop"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (UnOp unop e)\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
memo_rbt_trancl_def: fixes r :: "('b \<times> 'b) list" shows "memo_rbt_trancl r = (let tr = trancl_rbt_impl r; rm = rm.to_map (map (\<lambda>a. (a, tr [a])) (((rs.to_list \<circ>\<circ> set_op_from_list) rs_ops \<circ>\<circ> map) fst r)) in (\<lambda>a. case rm.lookup a rm of None \<Rightarrow> rs.empty () | Some as \<Rightarrow> as))"
norm_pos: fixes v :: "(nat \<Rightarrow> real) \<times> nat" shows "0 \<le> \<parallel>v\<parallel>"
restrict_def: fixes G :: "('b, 'c) labeled_graph" shows "restrict G = LG {(l, v1, v2). (l, v1, v2) \<in> edges G \<and> v1 \<in> vertices G \<and> v2 \<in> vertices G} (vertices G)"
vec_of_poly_rev_shifted_scalar_prod: fixes p :: "'b poly"   and m :: "nat"   and v :: "'b vec"   and n :: "nat"   and j :: "nat" assumes "degree p = m"   and "dim_vec v = n"   and "j < m + n" shows "vec_of_poly_rev_shifted p n (n + m - Suc j) \<bullet> v = coeff (p * poly_of_vec v) j"
weak_is_transitive: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "rational_preference carrier relation"   and "x \<in> carrier \<and> y \<in> carrier \<and> z \<in> carrier"   and "x \<succeq>[relation] y"   and "y \<succeq>[relation] z" shows "x \<succeq>[relation] z"
show_lawI: fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and x :: "'a" assumes "\<And>p y z. s p x (y @ z) = s p x y @ z" shows "show_law s x"
In0_UN1: fixes f :: "'h \<Rightarrow> ('f, 'g) Old_Datatype.node set" shows "In0 (\<Union> (range f)) = (\<Union>x. In0 (f x))"
coprime_not_unit_not_dvd: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a dvd b"   and "comm_monoid_mult_class.coprime b c"   and "\<not> is_unit a" shows "\<not> a dvd c"
absorb_iff1: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a + b = a)"
cbelow_on_first_in: fixes A :: "'c set"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "cbelow_on A ord z rel a b" shows "a \<in> A"
ground_PfP: fixes y :: "tm" shows "ground_fm (PfP y) = ground y"
npD: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>np v" shows "u \<le>p v"
higher_differentiable_on_cong: fixes S :: "'b set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "open S"   and "S = T"   and "\<And>x. x \<in> T \<Longrightarrow> f x = g x" shows "higher_differentiable_on S f n = higher_differentiable_on T g n"
ccBinds_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "ccBinds \<Gamma> = (\<Lambda> i. \<Squnion>v\<mapsto>e\<in>map_of \<Gamma>. ccBind v e\<cdot>i)"
image_spec__eq: fixes P :: "'p"   and P' :: "'p"   and A :: "'p" assumes "P P' ReflectL A A" shows "P = P'"
scale_scale: fixes a :: "'c"   and b :: "'c"   and x :: "'b \<Rightarrow>\<^sub>0 'c" shows "a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
cycle_free_diag_equiv: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "(\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs) = cycle_free M n"
finsum_reindex_bij_witness: fixes S :: "'c set"   and i :: "'d \<Rightarrow> 'c"   and j :: "'c \<Rightarrow> 'd"   and T :: "'d set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"   and "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"   and "\<And>b. b \<in> S \<Longrightarrow> g b \<in> carrier R"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "finsum R g S = finsum R h T"
subgraph_H: shows "subgraph (with_proj H) (with_proj K)"
ccw': shows "ccw' 0 ((1 - x) *\<^sub>R (s - t)) (p - t)"
mvar_existence_ivl_eq_existence_ivl: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> Y.existence_ivl0 x0" shows "Y.mvar.existence_ivl x0 t = (\<lambda>_. Y.existence_ivl0 x0)"
domain_permutation_def: fixes A :: "'a set"   and B :: "'b set" shows "domain_permutation A B = {(f, f'). (f, f') \<in> (A \<rightarrow>\<^sub>E B) \<times> (A \<rightarrow>\<^sub>E B) \<and> (\<exists>p. p permutes A \<and> (\<forall>x\<in>A. f x = f' (p x)))}"
cp: shows "lookup p (t \<oplus> lt f) \<noteq> (0::'b)"
loc1_a_s_4: shows "\<forall>loc a' s'. a' \<in>\<^sub>A frontier (c_pts c loc) \<longrightarrow> s' \<in>\<^sub>A path_summary loc loc2 \<longrightarrow> \<not> results_in a' s' < b"
