hcju/isabelleps huggingface dataset:
CondRed: fixes 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 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"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow> \<langle>if (e') e\<^sub>1 else e\<^sub>2,s'\<rangle>"
powr_add: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "x powr (a + b) = x powr a * x powr b"
measure_eqI_generator_eq: fixes E :: "'d set set"   and \<Omega> :: "'d set"   and M :: "'d measure"   and N :: "'d measure"   and A :: "nat \<Rightarrow> 'd set" assumes "Int_stable E"   and "E \<subseteq> Pow \<Omega>"   and "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"   and "sets M = sigma_sets \<Omega> E"   and "sets N = sigma_sets \<Omega> E"   and "range A \<subseteq> E"   and "\<Union> (range A) = \<Omega>"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "M = N"
sinks_aux_single_dom: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and xs :: "'f list" shows "sinks_aux I D {u} xs = insert u (sinks I D u xs)"
h5: shows "invertible_mat (get_matrix (reduce_system p (qs, matrix, subsets, signs)))"
map_spmf_bind_spmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'e spmf"   and g :: "'e \<Rightarrow> 'd spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
hd_singl: fixes a :: "'a" shows "hd (singl a) = a"
yy: 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" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst" shows "Generic_Syntax.Variable var 11 \<in> var"
not_disj: shows "verts G \<inter> verts H \<noteq> {}"
dist_triangle_lt: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and z :: "'b"   and y :: "'b"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist x z + dist y z < e" shows "dist x y < e"
pushout_step_def: fixes t :: "'e itself"   and R :: "('f, 'g) labeled_graph \<times> ('f, 'g) labeled_graph"   and G\<^sub>1 :: "('f, 'h) labeled_graph"   and G\<^sub>2 :: "('f, 'h) labeled_graph" shows "pushout_step t R G\<^sub>1 G\<^sub>2 \<equiv> subgraph G\<^sub>1 G\<^sub>2 \<and> (\<exists>f\<^sub>1 f\<^sub>2. graph_homomorphism (fst R) G\<^sub>1 f\<^sub>1 \<and> graph_homomorphism (snd R) G\<^sub>2 f\<^sub>2 \<and> f\<^sub>1 \<subseteq> f\<^sub>2 \<and> weak_universal t R G\<^sub>1 G\<^sub>2 f\<^sub>1 f\<^sub>2)"
iT_Div_Max_card: fixes d :: "nat"   and I :: "nat set" assumes "0 < d"   and "finite I" shows "card (I \<oslash> d) = (\<Sum>k\<le>Max I div d. if I \<inter> [k * d\<dots>,d - Suc 0] = {} then 0 else Suc 0)"
nn_integral_pmf_of_set: fixes S :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "S \<noteq> {}"   and "finite S" shows "integral\<^sup>N (measure_pmf (pmf_of_set S)) f = sum f S / of_nat (card S)"
C1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<Psi> \<rhd> Q \<sim> R" shows "(\<Psi>, P, R) \<in> Rel'"
cases: fixes a :: "Regions.intv" assumes "Regions.isGreater a" obtains uu_ :: "nat" where "a = Regions.intv.Greater uu_"
le_less: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.order less_eq less" shows "less_eq x y = (less x y \<or> x = y)"
has_dens: fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (state_measure (V \<union> V') \<Gamma>)" shows "has_subprob_density (M \<rho>) (stock_measure t) (f \<rho>)"
hyps_2: fixes \<P> :: "('a list \<times> nat) multiset \<Rightarrow> 'a" assumes "\<P> \<in> carrier M" shows "carrier_coeff \<P>"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
