hcju/isabelleps huggingface dataset:
tangent_space_derivation: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "X \<in> tangent_space p"   and "f \<in> diff_fun_space"   and "g \<in> diff_fun_space" shows "X (f * g) = f p * X g + g p * X f"
ZFfunDomainOpair: fixes f :: "ZF"   and x :: "ZF" assumes "isFun f"   and "x |\<in>| HOLZF.Domain f" shows "Opair x (f \<acute> x) |\<in>| f"
simps_1: fixes C :: "String.literal" shows "class_type_of (Class_type C) = C"
ncols_le: fixes A :: "'b matrix"   and n :: "nat" shows "(ncols A \<le> n) = (\<forall>j i. n \<le> i \<longrightarrow> Rep_matrix A j i = (0::'b))"
replicate_eqI: fixes xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "length xs = n"   and "\<And>y. y \<in> set xs \<Longrightarrow> y = x" shows "xs = replicate n x"
wt_\<Phi>: shows "Ik.wtPB \<Phi>"
MkIde_Cod: fixes a :: "(bool, 'arr list) concrete_category.arr" assumes "arr a" shows "MkIde (Cod a) = cod a"
trans: fixes a :: "'g"   and b :: "'g"   and c :: "'g" assumes "a \<le> b"   and "b \<le> c" shows "a \<le> c"
singular_cycle: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_relcycle p X {} c = (singular_chain p X c \<and> chain_boundary p c = 0)"
comm_neutral: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k"   and a :: "'k" assumes "comm_monoid f z" shows "f a z = a"
r_def: shows "r \<equiv> run (delta \<Sigma>) (initial \<phi>) w"
map_cong0: fixes x :: "'b pmf"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> f z = g z" shows "map_pmf f x = map_pmf g x"
range_sbintrunc: shows "range (signed_take_bit 31) = {i. - (2 ^ 31) \<le> i \<and> i < 2 ^ 31}"
half_open_segment_subset_1: fixes t0 :: "'a"   and t1 :: "'a" shows "{t0--<t1} \<subseteq> {t0--t1}"
list_all2_list_all3I: fixes xs :: "'d list"   and ys :: "'e list"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and zs :: "'f list" assumes "length xs = length ys"   and "list_all2 (\<lambda>(x, y). P x y) (zip xs ys) zs" shows "list_all3 P xs ys zs"
composite_simps_3: shows "local.dom (\<mu> \<star> \<nu>) = local.dom \<mu> \<star> local.dom \<nu>"
minus_p64_def: fixes p :: "uint64"   and x :: "uint64"   and y :: "uint64" shows "minus_p64 p x y \<equiv> if y \<le> x then x - y else x + p - y"
arr_cod: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.arr (hom'.cod f)"
isFault_simps_3: fixes f :: "'b" shows "isFault (Fault f) = True"
exhaust: fixes y :: "'c nres" obtains "y = FAILi"   | x2 :: "'c set" where "y = RES x2"
eta_step_imp_eta_reducible: fixes t :: "Core.term"   and t' :: "Core.term" assumes "t \<rightarrow>\<^sub>\<eta> t'" shows "eta_reducible t"
comp_inv_arr': fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.iso f" shows "S.inv f \<cdot>\<^sub>S f = S.dom f"
e_app_intro: fixes f :: "(val \<times> val) list"   and e1 :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v2 :: "val"   and e2 :: "exp"   and v2' :: "val"   and v3' :: "val"   and v3 :: "val" assumes "VFun f \<in> E e1 \<rho>"   and "v2 \<in> E e2 \<rho>"   and "(v2', v3') \<in> set f"   and "v2' \<sqsubseteq> v2"   and "v3 \<sqsubseteq> v3'" shows "v3 \<in> E (EApp e1 e2) \<rho>"
ifbd_bot_pres: shows "bot_pres bd\<^sup>-\<^sub>\<F>"
simps_25: fixes f1 :: "b_e \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "cl \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> e list \<Rightarrow> e list \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> v list \<Rightarrow> e list \<Rightarrow> 'a"   and x1 :: "b_e" shows "(case $x1 of $x \<Rightarrow> f1 x | Trap \<Rightarrow> f2 | Callcl x \<Rightarrow> f3 x | Label x xa xb \<Rightarrow> f4 x xa xb | Local x xa xb xc \<Rightarrow> f5 x xa xb xc) = f1 x1"