Loop: fixes G :: "'b triple set"   and P :: "'b assn"   and b :: "state \<Rightarrow> bool"   and c :: "com" assumes "G|-{P &> b}. c .{P}" shows "G|-{P}. WHILE b DO c .{P &> Not \<circ> b}"
U'def: shows "gram_schmidt_sub n ((adjuster n w us + w) # us) ws = us'"
P: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "locale_poly P" shows "P x"
negative_part_def: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "np f x = (if (0::'b) \<le> f x then 0::'b else - f x)"
redT_invariant: 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 t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   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"   and I :: "'t \<Rightarrow> unit option" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "ts_inv (\<lambda>_. P) I (thr s) (shr s)" shows "ts_inv (\<lambda>_. P) (upd_invs I (\<lambda>_. P) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>) (thr s') (shr s')"
sum_mult_product: fixes h :: "nat \<Rightarrow> 'b"   and A :: "nat"   and B :: "nat" shows "sum h {..<A * B} = (\<Sum>i<A. \<Sum>j<B. h (j + i * B))"
PP: fixes t :: "'trm" assumes "t \<in> trm" shows "PP t \<in> fmla"
SubstAtomicP_Mem: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SubstAtomicP v i (Q_Mem x y) (Q_Mem x' y')"
ControlSystemArch_def: fixes s :: "nat \<Rightarrow> nat list" shows "ControlSystemArch s \<equiv> \<exists>x z y. SteamBoiler x s y \<and> Controller y z \<and> Converter z x"
borel_measurable_restrict_space_iff: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "\<Omega> \<inter> space M \<in> sets M" shows "(f \<in> borel_measurable (restrict_space M \<Omega>)) = ((\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) \<in> borel_measurable M)"
subst_trm_fresh_var: fixes x :: "vrs"   and u :: "trm"   and t :: "trm" assumes "x \<sharp> u" shows "x \<sharp> t[x \<mapsto> u]"
welldefined_signs1: shows "well_def_signs (length qs1) signs1"
C_eq_Sets: fixes p :: "((int \<times> int) set set, int) Combinators list"   and s :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "singleCombinators p"   and "wellformed_policy1_strong p"   and "wellformed_policy1_strong s"   and "wellformed_policy3 p"   and "wellformed_policy3 s"   and "allNetsDistinct p"   and "set p = set s" shows "C (list2FWpolicy p) x = C (list2FWpolicy s) x"
contractible_space_def: fixes X :: "'a topology" shows "contractible_space X \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
cone_closed_monom_mult: fixes a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'b" assumes "a \<in> cone (h, U)"   and "t \<in> .[U]" shows "local.punit.monom_mult c t a \<in> cone (h, U)"
nff_\<alpha>_def: fixes dflt :: "'b"   and l :: "'b list"   and i :: "nat" shows "nff_\<alpha> dflt l i \<equiv> if i < length l then l ! i else dflt"
simps_5: fixes t :: "'c local.term" shows "\<^bold>\<lfloor>\<^bold>\<l>\<^bold>[t\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
OR_lower: fixes x :: "int"   and y :: "int" assumes "0 \<le> x"   and "0 \<le> y" shows "0 \<le> x OR y"
is_cnf_BigAnd: fixes ls :: "'b formula list" shows "is_cnf \<^bold>\<And>ls = (\<forall>l\<in>set ls. is_cnf l)"
assms_2: shows "fst (rep_nat_term u) = fst (rep_nat_term a)"
preserves_dom: fixes f :: "'a" assumes "C.arr f" shows "C.dom (C.map f) = C.map (C.dom f)"
get_component_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "ptr' \<in> set c" shows "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c"
wf_bs_insensitive: fixes bs :: "'b list"   and bs' :: "'b list"   and p :: "'b poly" assumes "length bs = length bs'" shows "wf_bs bs p = wf_bs bs' p"
yun_wrelD: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly" assumes "yun_wrel F c f" shows "of_int_poly F = Polynomial.smult c f"
assms_1: fixes x :: "'a" assumes "x \<in> A" shows "bounded (range (\<lambda>n. f n x))"
permute_bool_def: fixes p :: "perm"   and b :: "bool" shows "p \<bullet> b = b"
rep_eq: fixes x :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and xa :: "('a, 'c) RBT.rbt" shows "impl_of (RBT.map x xa) = RBT_Impl.map x (impl_of xa)"
trm_simps_1: fixes f :: "'a" assumes "arr f" shows "arr \<t>[f]"