eq: shows "x = 2 * u' * v' \<and> y = u'\<^sup>2 - v'\<^sup>2 \<or> x = u'\<^sup>2 - v'\<^sup>2 \<and> y = 2 * u' * v'"
step_Normal_elim_cases_10: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> (Throw, Normal s) \<rightarrow> u" shows "P"
preserves_reflects_arr: fixes f :: "'a" shows "D.C.arr (D'.cones_map f \<tau>o\<chi>.map f) = D.J.arr f"
unity_periodic_arithmetic_mult: fixes k :: "nat"   and m :: "int" shows "periodic_arithmetic (\<lambda>n. unity_root k (m * int n)) k"
canE_unitor_4: fixes f :: "'a E.term" assumes "E.Ide f" shows "B.can (E.Trg f \<^bold>\<star> f) f = B.lunit' (EVAL f)"
antisym_conv2: fixes x :: "'c word"   and y :: "'c word" assumes "x \<le>s y" shows "(\<not> x <s y) = (x = y)"
subseqs_of_subseq: fixes ys :: "'c list"   and xs :: "'c list" assumes "ys \<in> set (subseqs xs)" shows "set (subseqs ys) \<subseteq> set (subseqs xs)"
integrable: shows "AE v in stock_measure (\<Gamma> x). integrable (state_measure (set (filter (\<lambda>y. y \<noteq> x) vs)) \<Gamma>) (\<lambda>\<sigma>. extract_real (cexpr_sem (merge (set (filter (\<lambda>y. y \<noteq> x) vs)) (insert x (set vs')) (\<sigma>, \<rho>(x := v))) \<delta>))"
leg1_is_map: shows "is_left_adjoint (tab\<^sub>1 r)"
Torder_adjunct_ord: fixes a :: "'a" assumes "a \<notin> carrier D" shows "Torder (adjunct_ord D a)"
transfer_bool: fixes p :: "bool" shows "p \<equiv> \<forall>\<^sub>F n in \<U>. p"
separatedin_empty_1: fixes X :: "'c topology"   and S :: "'c set" shows "separatedin X S {} = (S \<subseteq> topspace X)"
bisim1_refl: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list" shows "P,e,h \<turnstile> (e, xs) \<leftrightarrow> ([], xs, 0, None)"
monad_contT': shows "monad Monad_Overloading.return Monad_Overloading.bind"
tendsto_add: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
card_lists_distinct_length_eq: fixes A :: "'g set"   and k :: "nat" assumes "finite A"   and "k \<le> card A" shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1..card A}"
LINEAR_def: fixes b :: "'g"   and r :: "('f + 'g + 'h, 'i) Old_Datatype.node set" shows "LINEAR b r = In1 (In0 (Scons (Old_Datatype.Leaf (Inr (Inl b))) r))"
matrix_vector_mul_component: fixes A :: "((real, 'b) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and x :: "(real, 'b) Finite_Cartesian_Product.vec"   and k :: "'a" shows "(A *v x) $h k = A $h k \<bullet> x"
addr_in_ipset_from_cidr_code: fixes addr :: "'a word"   and pre :: "'a word"   and len :: "nat" shows "(addr \<in> ipset_from_cidr pre len) = (pre && (mask len << LENGTH('a) - len) \<le> addr \<and> addr \<le> pre || mask (LENGTH('a) - len))"
pmat_extension_def: fixes m :: "'b mat" shows "d.pmat_extension m = d.ptensor_mat m (1\<^sub>m d.d2)"
s: shows "supp_fun s \<subseteq> supp_fun s \<union> supp_fun t"
state_io_rel_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('c, 'd) state" shows "state_io_rel P m = (\<forall>s. P s (snd (run_state m s)))"
indets_def: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "indets p = \<Union> (keys ` keys p)"
mult_lres_sub_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * (y / z) \<le> x * y / z"
simps_9: fixes v :: "poly"   and vc :: "nat"   and vb :: "poly" shows "head (CN v (Suc vc) vb) = CN v (Suc vc) vb"
tpart1_Un_tpart2: fixes V :: "Preliminaries.var set" assumes "tinfinite V" shows "tpart1 V \<union> tpart2 V = V"