make_one_new_out_of_two: fixes x :: "'a"   and y :: "'b" assumes "\<exists>n1. new_tv n1 x"   and "\<exists>n2. new_tv n2 y" shows "\<exists>n. new_tv n x \<and> new_tv n y"
binary_functor_axioms: shows "binary_functor (\<cdot>) (\<cdot>) (\<cdot>) local.Prod"
TR_sub_carr: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR1 R M N P \<union> TR2 R M N P \<union> TR3 R M N P \<union> TR4 R M N P \<subseteq> carrier P"
simps: fixes x :: "natural \<times> 'a"   and xs :: "(natural \<times> 'a) list"   and i :: "natural" shows "Random.pick (x # xs) i = (if i < fst x then snd x else Random.pick xs (i - fst x))"
intros_1: fixes a :: "'d set"   and A :: "'d set set"   and sp :: "'d set" assumes "a \<in> A" shows "a \<in> sigma_sets sp A"
partn_lst_VWF_imp_partn_lst_eq: fixes \<beta> :: "V"   and \<alpha> :: "V list"   and n :: "nat"   and U :: "'a set"   and r :: "('a \<times> 'a) set" assumes "partn_lst_VWF \<beta> \<alpha> n"   and "ordertype U r = \<beta>"   and "small U"   and "wf r"   and "trans r"   and "total_on U r" shows "Partitions.partn_lst r U \<alpha> n"
sup_idem: fixes x :: "'a upred" shows "(x \<or> x) = x"
vector_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "vector x"   and "vector y" shows "vector (x \<squnion> y)"
content_x_minus_const_int: fixes c :: "int" shows "content [:c, 1:] = 1"
product_over_sum: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CEM"   and "O x y \<and> O x z" shows "x \<otimes> (y \<oplus> z) = x \<otimes> y \<oplus> x \<otimes> z"
square_test_def: fixes n :: "nat" shows "square_test n = (n mod 64 \<in> q64 \<and> (let r = n mod 45045 in r mod 63 \<in> q63 \<and> r mod 65 \<in> q65 \<and> r mod 11 \<in> q11 \<and> n = (Discrete.sqrt n)\<^sup>2))"
bij_\<phi>: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>" shows "bij_betw \<phi> (Collect (category.terminal S)) (Collect (category.terminal S'))"
subformulas\<^sub>\<mu>_finite: fixes \<phi> :: "'a ltln" shows "finite (subformulas\<^sub>\<mu> \<phi>)"
same_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a list" shows "same f A \<equiv> \<forall>a1\<in>set A. \<forall>a2\<in>set A. f a1 = f a2"
IVT2': fixes f :: "'b \<Rightarrow> 'a"   and b :: "'b"   and y :: "'a"   and a :: "'b" assumes "f b \<le> y"   and "y \<le> f a"   and "a \<le> b"   and "continuous_on {a..b} f" shows "\<exists>x\<ge>a. x \<le> b \<and> f x = y"
Domain_unfold: fixes r :: "('l \<times> 'm) set" shows "Domain r = {x. \<exists>y. (x, y) \<in> r}"
subst_bv2_lt: fixes j :: "nat"   and i :: "nat"   and u :: "Core.term list" assumes "j < i" shows "subst_bvs1' (Bv j) i u = Bv j"
fps_lr_inverse_mult_divring_2: fixes f :: "'b fps"   and g :: "'b fps" shows "fps_right_inverse (f * g) (inverse ((f * g) $ 0)) = fps_right_inverse g (inverse (g $ 0)) * fps_right_inverse f (inverse (f $ 0))"
finite_if_finite_subsets_card_bdd: fixes F :: "'a set"   and C :: "nat" assumes "\<And>G. \<lbrakk>G \<subseteq> F; finite G\<rbrakk> \<Longrightarrow> card G \<le> C" shows "finite F \<and> card F \<le> C"
simps_2: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
timesul: shows "wtimes u\<^sub>1 l\<^sub>2 \<equiv>\<^sub>E ru1 * rl2"
mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> H"   and "b \<in> H" shows "a \<otimes> b \<in> H"
row_iarray_def: fixes k :: "nat"   and A :: "'a iarray iarray" shows "row_iarray k A = A !! k"
