hcju/isabelleps huggingface dataset:
isomorphic_implies_ide_2: fixes f :: "'d"   and g :: "'d" assumes "D.isomorphic f g" shows "D.ide g"
of_nat_inverse: fixes r :: "nat"   and a :: "'a word" assumes "word_of_nat r = a"   and "r < 2 ^ LENGTH('a)" shows "unat a = r"
map_converter_of_callee: fixes f :: "'a \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'b"   and h :: "'g \<Rightarrow> 'c"   and k :: "'d \<Rightarrow> 'h"   and callee :: "'i \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'i, 'g, 'h) gpv"   and s :: "'i" shows "map_converter f g h k (CNV callee s) = CNV (map_fun id (map_fun f (map_gpv' (map_prod g id) h k)) callee) s"
wt_subst_trm': fixes \<sigma> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>"   and "\<Gamma> s = \<Gamma> t" shows "\<Gamma> (s \<cdot> \<sigma>) = \<Gamma> (t \<cdot> \<sigma>)"
arc_def: fixes g :: "real \<Rightarrow> 'b" shows "arc g = (path g \<and> inj_on g {0..1})"
image_ideal_eq_Int: fixes h :: "'b \<Rightarrow> 'b"   and B :: "'b set" assumes "\<And>x y. h (x + y) = h x + h y"   and "\<And>x y. h (x * y) = h x * h y"   and "\<And>x. h (h x) = h x" shows "h ` ideal B = ideal (h ` B) \<inter> range h"
four_types: shows "At = EqUni (a, b, c) \<or> At = LessUni (a, b, c) \<or> At = LeqUni (a, b, c) \<or> At = NeqUni (a, b, c)"
forall_2: fixes P :: "2 \<Rightarrow> bool" shows "(\<forall>i. P i) = (P 1 \<and> P 2)"
tendsto: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd"   and a :: "'d"   and F :: "'f filter" assumes "bounded_linear f"   and "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
intro: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme" assumes "category AA"   and "category BB" shows "two_cats AA BB"
set_s_mtf: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list"   and n :: "nat" assumes "MTF_Off as rs s0" shows "set (MTF_Off.s_mtf rs s0 n) = set s0"
create_opt_sol_in_LP_2: fixes X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "[y\<^sub>v*A]=c"
AS_VALID_MPLS_VALID: fixes as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" assumes "as \<in> valid_plans PROB" shows "Inf (PLS s as) \<in> MPLS_s PROB s"
cod_assoc: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set"   and b :: "'b local.term set"   and c :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a"   and "partial_magma.ide (free_monoidal_category.comp C) b"   and "partial_magma.ide (free_monoidal_category.comp C) c" shows "partial_magma.cod (free_monoidal_category.comp C) (free_monoidal_category.assoc\<^sub>F\<^sub>M\<^sub>C C a b c) = free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C a (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C b c)"
a1: shows "low_equal s1 s2 \<and> (fst instr = load_store_type STB \<or> fst instr = load_store_type STH \<or> fst instr = load_store_type ST \<or> fst instr = load_store_type STD) \<and> t1 = snd (fst (store_sub1 instr rd 0 s1)) \<and> t2 = snd (fst (store_sub1 instr rd 0 s2))"
collision_within_p: assumes "s\<^sub>o \<le> ego.s_stop"   and "ego.s_stop < other.s_stop" shows "collision {0..} = (\<exists>t\<ge>0. ego.p t = other.p t \<and> t < ego.t_stop \<and> t < other.t_stop)"
finite_fourier_poly'_conv_finite_fourier_poly: fixes ws :: "nat \<Rightarrow> complex"   and k :: "nat" shows "finite_fourier_poly' ws k = finite_fourier_poly (map ws [0..<k])"
leg0_composite_1: shows "\<mu>\<nu>\<pi>.leg0 = \<pi>.leg0 \<cdot> Prj\<^sub>0\<^sub>0"