distinct_11: fixes x31 :: "bexp"   and x32 :: "bexp"   and x41 :: "aexp"   and x42 :: "aexp" shows "And x31 x32 \<noteq> Less x41 x42"
log2_def: shows "log2 = (\<lambda>x. LEAST z. x < 2 ^ (z + 1))"
intros_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e" shows "\<Gamma>\<turnstile> \<langle>Basic f,Normal s\<rangle> \<Rightarrow> Normal (f s)"
maximal_independent_subset: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "vector_space scale" obtains B :: "'b set" where "B \<subseteq> V" and "\<not> module.dependent scale B" and "V \<subseteq> module.span scale B"
solution_solves_ode: shows "(solution solves_ode f) T X"
prj_simps_10: shows "C.arr \<mu>\<nu>\<pi>.Prj\<^sub>1"
simple_closed_path_wn1: fixes e :: "real"   and p :: "real \<Rightarrow> complex"   and a :: "complex" assumes "0 < e"   and "simple_path (p +++ linepath (a - complex_of_real e) (a + complex_of_real e))"   and "pathstart p = a + complex_of_real e"   and "pathfinish p = a - complex_of_real e"   and "ball a e \<inter> path_image p = {}" obtains z :: "complex" where "z \<in> inside (path_image (p +++ linepath (a - complex_of_real e) (a + complex_of_real e)))" and "cmod (winding_number (p +++ linepath (a - complex_of_real e) (a + complex_of_real e)) z) = 1"
natural_transformation_def: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme"   and F :: "('b, 'c, 'e, 'f) functor"   and G :: "('b, 'c, 'e, 'f) functor"   and u :: "'b \<Rightarrow> 'f" shows "u : F \<Rightarrow> G in Func (AA , BB ) \<equiv> two_cats AA BB \<and> natural_transformation_axioms AA BB F G u"
assms_4: shows "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"
invariant_arbitraryI: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool" assumes "\<And>s. s \<in> reachable A I \<Longrightarrow> P s" shows "A \<TTurnstile> (I \<rightarrow>) P"
x: shows "\<not> is_zero_iarray (Gauss_Jordan_iarrays (matrix_to_iarray A) !! mod_type_class.to_nat x)"
ctr_transfer_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(R ===> list_all2 R ===> list_all2 R) (#) (#)"
language_def: shows "ST.language \<equiv> {xs. ST.nextl (nfa.init \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr>) xs \<inter> nfa.final \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr> \<noteq> {}}"
green_typeI_cube_axioms_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real" shows "green_typeI_cube_axioms i j twoC F \<equiv> typeI_twoCube twoC \<and> valid_two_cube twoC \<and> analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j"
mem\<^sub>2'_def: shows "mem\<^sub>2' = subst (to_partial mem\<^sub>2 |` differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k) mem\<^sub>h"
rk2_fas_err_nth_def: fixes rkp :: "floatarith"   and x0 :: "floatarith list"   and h :: "floatarith"   and cx :: "floatarith list"   and s2 :: "floatarith"   and i :: "nat" shows "rk2_fas_err_nth rkp x0 h cx s2 i = h ^\<^sub>e 3 / R\<^sub>e 6 * (ode_d_fa_nth 1 cx (ode_fa cx) i + ode_d_fa_nth 0 cx (ode_d_fa 0 cx (ode_fa cx)) i) - h ^\<^sub>e 3 * rkp / R\<^sub>e 4 * ode_d_fa_nth 1 (euler_incr_fas x0 (s2 * h * rkp) x0) (ode_fa x0) i"
le_imp_diff_is_add: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "a \<le> b" shows "(b - a = c) = (b = c + a)"