sel: fixes x2 :: "'e" shows "the (Some x2) = x2"
natural_transformation_axioms: shows "natural_transformation A B G F \<tau>'.map"
assms_2: assumes "h \<noteq> 0" shows "U \<subseteq> X"
ctermsE: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>"   and "p \<in> cterms \<Gamma>" obtains pn :: "'e" where "p \<in> ctermsl (\<Gamma> pn)" and "not_call p"
hh: fixes e :: "nat" shows "eval (Cn 1 (Pr 1 Z g) [Cn 1 r_length [recf.Id 1 0], recf.Id 1 0]) [e] \<down>= (if \<forall>i<e_length e. e_nth e i = 0 then e_length e else GREATEST i. i < e_length e \<and> e_nth e i \<noteq> 0)"
set_intros_2: fixes x22 :: "'a"   and x21 :: "linear_poly" shows "x22 \<in> set_ns_constraint (GEQ_ns x21 x22)"
auth_prikey_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_prikey" shows "Auth_PriKey n \<notin> spied s"
poly_lift_hom_def: fixes R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and \<phi> :: "'c \<Rightarrow> 'd" shows "poly_lift_hom R S \<phi> = eval R (UP S) (to_polynomial S \<circ> \<phi>) (X_poly S)"
goodInp_def: fixes inp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option" shows "goodInp inp \<equiv> liftAll good inp \<and> |{i. inp i \<noteq> None}| <o |UNIV|"
2: shows "S'.set (S'.dom (\<Phi> m)) \<subseteq> S'.set (S'.cod (\<Phi> m))"
K_cfg_trans': fixes l :: "'s"   and u :: "'c \<Rightarrow> real"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and cfg' :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and l' :: "'s"   and u' :: "'c \<Rightarrow> real" assumes "repcs (l, u) cfg \<in> MDP.cfg_on (l, u)"   and "cfg' \<in> set_pmf (K_cfg (repcs (l, u) cfg))"   and "state cfg' = (l', u')"   and "(l, u) \<in> S"   and "cfg \<in> R_G.valid_cfg"   and "abss (l, u) = state cfg" shows "trans' u u'"
sorted_wrt_insort_wrt: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "transp r"   and "\<And>a. r a x \<or> r x a" shows "sorted_wrt r (insort_wrt r x xs) = sorted_wrt r xs"
idempotent_one_closed: shows "idempotent (1::'a)"
simps_2: fixes x :: "char list"   and e :: "aexp" shows "lvars (x ::= e) = {x}"
S_separates: fixes xs :: "'a list" assumes "v0 \<leadsto>xs\<leadsto> v1" shows "set xs \<inter> S \<noteq> {}"
simps_2: shows "Rtimeout_error \<noteq> Rtype_error"
not_Carmichael_number: shows "\<not> Carmichael_number n \<or> n < 3"
RG_one_closed: shows "1 \<in> Supgroup.group_ring"
psums_replicate_0_append: fixes n :: "nat"   and xs :: "'a list" shows "psums (replicate n (0::'a) @ xs) = replicate n (0::'a) @ psums xs"
degree_def: fixes p :: "'a poly" shows "degree p = (LEAST n. \<forall>i>n. poly.coeff p i = (0::'a))"
even_card_set: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "finite A"   and "\<forall>x\<in>A. f x \<in> A \<and> f x \<noteq> x \<and> f (f x) = x" shows "even (card A)"
Max_ge: fixes A :: "complex set"   and x :: "complex" assumes "finite A"   and "x \<in> A" shows "x \<le>\<^sub>\<complex> complex_lex.Max A"
last_prog: shows "last_prog p\<^sub>s\<^sub>b sb = p\<^sub>s\<^sub>b"
rw_add_strict_def: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and q :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "rw_add_strict p q = (fst p \<prec>\<^sub>t fst q)"
f3: shows "(0 \<le> 4294967296 + - 1 * uint w + - 1 * ((4294967296 + - 1 * uint w) mod 4294967296)) = (uint w + (4294967296 + - 1 * uint w) mod 4294967296 \<le> 4294967296)"
hcomp_simps_4: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.hseq \<nu> \<mu>" shows "D.cod (\<nu> \<star>\<^sub>D \<mu>) = D.cod \<nu> \<star>\<^sub>D D.cod \<mu>"