sub2_grows_triangle: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and A :: "'a mat"   and l :: "nat"   and r :: "'a"   and r' :: "'a"   and A' :: "'a mat" assumes "det_selection_fun sel_fun"   and "mute_fun mf"   and "sub2 sel_fun mf (dim_row A) l (r, A) = (r', A')"   and "triangular_to l A"   and "l < dim_col A"   and "l < dim_row A" shows "triangular_to (Suc l) A'"
intros_1: fixes c :: "secprop set"   and A :: "agent"   and B :: "agent"   and M :: "atom list"   and H :: "chmsg set"   and At :: "atom" assumes "StatCh c A B (Msg M) \<in> H"   and "At \<in> set M" shows "At \<in> atoms H"
principal_inject: fixes A :: "'b set"   and B :: "'b set" shows "(principal A = principal B) = (A = B)"
integral_prod_continuous: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b"   and f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "continuous_on (cbox (a, c) (b, d)) f" shows "integral (cbox (a, c) (b, d)) f = integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y)))"
mkKripke_simps_3: fixes ws :: "'b set"   and rels :: "'d \<Rightarrow> ('b \<times> 'b) set"   and val :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "valuation (mkKripke ws rels val) = val"
pmat_extension_carrier: fixes m :: "'a mat" shows "st.pmat_extension m \<in> carrier_mat st.d0 st.d0"
cut_ge_finite: fixes I :: "'a set"   and t :: "'a" assumes "finite I" shows "finite (I \<down>\<ge> t)"
at_vchop_dist2: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<psi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<exists>va u. v=va--u \<and> (\<forall>v'. va = c > v' \<longrightarrow> \<phi> ts v') \<and> (\<forall>v'. u = c > v' \<longrightarrow> \<psi> ts v')) \<longrightarrow> (\<forall>v'. v = c > v' \<longrightarrow> (\<exists>v u. v'=v--u \<and> \<phi> ts v \<and> \<psi> ts u))"
integral_mono_AE': fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. g x \<le> f x"   and "AE x in M. 0 \<le> f x" shows "integral\<^sup>L M g \<le> integral\<^sup>L M f"
Diagonalize_COD: fixes f :: "'c local.term set" assumes "arr f" shows "\<^bold>\<lfloor>COD f\<^bold>\<rfloor> = Cod \<^bold>\<lfloor>rep f\<^bold>\<rfloor>"
emsapss_decode_help1_def: fixes mHash :: "bit list"   and EM :: "bit list"   and emBits :: "nat" shows "emsapss_decode_help1 mHash EM emBits = (if emBits < length mHash + sLen + 16 then False else emsapss_decode_help2 mHash EM emBits)"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_subst_cons: fixes l :: "'f strand_label"   and b :: "('d, 'e) stateful_strand_step"   and A :: "('f strand_label \<times> ('d, 'e) stateful_strand_step) list"   and \<delta> :: "'e \<Rightarrow> ('d, 'e) Term.term" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, b) # A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) \<union> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
card_surjective_functions_domain_and_range_permutation: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // domain_and_range_permutation A B) = Partition (card A) (card B)"
impl_refine_intro: fixes P\<^sub>1 :: "'b upred"   and Q\<^sub>1 :: "'b upred"   and Q\<^sub>2 :: "'b upred"   and P\<^sub>2 :: "'b upred" assumes "Q\<^sub>1 \<sqsubseteq> P\<^sub>1"   and "P\<^sub>2 \<sqsubseteq> (P\<^sub>1 \<and> Q\<^sub>2)" shows "(P\<^sub>1 \<Rightarrow> P\<^sub>2) \<sqsubseteq> (Q\<^sub>1 \<Rightarrow> Q\<^sub>2)"
irrlist_perm_cong: fixes as :: "'a list"   and bs :: "'a list" assumes "as <~~> bs"   and "\<forall>a\<in>set as. irreducible G a" shows "\<forall>a\<in>set bs. irreducible G a"