id_eq_prop_prop_9_b: fixes v :: "i"   and p :: "\<o>"   and q :: "\<o>"   and r :: "\<o>" shows "[p \<^bold>= q \<^bold>& q \<^bold>= r \<^bold>\<rightarrow> p \<^bold>= r in v]"
\<Z>_\<alpha>_\<alpha>\<omega>: fixes \<alpha> :: "V" assumes "\<Z> \<alpha>" shows "\<alpha> \<in>\<^sub>\<circ> \<alpha> + \<omega>"
ensures_def: fixes A :: "'c set"   and B :: "'c set" shows "A ensures B \<equiv> (A - B co A \<union> B) \<inter> transient (A - B)"
not_disj_deMorgans: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not> Q)"
Append_preserves_finite: fixes rhs :: "'b trm set"   and r :: "'b rexp" assumes "finite rhs" shows "finite (Append_rexp_rhs rhs r)"
hm_iterate_impl: shows "imp_map_iterate is_hashmap hm_is_it hm_it_init hm_it_has_next hm_it_next"
simps_5: fixes f1 :: "'b"   and f2 :: "'d \<Rightarrow> 'b"   and x2 :: "'d" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
smalleraux2b: fixes a :: "'b"   and b :: "'b"   and l :: "'b set list"   and c :: "'b"   and d :: "'b"   and y :: "('b, 'c) Combinators"   and p :: "'c" assumes "{a, b} \<in> set l"   and "{c, d} \<in> set l"   and "{a, b} \<noteq> {c, d}"   and "y = DenyAllFromTo a b"   and "smaller (AllowPortFromTo c d p) y l" shows "\<not> smaller y (AllowPortFromTo c d p) l"
zmset_of_def: shows "zmset_of \<equiv> map_fun count abs_zmultiset (\<lambda>f. (Abs_multiset f, {#}))"
nonneg_weights: fixes v :: "'V"   and w :: "'W"   and v' :: "'V" assumes "(v, w, v') \<in> E" shows "(0::'W) \<le> w"
max: fixes C :: "'a set" assumes "chain_on P C {x \<in> A. \<not> Q x}"   and "M \<subseteq> C" shows "M = C"
assms_2: shows "s \<rightarrow>\<^sup>* v"
row_vec_mat_Tensor_prelim: fixes v :: "'a list"   and M :: "'a list list"   and nr :: "nat" shows "\<forall>i. i < length v * row_length M \<and> mat nr (length M) M \<longrightarrow> row (vec_mat_Tensor v M) i = local.times (v ! (i div row_length M)) (row M (i mod row_length M))"
assms_1: shows " t \<longrightarrow>cbv* v"
elims: fixes x :: "'a list"   and y :: "'a list" assumes "remdups_adj x = y" obtains "xa__ = []" and "ya__ = []"   | x :: "'a" where "xa__ = [xb__]" and "ya__ = [xb__]"   | x :: "'a" and y :: "'a" and xs :: "'a list" where "xa__ = x # y # xs" and "ya__ = (if x = y then remdups_adj (x # xs) else x # remdups_adj (y # xs))"
tensor_mat_positive: fixes m1 :: "complex mat"   and m2 :: "complex mat" assumes "m1 \<in> carrier_mat ps_P.d1 ps_P.d1"   and "m2 \<in> carrier_mat ps_P.d2 ps_P.d2"   and "positive m1"   and "positive m2" shows "positive (ps_P.tensor_mat m1 m2)"
pigeonhole: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and k :: "nat" assumes "finite A"   and "B \<subseteq> A"   and "C \<subseteq> A"   and "card A + k \<le> card B + card C" shows "k \<le> card (B \<inter> C)"
card_orbit_eq: fixes f :: "'a perm"   and a :: "'a" shows "card (orbit f a) = order f a"
rev_conv_fold: fixes xs :: "'k list" shows "rev xs = fold (#) xs []"
assms_19: shows "InvariantClCurrentLevel (getCl state) (getM state)"
Smith_normal_form_uniqueness_aux: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and P :: "'b mat"   and Q :: "'b mat"   and B :: "'b mat"   and P' :: "'b mat"   and Q' :: "'b mat"   and B' :: "'b mat"   and k :: "nat" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "Q \<in> carrier_mat n n"   and "invertible_mat P"   and "invertible_mat Q"   and "P * A * Q = B"   and "Smith_normal_form_mat B"   and "P' \<in> carrier_mat m m"   and "Q' \<in> carrier_mat n n"   and "invertible_mat P'"   and "invertible_mat Q'"   and "P' * A * Q' = B'"   and "Smith_normal_form_mat B'"   and "k < min m n" shows "\<forall>i\<le>k. B $$ (i, i) dvd B' $$ (i, i) \<and> B' $$ (i, i) dvd B $$ (i, i)"