infinite_halfspace_Im_gt: fixes b :: "real" shows "infinite {x. b < Im x}"
2: fixes P :: "'a mat"   and A :: "'a mat"   and Q :: "'a mat" assumes "P \<in> carrier_mat CARD('nr) CARD('nr) \<and> A \<in> carrier_mat CARD('nr) CARD('nc) \<and> Q \<in> carrier_mat CARD('nc) CARD('nc)" shows "Domainp HMA_M3 (P, A, Q)"
bl_word_or: fixes v :: "'b word"   and w :: "'b word" shows "to_bl (v OR w) = map2 (\<or>) (to_bl v) (to_bl w)"
finite_States: fixes A :: "('a, 'b, 'c) seqauto" shows "finite (States A)"
permute_Abs_3: fixes p :: "perm"   and as :: "atom list"   and x :: "'a" shows "p \<bullet> ([as]lst. x) = [p \<bullet> as]lst. p \<bullet> x"
subtopology_Times: fixes X :: "'b topology"   and Y :: "'c topology"   and S :: "'b set"   and T :: "'c set" shows "subtopology (prod_topology X Y) (S \<times> T) = prod_topology (subtopology X S) (subtopology Y T)"
assms_3: shows "w \<le>p r \<cdot> w"
IN_plus_flow: fixes \<Delta> :: "('a, 'b) graph_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> f e = 0"   and "\<And>x y. edge \<Delta> x y \<Longrightarrow> g (y, x) \<le> f (x, y)"   and "d_OUT g x \<noteq> \<top>" shows "d_IN (f \<oplus>\<^bsub>\<Delta>\<^esub> g) x = d_IN f x + (\<Sum>\<^sup>+ y. g (y, x) * indicator \<^bold>E\<^bsub>\<Delta>\<^esub> (y, x)) - (\<Sum>\<^sup>+ y. g (x, y) * indicator \<^bold>E\<^bsub>\<Delta>\<^esub> (y, x))"
prime_aux: fixes p :: "nat" assumes "prime p" shows "g_prime_power p 1 = g_prime p"
aff_dim_le_DIM: fixes S :: "'b set" shows "aff_dim S \<le> int DIM('b)"
size_cartesian_product_singleton: fixes a :: "'a"   and B :: "'b multiset" shows "size ({#a#} \<times># B) = size B"
ldropn_Suc_LCons: fixes n :: "nat"   and x :: "'c"   and xs :: "'c llist" shows "ldropn (Suc n) (LCons x xs) = ldropn n xs"
inputPres: fixes P :: "pi"   and x :: "name"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name" assumes "\<forall>y. (P[x::=y], Q[x::=y]) \<in> Rel"   and "eqvt Rel" shows "a<x>.P \<leadsto><Rel> a<x>.Q"
cases: fixes x :: "nat \<times> nat \<times> atom" obtains l :: "nat" and u :: "nat" where "x = (l__, u__, Less 0 0)"   | l :: "nat" and u :: "nat" and j :: "nat" where "x = (la__, ua__, Less 0 (Suc j__))"   | l :: "nat" and u :: "nat" and i :: "nat" where "x = (lb__, ub__, Less (Suc i__) 0)"   | l :: "nat" and u :: "nat" and i :: "nat" and j :: "nat" where "x = (lc__, uc__, Less (Suc ia__) (Suc ja__))"   | l :: "nat" and u :: "nat" where "x = (ld__, ud__, Eq 0 0)"   | l :: "nat" and u :: "nat" and v :: "nat" where "x = (le__, ue__, Eq 0 (Suc v__))"   | l :: "nat" and u :: "nat" and v :: "nat" where "x = (lf__, uf__, Eq (Suc v) 0)"   | l :: "nat" and u :: "nat" and i :: "nat" and j :: "nat" where "x = (l, u, Eq (Suc i) (Suc j))"
Irr_Mon_def: shows "Irr_Mon = {x. irreducible x \<and> monic x}"
sortedByWeight_Cons_imp_forall_weight_ge: fixes t :: "'a tree"   and ts :: "'a tree list" assumes "sortedByWeight (t # ts)" shows "\<forall>u\<in>set ts. weight t \<le> weight u"
cosh_diff: fixes x :: "'a"   and y :: "'a" shows "cosh (x - y) = cosh x * cosh y - sinh x * sinh y"
elims_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list"   and y :: "bool" assumes "class.linorder less_eq less"   and "linorder.strict_sorted less x = y" obtains "x = []" and "y"   | xa :: "'a" and ys :: "'a list" where "x = xa # ys" and "y = (Ball (set ys) (less xa) \<and> linorder.strict_sorted less ys)"
dom_subid_aux2'': fixes y :: "'a"   and x :: "'a" shows "ar (ar y) \<cdot> x \<le> x"
