hcju/isabelleps huggingface dataset:
sup_assertion: fixes one :: "'c"   and times :: "'c \<Rightarrow> '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 dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "y \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "sup x y \<in> mbt_algebra.assertion one times inf less_eq dual top"
ground_ord_resolve_ground: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and E :: "'a literal multiset" assumes "selection S"   and "ground_resolution_with_selection.ord_resolve S CAs DA AAs As E"   and "is_ground_cls_list CAs"   and "is_ground_cls DA" shows "is_ground_cls E"
card_le_Suc_iff: fixes n :: "nat"   and A :: "'a set" shows "(Suc n \<le> card A) = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
simps_2: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "typeof\<^bsub>h\<^esub> Null = \<lfloor>NT\<rfloor>"
fps_mult_nth: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "(f * g) $ n = (\<Sum>i = 0..n. f $ i * g $ (n - i))"
eq: shows "fds_deriv (fds moebius_mu) = fds moebius_mu * fds mangoldt"
vector_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and i :: "'c"   and j :: "'c"   and k :: "'c" assumes "matrix_bounded_idempotent_semiring.vector f" shows "f (i, j) = f (i, k)"
fcs_proj_arcs_GH: fixes a :: "'b" assumes "a \<in> arcs H" shows "H.face_cycle_set (proj_arcs_G (proj_arcs_H a)) = H.face_cycle_set a"
Rep_le_Cancel: fixes x :: "hf"   and y :: "hf"   and z :: "hf" assumes "x \<squnion> RepFun y ((@+) x) \<le> x \<squnion> RepFun z ((@+) x)" shows "RepFun y ((@+) x) \<le> RepFun z ((@+) x)"
simps_3: fixes x1 :: "'b"   and x21 :: "'b"   and x22 :: "'b" shows "Vertex x1 \<noteq> Edge x21 x22"
index_update_vec2: fixes i' :: "nat"   and i :: "nat"   and v :: "'c vec"   and a :: "'c" assumes "i' \<noteq> i" shows "(v |\<^sub>v i \<mapsto> a) $ i' = v $ i'"
simps_2: fixes d :: "nat"   and l :: "nat"   and rA :: "'a \<times> 'a mat" shows "local.sub3 d (Suc l) rA = local.sub2 d l (local.sub3 d l rA)"
simps_3: fixes x :: "int"   and y :: "int" shows "Num_arc x * Num_arc y = Num_arc (x + y)"
converse_tranclpE: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and z :: "'d" assumes "r\<^sup>+\<^sup>+ x z" obtains "r x z"   | y :: "'d" where "r x y" and "r\<^sup>+\<^sup>+ y z"
satisfies_properties_sys1: shows "satisfies_properties p qs1 (get_subsets (calculate_data p qs1)) (get_signs (calculate_data p qs1)) (get_matrix (calculate_data p qs1))"
IN_plus_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g" shows "d_IN (plus_current f g) x = (if x \<in> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) then d_IN f x else d_IN g x)"
simulation_refl: fixes G :: "('h, 'i) graph_rec_scheme" assumes "graph G" shows "simulation Id G G"
UNIV_eq_I: fixes A :: "'e set" assumes "\<And>x. x \<in> A" shows "UNIV = A"
sg_inc_set_mult: fixes G :: "('e, 'f) Group_scheme"   and L :: "'e set"   and H :: "'e set"   and K :: "'e set" assumes "Group G"   and "G \<guillemotright> L "   and "H \<subseteq> L"   and "K \<subseteq> L" shows "H \<diamondop>\<^bsub>G\<^esub> K \<subseteq> L"
cmp_simps_4: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.dom (\<Phi>\<^sub>H (f, g)) = H f \<star>\<^sub>D H g"
real_of_nat_div4: fixes n :: "nat"   and x :: "nat" shows "real (n div x) \<le> real n / real x"
degree_m_le_degree: fixes f :: "int poly" shows "pl.degree_m f \<le> degree f"
find_succ: fixes m' :: "('a, 'b) insert_message" assumes "m' \<in> insert_messages M \<union> {m}" shows "S m' \<in> set (ext_ids s)"
K_ImpI: fixes p :: "'b fm"   and G :: "'b fm list"   and q :: "'b fm" assumes "\<turnstile> imply (p # G) q" shows "\<turnstile> imply G (p \<^bold>\<longrightarrow> q)"
is_null_anga_out: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "a A B C"   and "QCongANullAcute a" shows "B Out A C"
z_iff: shows "\<forall>x. poly f x = 0 \<longrightarrow> poly (prod_list fss) x = 0"