parTrace_not_empty: fixes cfl :: "(('test, 'atom) com \<times> 'state) list" assumes "parTrace cfl" shows "cfl \<noteq> []"
m_def: shows "m = length (chop (Suc (2 * d)) xs)"
disc_12: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" shows "\<not> isRecv (Send x21 x22 x23 x24 x25 x26)"
good_context_1: fixes s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "good_context s \<and> s' = s \<and> get_trap_set s' \<noteq> {} \<and> reset_trap_val s' = False \<and> get_ET (cpu_reg_val PSR s') = 0" shows "False"
lm045: fixes a :: "('a \<times> 'b set) set" shows "\<Union> {{(fst pair, {y}) |y. y \<in> snd pair} |pair. pair \<in> a} = {(fst pair, {y}) |y pair. y \<in> snd pair \<and> pair \<in> a}"
gcd_mod: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_euclidean_semiring divide plus minus zero times one modulo euclidean_size unit_factor normalize"   and "a \<noteq> zero" shows "normalization_euclidean_semiring.gcd zero modulo normalize a (modulo b a) = normalization_euclidean_semiring.gcd zero modulo normalize b a"
simps_1: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and project :: "'c \<Rightarrow> 'c"   and embed :: "'c \<Rightarrow> 'c list"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and r :: "'d rexp" assumes "Pi_Derivatives.embed \<Sigma> project embed" shows "embed.lderivs lookup embed [] r = r"
fgroup_by_nonempty_inner: fixes b :: "'d"   and cs :: "'e fset"   and f :: "'f \<Rightarrow> 'd \<times> 'e"   and as :: "'f fset" assumes "(b, cs) |\<in>| fgroup_by f as" obtains c :: "'e" where "c |\<in>| cs"
BRBangCases': fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, !Q) \<in> bangRel Rel" obtains P :: "pi" and Q :: "pi" where "Pa__ = !P" and "!Qa__ = !Q" and "(P, Q) \<in> Rel"
leD: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "\<not> x < y"
scaleR_eq_0_iff: fixes a :: "real"   and x :: "'c" shows "(a *\<^sub>R x = (0::'c)) = (a = 0 \<or> x = (0::'c))"
yun_erel_def: fixes F :: "int poly"   and f :: "rat poly" shows "yun_erel F f = (\<exists>c. yun_rel F c f)"
clAB: shows "closed_csubspace (A +\<^sub>M B)"
\<Gamma>'_sel_2: shows "A \<Gamma>' = A \<Gamma>"
equiv_order_trans: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "semiattractive A less_eq"   and "equivpartp less_eq x y"   and "less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
word_eq_rbl_eq: fixes x :: "'a word"   and y :: "'a word" shows "(x = y) = (rev (to_bl x) = rev (to_bl y))"
induct: fixes P :: "name list \<Rightarrow> nterm \<Rightarrow> bool"   and a0 :: "name list"   and a1 :: "nterm" assumes "\<And>\<Gamma> name. P \<Gamma> (Nconst name)"   and "\<And>\<Gamma> name. P \<Gamma> (Nvar name)"   and "\<And>\<Gamma> t u. \<lbrakk>P \<Gamma> t; P \<Gamma> u\<rbrakk> \<Longrightarrow> P \<Gamma> (t $\<^sub>n u)"   and "\<And>\<Gamma> x t. P (x # \<Gamma>) t \<Longrightarrow> P \<Gamma> (\<Lambda>\<^sub>n x. t)" shows "P a0 a1"
v1': shows "v = punit.lt (q g1) \<oplus> lt g1"
simps: fixes u :: "nat"   and v :: "nat" shows "mk_uedge (u, v) = {u, v}"
preserves_arr: fixes f :: "'j" assumes "J.arr f" shows "S.arr (U.map f)"
finite_cone_carrier: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n" shows "finite_cone A \<subseteq> carrier_vec n"