mon_c_def: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and c :: "'g list multiset" shows "mon_c fg c \<equiv> \<Union> {mon_s fg s |s. s \<in># c}"
auth_shakey_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_shakey" shows "Key (Auth_ShaKey n) \<notin> spied s"
insert_lclss_of_state_2: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "insert (C, Processed) (lclss_of_state (N, P, Q)) = lclss_of_state (N, P \<union> {C}, Q)"
unbounded_dense_linorder_axioms: shows "class.unbounded_dense_linorder (\<le>) (<)"
MtransT_invert2: fixes c :: "('test, 'atom) com"   and s :: "'state"   and s'' :: "'state" assumes "(c, s) \<rightarrow>*t s''" obtains c' :: "('test, 'atom) com" and s' :: "'state" where "(c, s) \<rightarrow>*c (c', s')" and "(c', s') \<rightarrow>t s''"
elims: fixes x :: "ereal"   and y :: "ereal" assumes "- x = y" obtains r :: "real" where "x = ereal r" and "y = ereal (- r)"   | "x = PInfty" and "y = MInfty"   | "x = MInfty" and "y = PInfty"
poly_deg_plus_le: fixes p :: "('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd" shows "poly_deg (p + q) \<le> max (poly_deg p) (poly_deg q)"
cases: fixes a :: "nat" assumes "sum_of_2_squares_nat a" obtains aa :: "nat" and b :: "nat" where "a = aa\<^sup>2 + b\<^sup>2"
Act_def: shows "Act \<equiv> (id ---> id ---> Rep_formula ---> Abs_formula) Act\<^sub>\<alpha>"
map_transform_fundef_cong: fixes m1 :: "(Vars.var \<times> exp) list"   and t1 :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and t2 :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae1 :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and ae2 :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and m2 :: "(Vars.var \<times> exp) list" assumes "\<And>x e a. (x, e) \<in> set m1 \<Longrightarrow> t1 a e = t2 a e"   and "ae1 = ae2"   and "m1 = m2" shows "map_transform t1 ae1 m1 = map_transform t2 ae2 m2"
in_set_conv_decomp_first: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys \<cdot> x # zs \<and> x \<notin> set ys)"
agents_eq: shows "agents = {i. R i \<noteq> (\<lambda>_ _. False)}"
strip_guards_simps_3: fixes F :: "'f set"   and r :: "('d \<times> 'd) set" shows "strip_guards F (Spec r) = Spec r"
conv_fB: shows "infsetsum'_converges fB (A \<union> B)"
assms_1: shows "isUnitClause uClause uLiteral valuation"
case_2: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d" shows "(case Point x21 x22 of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
chamber_image_and_complement_2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and D :: "'b set" assumes "ThinChamberComplexFolding X f"   and "D \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X" shows "ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X = {B \<in> ChamberComplex.chamber_system X. order.greater (ChamberComplex.chamber_distance X B (f ` D)) (ChamberComplex.chamber_distance X B D)}"
productive_imp_not_interp: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset" assumes "production N C \<noteq> {}" shows "\<not> interp N C \<Turnstile> C"
field_differentiable_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b filter" assumes "\<And>i. i \<in> I \<Longrightarrow> f i field_differentiable F" shows "(\<lambda>z. \<Sum>i\<in>I. f i z) field_differentiable F"
simps_5: fixes f :: "'a \<Rightarrow> 'a fmUni"   and \<phi> :: "'a fmUni"   and \<psi> :: "'a fmUni" shows "map_atomUni f (OrUni \<phi> \<psi>) = OrUni (map_atomUni f \<phi>) (map_atomUni f \<psi>)"