residual_is_proper: assumes "multiplicity bl = 1" shows "proper_design (bl\<^sup>c) \<B>\<^sup>R"
supp_h_diff_edges: fixes i :: "nat" shows "support_flow (h_diff i) \<subseteq> \<^bold>E"
in_homI: fixes f :: "'a2"   and a :: "'a2"   and b :: "'a2" assumes "C2.arr f"   and "C2.dom f = a"   and "C2.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>2 b\<guillemotright>"
add_mult_distrib2: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "k * (m + n) = k * m + k * n"
simps_3: shows "sch_to_nat Base_fst = c_pair 2 0"
has_bochner_integral_integral_eq: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "has_bochner_integral M f x" shows "integral\<^sup>L M f = x"
cring_simprules_2: fixes R :: "('a, 'b) ring_scheme" assumes "cring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
meet_compl_bot: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<^bold>- x \<sqinter> x = \<bottom>"
hfun_arr_to_hfun: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "hfun (ide_to_hf (local.dom f)) (ide_to_hf (cod f)) (arr_to_hfun f)"
assms_2: shows "ui_edge ` out_edges (red prb') (src re) \<subset> out_edges (black prb) (fst (src re))"
right_idem: fixes a :: "'b"   and b :: "'b" shows "sup (sup (a::'b) (b::'b)) b = sup a b"
element_exists: fixes A :: "'a set" assumes "A \<in> Partition" shows "\<exists>a\<in>G. a \<in> A"
insertion_add_ind_basecase: fixes L :: "'a list"   and var :: "nat"   and x :: "'a"   and p :: "'a mpoly" shows "insertion ((!) (L[var := x])) (\<Sum>i\<le>0. isolate_variable_sparse p var i * Var var ^ i) = (\<Sum>i = 0..<0 + 1. insertion ((!) (L[var := x])) (isolate_variable_sparse p var i * Var var ^ i))"
ts_tuple_rel_filter: fixes P :: "nat \<Rightarrow> bool"   and xs :: "(nat \<times> 'a option list set) list" shows "ts_tuple_rel (set (filter (\<lambda>(t, X). P t) xs)) = {(t, X). (t, X) \<in> ts_tuple_rel (set xs) \<and> P t}"
distinct_1: fixes x1 :: "'d"   and x21 :: "'e"   and x22 :: "'f" shows "Pure x1 \<noteq> IO x21 x22"
mkM_simps_1: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'e Trace set" assumes "PreEnvironmentJView jkbp jview" shows "worlds (PreEnvironmentJView.mkM envVal jview T) = T"
constrains_imp_Constrains: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> A co A'" shows "F \<in> A Co A'"
subst_subterm: fixes f :: "'a"   and T :: "('a, 'b) Term.term list"   and t :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "Fun f T \<sqsubseteq> t \<cdot> \<theta>" shows "(\<exists>S. Fun f S \<sqsubseteq> t \<and> Fun f S \<cdot> \<theta> = Fun f T) \<or> (\<exists>s\<in>subst_range \<theta>. Fun f T \<sqsubseteq> s)"
chain_length_pos: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a llist" assumes "chain R xs" shows "0 < llength xs"
eta_star_preserves_typ_of1: fixes r :: "Core.term"   and s :: "Core.term"   and Ts :: "typ list"   and T :: "typ" assumes "r \<rightarrow>\<^sub>\<eta>\<^sup>* s"   and "typ_of1 Ts r = Some T" shows "typ_of1 Ts s = Some T"
SconsI: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" assumes "M \<in> sexp"   and "N \<in> sexp" shows "Scons M N \<in> sexp"