cases: fixes a1 :: "char list \<Rightarrow> ty option"   and a2 :: "expr list"   and a3 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a4 :: "expr list"   and a5 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P,a1 \<turnstile> \<langle>a2,a3\<rangle> [\<rightarrow>] \<langle>a4,a5\<rangle>" obtains E :: "char list \<Rightarrow> ty option" and e :: "expr" and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and e' :: "expr" and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and es :: "expr list" where "a1 = Ea__" and "a2 = e # es__" and "a3 = s__" and "a4 = e' # es__" and "a5 = s'__" and "P,Ea__ \<turnstile> \<langle>e,s__\<rangle> \<rightarrow> \<langle>e',s'__\<rangle>"   | E :: "char list \<Rightarrow> ty option" and es :: "expr list" and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and es' :: "expr list" and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and v :: "val" where "a1 = E" and "a2 = Val v # es" and "a3 = s" and "a4 = Val v # es'" and "a5 = s'" and "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>"
trans: fixes y :: "'c"   and x :: "'c"   and z :: "'c" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
correctness_def: fixes m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool" shows "OT_14.correctness m1 m2 \<equiv> protocol_14_OT m1 m2 = funct_OT_14 m1 m2"
Sent_imp_hfthm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "x \<in> Sent" shows "\<exists>A. x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> {} \<turnstile> A"
trans2: fixes p :: "'b"   and t :: "'b"   and r :: "'b"   and q :: "'b" assumes "p \<parallel> t"   and "t \<parallel> r"   and "r \<parallel> q" shows "\<not> p \<parallel> q"
loc_reach_mem_diff_subset: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and c :: "'Com" assumes "\<forall>x. var_asm_not_written mds x \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x \<and> dma mem\<^sub>1 x = dma mem\<^sub>2 x"   and "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>c, mds, mem\<^sub>1\<rangle>" shows "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>c, mds, mem\<^sub>2\<rangle>"
trms\<^sub>s\<^sub>s\<^sub>t_cons: fixes a :: "('c, 'd) stateful_strand_step"   and A :: "('c, 'd) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t (a # A) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> trms\<^sub>s\<^sub>s\<^sub>t A"
assms_2: shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) local.map a d x"
card_A_eq': shows "{..<card A} = {..<m} \<union> {m..<m + n} \<union> {m + n..<card A}"
poly_of_list_i_def: fixes ops :: "'b arith_ops_record" shows "poly_of_list_i ops = strip_while ((=) (arith_ops_record.zero ops))"
read_only_unowned_axioms: shows "read_only_unowned \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
inj: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetIso less_eq less less_eq' less' P f" shows "inj_on f P"
Quotient_total_abs_eq_iff: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "Quotient R Abs Rep T"   and "reflp R" shows "(Abs x = Abs y) = R x y"
scope_find: fixes v :: "'a"   and outPorts :: "'b \<Rightarrow> 'c fset"   and vertices :: "'a fset"   and nodeOf :: "'a \<Rightarrow> 'b"   and edges :: "(('a \<times> 'c) \<times> 'a \<times> 'd) set"   and ps :: "'a \<times> 'd"   and v' :: "'a"   and pth :: "(('a \<times> 'c) \<times> 'a \<times> 'd) list" assumes "v \<in> Pre_Port_Graph.scope outPorts vertices nodeOf edges ps"   and "Vertex_Graph.terminal_vertex outPorts vertices nodeOf v'"   and "Pre_Port_Graph.path edges v v' pth" shows "ps \<in> snd ` set pth"
b: fixes p :: "'a"   and q :: "'a" shows "((p, q) \<in> b) = (\<exists>t. p \<parallel> t \<and> t \<parallel> q)"
perzeta_frac: fixes q :: "real" shows "perzeta (frac q) = perzeta q"