simps: fixes f :: "primfun"   and e :: "real" shows "eval_primfun (f, e) = (\<lambda>x. eval_primfun' f x powr e)"
lexordp_simps_3: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "ord.lexordp less (x # xs) (y # ys) = (less x y \<or> \<not> less y x \<and> ord.lexordp less xs ys)"
spr_abs_conv: fixes v :: "'s"   and t :: "'s Trace"   and a :: "'a" shows "(v \<in> spr_abs t) = (\<exists>t'. t' \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jview a t \<and> v = tLast t')"
inorder_insert: fixes t :: "'b tree"   and x :: "'b" assumes "True \<and> Sorted_Less.sorted (inorder t)" shows "inorder (insert x t) = ins_list x (inorder t)"
nterm_to_term: fixes t :: "nterm"   and \<Gamma> :: "name list" assumes "no_abs t"   and "fdisjnt (fset_of_list \<Gamma>) (frees t)" shows "nterm_to_term \<Gamma> t = convert_term t"
get_lower_bound_1: fixes U :: "'a set"   and V :: "'a set" assumes "U \<in> I"   and "V \<in> I" shows "get_lower_bound U V \<in> I"
ae_filter_def: fixes M :: "'c measure" shows "ae_filter M = (\<Sqinter>N\<in>null_sets M. principal (space M - N))"
one_div_two_eq_zero: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide one (numeral.numeral one plus (num.Bit0 num.One)) = zero"
encF_fmla: 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 prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv enc F encF N ssap"   and "f \<in> F" shows "encF f \<in> fmla"
that: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "abc = [a, b, c]\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "c \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "thesis"
i_set_mult_imp_not_empty: fixes I :: "nat set"   and k :: "nat" assumes "I \<in> i_set_mult k" shows "I \<noteq> {}"
Ri_mono: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and N :: "'b literal multiset set"   and N' :: "'b literal multiset set" assumes "redundancy_criterion \<Gamma> Rf Ri"   and "N \<subseteq> N'" shows "Ri N \<subseteq> Ri N'"
th: fixes a :: "'a fps"   and b :: "'a fps" shows "(dist a b = 0) = (a = b)"
contra: shows "(\<forall>v\<in>kon_graph.V. even (degree v kon_graph)) \<or> num_of_odd_nodes kon_graph = 2"
le_lin: fixes p :: "tm" assumes "tmbound0 p" shows "islin (le p)"
add_Pair: fixes a :: "'a"   and b :: "'b"   and c :: "'a"   and d :: "'b" shows "(a, b) + (c, d) = (a + c, b + d)"
mod_poly_one_main_list: fixes q :: "'b list"   and r :: "'b list"   and d :: "'b list"   and n :: "nat" shows "snd (divmod_poly_one_main_list q r d n) = mod_poly_one_main_list r d n"
sup_bot_right: fixes x :: "'a upred" shows "(x \<or> false) = x"
vertex_2_list_def: shows "vertex_2 \<equiv> [vertex_1]"
up_to_def: fixes J :: "nat set" shows "up_to J = (LEAST n. \<forall>i\<ge>n. i \<notin> J)"
matrix_transpose_mul: fixes A :: "(('b, 'e) vec, 'c) vec"   and B :: "(('b, 'd) vec, 'e) vec" shows "Finite_Cartesian_Product.transpose (A ** B) = Finite_Cartesian_Product.transpose B ** Finite_Cartesian_Product.transpose A"
ns_mul_ext_bottom_uniqueness: fixes A :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "({#}, A) \<in> ns_mul_ext ns s" shows "A = {#}"
Un: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> null_sets M"   and "b \<in> null_sets M" shows "a \<union> b \<in> null_sets M"
cs_inj: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "\<pi> n \<noteq> IFC_def.return prob"   and "IFC_def.cs prob \<pi> n = IFC_def.cs prob \<pi> n'" shows "n = n'"
h'_outside: fixes e :: "'v \<times> 'v" assumes "e \<notin> \<^bold>E" shows "h' e = 0"