RInitInitThrow: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh' :: "sheap"   and a :: "nat"   and D :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, i)\<rfloor>"   and "sh' = sh(C \<mapsto> (sfs, Error))" shows "P \<turnstile> \<langle>RI (C,Throw a) ; D # Cs \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>RI (D,Throw a) ; Cs \<leftarrow> e,(h, l, sh'),b\<rangle>"
combine_inv2_1: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt" assumes "inv2 lt"   and "inv2 rt"   and "bheight lt = bheight rt" shows "bheight (combine lt rt) = bheight lt"
subcls_compareable: fixes A :: "qtname"   and X :: "qtname"   and G :: "prog"   and Y :: "qtname" assumes "G\<turnstile>A\<preceq>\<^sub>C X"   and "G\<turnstile>A\<preceq>\<^sub>C Y" shows "G\<turnstile>X\<preceq>\<^sub>C Y \<or> G\<turnstile>Y\<preceq>\<^sub>C X"
rb0: shows "r k (b $ 0) ^ k = b $ 0"
h1: shows "vars (liftPoly i j (MPoly_Type.monom m a)) = keys (lowerPowers i j m)"
lower_inv_eq: fixes x :: "'a" assumes "x \<in> carrier \<X>\<^bsub>G\<^esub>" shows "\<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> x) = x"
Utrans_P_def: fixes vars :: "nat set"   and U :: "complex mat" shows "ps_P.Utrans_P vars U = Utrans (mat_extension ps2_P.dims0 vars U)"
matrix_to_iarray_snd_echelon_form_of_upt_k_det: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" assumes "k < ncols A" shows "matrix_to_iarray (snd (echelon_form_of_upt_k_det bezout (1::'c, A) k)) = snd (echelon_form_of_upt_k_det_iarrays (1::'c, matrix_to_iarray A) k bezout)"
cases: fixes a1 :: "('g, 'h) locks \<times> (('h \<Rightarrow> ('x \<times> 'g \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'j wait_set_status option) \<times> 'h set"   and a2 :: "'h"   and a3 :: "('g, 'h, 'k, 'i, 'j, 'l) thread_action" assumes "actions_ok' a1 a2 a3" obtains s :: "('g, 'h) locks \<times> (('h \<Rightarrow> ('x \<times> 'g \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'j wait_set_status option) \<times> 'h set" and t :: "'h" and ta :: "('g, 'h, 'k, 'i, 'j, 'l) thread_action" where "a1 = s" and "a2 = t" and "a3 = ta" and "lock_ok_las' (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>" and "thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" and "cond_action_oks' s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>" and "wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>" and "interrupt_actions_ok' (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"
sequences_ne: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "sequences key xs \<noteq> []"
to_effect_def: fixes op :: "'a strips_operator" shows "to_effect op = map (\<lambda>v\<^sub>a. (v\<^sub>a, True)) (add_effects_of op) @ map (\<lambda>v\<^sub>d. (v\<^sub>d, False)) (delete_effects_of op)"
simps: fixes f :: "heap \<Rightarrow> ('d \<times> heap) option" shows "execute (Heap f) = f"
nuc_comp_prop: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "nucleus f" shows "f (f x \<cdot> f y) = f (x \<cdot> y)"
assert_bound_tableau_valuated: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<nabla> (assert_bound a s)"
star_unfold': fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
\<F>\<G>_subformulas\<^sub>\<nu>: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<F>\<G> \<phi> w \<subseteq> subformulas\<^sub>\<nu> \<phi>"
rel_G_eq: shows "rel_G (=) (=) (=) (=) (=) (=) = (=)"
r'_def: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" shows "r' x = (MIN j. (A *v x) $h j / x $h j)"
mult_cancel_left2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times c a = c) = (c = zero \<or> a = one)"
ESem_Let: fixes Fn :: "('a \<rightarrow> 'a) \<rightarrow> 'a"   and Fn_project :: "'a \<rightarrow> 'a \<rightarrow> 'a"   and B :: "bool discr \<rightarrow> 'a"   and B_project :: "'a \<rightarrow> 'a \<rightarrow> 'a \<rightarrow> 'a"   and tick :: "'a \<rightarrow> 'a"   and as :: "(Vars.var \<times> exp) list"   and body :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> 'a" shows "semantic_domain.ESem Fn Fn_project B B_project tick (Terms.Let as body)\<cdot>\<rho> = tick\<cdot>(semantic_domain.ESem Fn Fn_project B B_project tick body\<cdot>(has_ESem.HSem (semantic_domain.ESem Fn Fn_project B B_project tick) as\<cdot>\<rho>))"