execute_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and s :: "nat \<Rightarrow> nat option" shows "ast_problem.execute problem name s \<equiv> case ast_problem.lookup_operator problem name of Some (x, xa, effs, xb) \<Rightarrow> s ++ map_of (map (\<lambda>(uu_, x, uu_, v). (x, v)) (filter (ast_problem.eff_enabled problem s) effs))"
m: fixes x :: "'a" assumes "x \<in> set xs"   and "\<not> P x" shows "\<exists>i\<le>m. \<not> p ((f ^^ i) x)"
new_document_no_disconnected_nodes: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_disconnected_nodes new_document_ptr \<rightarrow>\<^sub>r []"
simps: fixes a :: "'node"   and n :: "'node"   and S :: "'node set" shows "(a \<in> obs n S) = (\<exists>as n'. a = n' \<and> n -as\<rightarrow>* n' \<and> (\<forall>nx\<in>set (sourcenodes as). nx \<notin> S) \<and> n' \<in> S)"
dX_range': fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> PdXXX.dX X d" shows "x \<in> X \<and> Xd x = d"
assoc'_simps_1: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.arr \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h]"
psubsetE: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subset> B" obtains "A \<subseteq> B" and "\<not> B \<subseteq> A"
rep_eq: fixes x :: "'b"   and xa :: "nat" shows "coeff (monom x xa) = (\<lambda>n. if xa = n then x else (0::'b))"
\<beta>: shows "\<beta> \<in> elts (ordertype M r)"
list_of_pdevs_all_nonzero: fixes xs :: "'a pdevs" shows "list_all (\<lambda>x. x \<noteq> (0::'a)) (map snd (list_of_pdevs xs))"
supp_Act: fixes f :: "'d"   and \<alpha> :: "'c"   and x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "supp (FL_Formula.Act f \<alpha> x) = supp f \<union> (supp \<alpha> \<union> supp x - bn \<alpha>)"
cases: fixes a :: "fm" assumes "a \<in> boolean_axioms" obtains A :: "fm" where "a = Aa__ IMP Aa__"   | A :: "fm" and B :: "fm" where "a = Ab__ IMP Ab__ OR B__"   | A :: "fm" where "a = Ac__ OR Ac__ IMP Ac__"   | A :: "fm" and B :: "fm" and C :: "fm" where "a = Ad__ OR Ba__ OR C__ IMP (Ad__ OR Ba__) OR C__"   | C :: "fm" and A :: "fm" and B :: "fm" where "a = C OR A IMP (C IMP B) IMP A OR B"
weakenS: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and a :: "'b seq_action" assumes "S' \<sigma> \<sigma>' a" shows "S \<sigma> \<sigma>' a"
simps_3: fixes I :: "interp"   and f :: "char" shows "term_sem I (Const f) = (\<lambda>\<omega>. Consts I f)"
tab_in_vhom': shows "\<guillemotleft>\<rho> \<cdot> \<phi> : g' \<Rightarrow> r \<star> f\<guillemotright>"
update_simps_17: fixes V :: "(nat, 'b) mapping"   and s :: "('a, 'b) state" shows "\<U> (\<V>_update V s) = \<U> s"
1: shows "X = w +o \<langle>S - {s}\<rangle>"
P1_OT_14_adv_bound: fixes funct_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and M :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool"   and D :: "'v_14_OT1 \<Rightarrow> bool spmf" shows "OT_14_sim.adv_P1 funct_14_OT M C D \<le> adv_14_OT"
join_def: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = (THE sup. is_sup x y sup)"
IH_1: assumes "wbt l" shows "wbt (Weight_Balanced_Trees.insert x l)"
config'_snoc: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and s :: "'state \<times> 'b"   and rs :: "'request list"   and r :: "'request" shows "config' A s (rs @ [r]) = Step A (config' A s rs) r"
app_inv: fixes t1 :: "('a, 'b) FingerTreeStruc"   and t2 :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t1"   and "FingerTreeStruc.ft_invar t2" shows "FingerTreeStruc.ft_invar (FingerTreeStruc.app t1 t2)"
dgrad_p_set_closed_monom_mult: fixes d :: "'a \<Rightarrow> nat"   and t :: "'a"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'c" assumes "dickson_grading d"   and "d t \<le> m"   and "p \<in> dgrad_p_set d m" shows "monom_mult c t p \<in> dgrad_p_set d m"
subst_lemma: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and T :: "LambdaType.type"   and e' :: "nat \<Rightarrow> LambdaType.type"   and u :: "dB"   and U :: "LambdaType.type"   and i :: "nat" assumes "e \<turnstile> t : T"   and "e' \<turnstile> u : U"   and "e = e'\<langle>i:U\<rangle>" shows "e' \<turnstile> t[u/i] : T"