transitivity: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" assumes "preference carrier relation" shows "trans relation"
rep_eq: fixes x :: "'b st_vec" shows "measure_pmf (pmf_of_st_vec x) = measure_of_st_vec x"
wgraph_specs_7: fixes g :: "'g"   and u :: "'v"   and v :: "'v"   and d :: "nat" assumes "invar g"   and "(u, v) \<notin> edges (\<alpha>g g)"   and "u \<noteq> v" shows "\<alpha>w (add_edge (u, v) d g) = (\<alpha>w g)({u, v} := d)"
assms_3: shows "\<pp>\<^sub>1[a, b] \<cdot> h = \<pp>\<^sub>1[a, b] \<cdot> h'"
elims: fixes x :: "floatarith"   and y :: "floatarith set" assumes "subterms x = y" obtains a :: "floatarith" and b :: "floatarith" where "x = Add a__ b__" and "y = insert (Add a__ b__) (subterms a__ \<union> subterms b__)"   | a :: "floatarith" and b :: "floatarith" where "x = Mult aa__ ba__" and "y = insert (Mult aa__ ba__) (subterms aa__ \<union> subterms ba__)"   | a :: "floatarith" and b :: "floatarith" where "x = Min ab__ bb__" and "y = insert (Min ab__ bb__) (subterms ab__ \<union> subterms bb__)"   | a :: "floatarith" and b :: "floatarith" where "x = Max ac__ bc__" and "y = insert (Max ac__ bc__) (subterms ac__ \<union> subterms bc__)"   | a :: "floatarith" and b :: "floatarith" where "x = Powr ad__ b" and "y = insert (Powr ad__ b) (subterms ad__ \<union> subterms b)"   | a :: "floatarith" where "x = Inverse ae__" and "y = insert (Inverse ae__) (subterms ae__)"   | a :: "floatarith" where "x = Cos af__" and "y = insert (Cos af__) (subterms af__)"   | a :: "floatarith" where "x = Arctan ag__" and "y = insert (Arctan ag__) (subterms ag__)"   | a :: "floatarith" where "x = Abs ah__" and "y = insert (Abs ah__) (subterms ah__)"   | a :: "floatarith" where "x = Sqrt ai__" and "y = insert (Sqrt ai__) (subterms ai__)"   | a :: "floatarith" where "x = Exp aj__" and "y = insert (Exp aj__) (subterms aj__)"   | a :: "floatarith" where "x = Ln ak__" and "y = insert (Ln ak__) (subterms ak__)"   | a :: "floatarith" and n :: "nat" where "x = al__ ^\<^sub>e n__" and "y = insert (al__ ^\<^sub>e n__) (subterms al__)"   | a :: "floatarith" where "x = Floor am__" and "y = insert (Floor am__) (subterms am__)"   | a :: "floatarith" where "x = Minus a" and "y = insert (Minus a) (subterms a)"   | "x = Pi" and "y = {Pi}"   | v :: "nat" where "x = Var v" and "y = {Var v}"   | n :: "float" where "x = Num n" and "y = {Num n}"
real_of_nat_div_aux: fixes x :: "nat"   and d :: "nat" shows "real x / real d = real (x div d) + real (x mod d) / real d"
Id_on_domain: fixes A :: "'e set"   and f :: "('e \<times> 'f) set" shows "Domain (Id_on A O f) = A \<inter> Domain f"
blaschke_a_to_zero: fixes a :: "complex" assumes "cmod a \<noteq> 1" shows "moebius_pt (blaschke a) (of_complex a) = 0\<^sub>h"
less_mult_imp_mod_less: fixes m :: "nat"   and n :: "nat"   and i :: "nat" assumes "m < n * i" shows "m mod i < i"
pfun_member_alt_def: fixes x :: "'a"   and y :: "'b"   and f :: "('a, 'b) pfun" shows "((x, y) \<in>\<^sub>p f) = (x \<in> pdom f \<and> f(x)\<^sub>p = y)"
run_state_get: fixes s :: "'c" shows "run_state get s = (s, s)"
ifte_thms_2: fixes e1 :: "'b"   and e2 :: "'b" shows "If FF then e1 else e2 = e2"