onorm_def: fixes f :: "'a \<Rightarrow> 'b" shows "onorm f = (SUP x. norm (f x) / norm x)"
size_mset_tree: fixes t :: "'b tree" shows "size (mset_tree t) = size t"
nat_filtration_null_set: fixes A :: "bool stream set"   and n :: "nat" assumes "A \<in> sets (nat_filtration n)"   and "0 < p"   and "p < 1"   and "emeasure M A = 0" shows "A = {}"
simps: fixes xs :: "('a \<times> 'b) list"   and ko :: "'a nat_term_order" shows "sort_oalist_ko_ntm (xs, ko) = (sort_oalist_ko_ntm' ko xs, ko)"
H_iso_loop: fixes f :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b"   and q :: "'b" assumes "mono f"   and "p \<sqinter> x \<le> f x" shows "x \<le> (fiter_id (\<lfloor>p\<rfloor> \<circ> f) \<circ> \<lfloor>q\<rfloor>) (x \<sqinter> q)"
sincreasing_sdrop: fixes s :: "'b stream"   and n :: "nat" assumes "sincreasing s" shows "sincreasing (sdrop n s)"
ereal_mult_infty: fixes a :: "ereal" shows "a * \<infinity> = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
mult_mm_sm: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and k :: "complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "A *\<^sub>m\<^sub>m (k *\<^sub>s\<^sub>m B) = k *\<^sub>s\<^sub>m (A *\<^sub>m\<^sub>m B)"
assms: shows "x \<in>\<^sub>\<circ> vrange A"
hyps: shows "R \<longmapsto> \<alpha> \<prec> R'"
sgn_eq_0_iff: fixes a :: "'b" shows "(sgn a = (0::'b)) = (a = (0::'b))"
comm_semiring_ow: shows "class.comm_semiring = comm_semiring_ow UNIV"
rep_eq: fixes x :: "('a, 'b) pp" shows "lookup_pp x = lookup (pp.mapping_of x)"
compl_sup: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (sup x y) = inf (uminus x) (uminus y)"
mono: fixes x :: "'a" shows "spmf.mono_body (\<lambda>while. if x \<notin> B then spmf_of_set A \<bind> while else return_spmf x)"
upset_to_downset3: fixes dual :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.ord_with_dual dual less_eq" shows "ord.upset less_eq x = dual ` ord.downset less_eq (dual x)"
card_finite_spectrum_1: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "finite (spectrum A)"
bool_to_ternary_simp4: fixes X :: "bool" shows "(eval_ternary_Not (bool_to_ternary X) = TernaryFalse) = X"
AVar: fixes G :: "prog"   and s0 :: "st"   and e1 :: "expr"   and n :: "nat"   and a :: "val"   and s1 :: "state"   and e2 :: "expr"   and i :: "val"   and s2 :: "state"   and v :: "vvar"   and s2' :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>a\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e2-\<succ>i\<midarrow>n\<rightarrow> s2"   and "(v, s2') = avar G i a s2" shows "G\<turnstile>Norm s0 \<midarrow>e1.[e2]=\<succ>v\<midarrow>n\<rightarrow> s2'"
cart_prod_singleton_right_gen: fixes A :: "'a multiset"   and b :: "'b"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in># A \<times># {#b#} \<Longrightarrow> P x = Q (fst x)" shows "filter_mset P (A \<times># {#b#}) = filter_mset Q A \<times># {#b#}"
axioms_2: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and b :: "'ah" assumes "constant_functor A B b" shows "category B"
restrict_restrict: fixes m :: "'b \<Rightarrow> 'c option"   and A :: "'b set"   and B :: "'b set" shows "m |` A |` B = m |` (A \<inter> B)"
ideal_ofI: fixes A :: "('c \<Rightarrow> 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "\<And>a. a \<in> A \<Longrightarrow> poly_eval a f = (0::'d)" shows "f \<in> \<I> A"
simps_1: fixes G :: "'a graph" shows "delete_edges_list G [] = G"