poincare_in_angle_def: fixes p :: "complex_homo"   and a :: "complex_homo"   and b :: "complex_homo"   and c :: "complex_homo" shows "poincare_in_angle p a b c = (b \<noteq> a \<and> b \<noteq> c \<and> p \<noteq> b \<and> (\<exists>x\<in>unit_disc. poincare_between a x c \<and> x \<noteq> a \<and> x \<noteq> c \<and> poincare_on_ray p b x))"
wf_matchP_aux_Nil: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex" shows "wf_matchP_aux \<sigma> V n R I r [] 0"
mem_Rep_star_iff: fixes X :: "nat \<Rightarrow> 'c"   and x :: "'c star" shows "(X \<in> Rep_star x) = (x = star_n X)"
section_def: fixes f :: "'c" shows "C.section f \<equiv> \<exists>g. C.ide (C g f)"
simps_2: fixes ev_counter :: "nat"   and more :: "'a" shows "thread_t.more \<lparr>ev_counter = ev_counter, \<dots> = more\<rparr> = more"
secure1_def: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L" shows "secure1 p G c H = (\<forall>s t. s , c \<Down> t  \<longrightarrow> Q p H s t)"
take_rows_def: fixes A :: "'b matrix"   and r :: "nat" shows "take_rows A r \<equiv> Abs_matrix (\<lambda>j i. if j < r then Rep_matrix A j i else (0::'b))"
rbt_comp_insert: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_insert c = ord.rbt_insert (lt_of_comp c)"
infinite_subset: fixes A :: "'e set"   and B :: "'e set" assumes "infinite A"   and "A \<subseteq> B" shows "infinite B"
pair_sigma_finite_axioms: shows "pair_sigma_finite M1 M2"
Block_def: shows "e1.Class = (\<lambda>a\<in>Sigma I \<FF>1. THE A. a \<in> A \<and> A \<in> e1.Partition)"
wellorder_iMin_lemma: fixes k :: "'a"   and I :: "'a set" assumes "k \<in> I" shows "iMin I \<in> I \<and> iMin I \<le> k"
ring_l_one: fixes x :: "'a" assumes "x \<in> carrier R" shows "1\<^sub>r \<cdot>\<^sub>r x = x"
Sup_UNIV: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup UNIV = top"
simps_2: fixes f :: "'c"   and Tl :: "'c trm list" shows "vars (Fn f Tl) = \<Union> (vars ` set Tl)"
rel_state_elim: fixes P :: "heap \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "(heap, 'g) state"   and g :: "'h Heap"   and heap :: "heap" assumes "heap_mem_defs.rel_state P R f g"   and "P heap" obtains heap' :: "heap" and v :: "'g" and v' :: "'h" where "run_state f heap = (v, heap')" and "execute g heap = Some (v', heap')" and "R v v'" and "P heap'"
is_generalised_output_of_does_not_subsume: fixes t' :: "transition"   and t :: "transition"   and r :: "nat"   and p :: "nat"   and v :: "value"   and c :: "nat \<Rightarrow>f value option" assumes "is_generalised_output_of t' t r p"   and "p < length (Outputs t)"   and "Outputs t ! p = L v"   and "c $ r \<noteq> Some v"   and "\<exists>i. can_take_transition t i c" shows "\<not> subsumes t' c t"
flrestrictionI: fixes a :: "V"   and A :: "V"   and b :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<bullet> A"
assms: shows "wf\<^sub>t\<^sub>r\<^sub>m (t \<cdot> \<delta>)"
from_nat_suc: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs (j + 1) = plus (mod_type.from_nat Abs j) one"
f3': fixes a :: "nat"   and c :: "nat"   and b :: "nat" assumes "a < c"   and "c < b" shows "f3 {a, b, c} = f {(a, b), (a, c)}"
verts_add_vert: fixes u :: "'a" shows "verts (add_vert u) = insert u (verts G)"
add_nonpos_nonpos: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># {#}"   and "b \<subseteq># {#}" shows "a + b \<subseteq># {#}"
n_nin: shows "n \<notin> set (rev (takeWhile ((\<noteq>) n) (rev p)))"
LSC_need_it_once_have_many: fixes A :: "'b literal"   and F :: "'b literal list"   and \<Gamma> :: "'b formula multiset" assumes "A \<in> set F"   and "form_of_lit A \<^bold>\<or> disj_of_clause (removeAll A F), \<Gamma> \<Rightarrow>\<^sub>n" shows "disj_of_clause F, \<Gamma> \<Rightarrow>\<^sub>n"
cancels_to_preserves_generators: fixes l :: "(bool \<times> 'd) list"   and l' :: "(bool \<times> 'd) list"   and gens :: "'d set" assumes "cancels_to l l'"   and "l \<in> lists (UNIV \<times> gens)" shows "l' \<in> lists (UNIV \<times> gens)"
Rel_app: fixes A :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'j"   and x :: "'g"   and y :: "'i" assumes "Transfer.Rel (A ===> B) f g"   and "Transfer.Rel A x y" shows "Transfer.Rel B (f x) (g y)"
convergent_subseq_convergent: fixes X :: "nat \<Rightarrow> 'c"   and f :: "nat \<Rightarrow> nat" assumes "convergent X"   and "strict_mono f" shows "convergent (X \<circ> f)"