val_case_stock_measurable: fixes t :: "pdf_type"   and c :: "'a"   and M :: "'a measure"   and g :: "bool \<Rightarrow> 'a"   and h :: "int \<Rightarrow> 'a"   and j :: "real \<Rightarrow> 'a"   and k :: "val \<Rightarrow> val \<Rightarrow> 'a" assumes "t = UNIT \<Longrightarrow> c \<in> space M"   and "\<And>b. t = BOOL \<Longrightarrow> g b \<in> space M"   and "\<And>i. t = INTEG \<Longrightarrow> h i \<in> space M"   and "t = REAL \<Longrightarrow> j \<in> borel \<rightarrow>\<^sub>M M"   and "\<And>t1 t2. t = PRODUCT t1 t2 \<Longrightarrow> (\<lambda>(x, y). k x y) \<in> stock_measure t1 \<Otimes>\<^sub>M stock_measure t2 \<rightarrow>\<^sub>M M" shows "(\<lambda>x. case x of UnitVal \<Rightarrow> c | BoolVal b \<Rightarrow> g b | IntVal i \<Rightarrow> h i | RealVal r \<Rightarrow> j r | <|y, z|> \<Rightarrow> k y z) \<in> stock_measure t \<rightarrow>\<^sub>M M"
equivclp_blinding_of: shows "equivclp bo = vimage2p h h (=)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and comp_res :: "comp_res" shows "P (case comp_res of LESS \<Rightarrow> f1 | EQUAL \<Rightarrow> f2 | GREATER \<Rightarrow> f3) = ((comp_res = LESS \<longrightarrow> P f1) \<and> (comp_res = EQUAL \<longrightarrow> P f2) \<and> (comp_res = GREATER \<longrightarrow> P f3))"
roots0: fixes p :: "'a poly" assumes "p \<noteq> 0"   and "degree p = 0" shows "{x. poly p x = (0::'a)} = {}"
exhaust_sel: fixes interval_tree :: "interval_tree" obtains "interval_tree = Tip"   | "interval_tree = Node (lsub interval_tree) (val interval_tree) (tmax interval_tree) (rsub interval_tree)"
exact_im_sub_ker1: fixes L :: "('i, 'a) Module"   and M :: "('j, 'a) Module"   and N :: "('k, 'a) Module"   and f :: "'i \<Rightarrow> 'j"   and g :: "'j \<Rightarrow> 'k"   and z1 :: "'k \<Rightarrow> 'l"   and e :: "'l"   and Z :: "('m, 'a, 'n) Module_scheme"   and x :: "'j \<Rightarrow> 'm" assumes "R module L"   and "R module M"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N"   and "z1 \<in> mHom R N (Zm R e)"   and "R module Z"   and "exact4 R L f M g N z1 (Zm R e)"   and "x \<in> mHom R M Z"   and "compos L x f = mzeromap L Z" shows "compos M (\<lambda>z\<in>carrier N. x (SOME y. y \<in> carrier M \<and> g y = z)) g = x"
eval_conjoin: fixes e :: "nat \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c list \<Rightarrow> bool"   and G :: "('a, 'b) form list" shows "list_all (eval e f g) G = eval e f g (conjoin G)"
pdevs_val_add_pdevs: fixes e :: "nat \<Rightarrow> real"   and X :: "'b pdevs"   and Y :: "'b pdevs" shows "pdevs_val e (add_pdevs X Y) = pdevs_val e X + pdevs_val e Y"
opp_chambers_bij: shows "bij_betw ((`) f) (folding.\<C> - f \<turnstile> folding.\<C>) (f \<turnstile> folding.\<C>)"
repeat_mset_empty: fixes n :: "nat" shows "repeat_mset n {#} = {#}"
bs_lower_bound: shows "\<exists>C>0. \<forall>b\<in>set bs. C < b"
grounding_of_clss_mono: fixes CC :: "'a literal multiset set"   and DD :: "'a literal multiset set" assumes "CC \<subseteq> DD" shows "grounding_of_clss CC \<subseteq> grounding_of_clss DD"
UNION_eq: fixes B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "\<Union> (B ` A) = {y. \<exists>x\<in>A. y \<in> B x}"
a_in_carr_fgmodule: fixes a :: "'e"   and A :: "'e set"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'a \<Rightarrow> 'e \<Rightarrow> 'e" assumes "a \<in> A" shows "a \<in> carrier (fgmodule R A z i f s)"
aeq_nat_eq: fixes n :: "ant"   and m :: "ant" assumes "n \<noteq> \<infinity>"   and "0 \<le> n"   and "m \<noteq> \<infinity>"   and "0 \<le> m" shows "(n = m) = (na n = na m)"
simps: fixes p :: "integer"   and x :: "integer"   and n :: "integer" shows "power_p_integer p x n = (if n \<le> 0 then 1 else let rec = power_p_integer p (mult_p_integer p x x) (n >> 1) in if n AND 1 = 0 then rec else mult_p_integer p rec x)"