swap_parent_eq_on: fixes arr1 :: "int \<Rightarrow> 'a"   and arr2 :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int"   and c :: "int" assumes "arr1 = arr2 on - {l..<r}"   and "l < c"   and "c < r" shows "arr1 = swap arr2 (parent l c) c on - {l..<r}"
ii_not_nonneg_Reals: shows "\<i> \<notin> \<real>\<^sub>\<ge>\<^sub>0"
interior_mono: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T" shows "interior S \<subseteq> interior T"
fmpred_of_list: fixes xs :: "('c \<times> 'd) list"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>k v. (k, v) \<in> set xs \<Longrightarrow> P k v" shows "fmpred P (fmap_of_list xs)"
IH: fixes T :: "tree" assumes "internal p T" shows "treesize (delete p T) < treesize T"
add_sign_intros_3: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "{#} \<subseteq># a"   and "{#} \<subset># b" shows "{#} \<subset># a + b"
simps_1: fixes h :: "heap" shows "typeof\<^bsub>h\<^esub> Unit = \<lfloor>Void\<rfloor>"
new_tv_le: fixes n :: "nat"   and m :: "nat"   and t :: "'a" assumes "n \<le> m"   and "new_tv n t" shows "new_tv m t"
sum_complement_is_complements_product: fixes x :: "i"   and y :: "i" assumes "x \<oplus> y \<noteq> u" shows "\<midarrow> (x \<oplus> y) = \<midarrow> x \<otimes> \<midarrow> y"
nn_integral_PInf_AE: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "f \<in> borel_measurable M"   and "integral\<^sup>N M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
defNode_code: fixes g :: "'g"   and v :: "'val" shows "defNode g v \<equiv> hd (filter (\<lambda>n. v \<in> allDefs g n) (\<alpha>n g))"
eval_lpoly_eq_dot_prod_split2: fixes x :: "rat vec"   and y :: "rat vec"   and b :: "'a vec"   and c :: "rat vec"   and X :: "(nat, rat) mapping" assumes "(x, y) = split_n_m_x (dim_vec b) (dim_vec c) X" shows "vec_to_lpoly (0\<^sub>v (dim_vec b) @\<^sub>v c) \<lbrace> \<langle>X\<rangle> \<rbrace> = c \<bullet> y"
sub_lists_mono: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b list" assumes "A \<subseteq> B"   and "x \<in> lists A" shows "x \<in> lists B"
OAlist_list_of_oalist: fixes xs :: "('a, 'b) oalist" shows "OAlist (list_of_oalist xs) = xs"
rep_eq: fixes x :: "'d \<Rightarrow> 'c"   and xa :: "'d" shows "un_B (pure_dual x xa) = x xa"
jonsson3c: fixes x :: "'a" assumes "\<forall>x. x \<rhd> (1::'a) \<rhd> (1::'a) = x" shows "x \<rhd> (1::'a) = (1::'a) \<lhd> x"
code: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "siterate f x = x ## siterate f (f x)"
star_decom: fixes x :: "'a list"   and A :: "'a list set" assumes "x \<in> star A"   and "x \<noteq> []" shows "\<exists>a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> A \<and> b \<in> star A"
sabove_halfspace_def: fixes sctn :: "'e sctn" shows "sabove_halfspace sctn = Collect (gt_halfspace sctn)"
cmiovi: shows "m\<inverse> O ov\<inverse> \<subseteq> b\<inverse>"
det_def: fixes P :: "'c process"   and Q :: "'c process" shows "P [+] Q \<equiv> Abs_process ({(s, X). s = [] \<and> (s, X) \<in> F P \<inter> F Q} \<union> {(s, X). s \<noteq> [] \<and> (s, X) \<in> F P \<union> F Q} \<union> {(s, X). s = [] \<and> s \<in> D P \<union> D Q} \<union> {(s, X). s = [] \<and> tick \<notin> X \<and> [tick] \<in> T P \<union> T Q}, D P \<union> D Q)"
distr_uniform_measure_count_space_inj: fixes f :: "'b \<Rightarrow> 'c"   and A' :: "'b set"   and A :: "'b set"   and B :: "'c set" assumes "inj_on f A'"   and "A' \<subseteq> A"   and "f ` A \<subseteq> B"   and "finite A'" shows "distr (uniform_measure (count_space A) A') (count_space B) f = uniform_measure (count_space B) (f ` A')"
path_conc: fixes E :: "('e \<times> 'e) set"   and u :: "'e"   and la :: "'e list"   and v :: "'e"   and lb :: "'e list"   and w :: "'e" assumes "path E u la v"   and "path E v lb w" shows "path E u (la @ lb) w"
rel_option_None2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b option" shows "rel_option P x None = (x = None)"
fcrb_def: fixes \<Gamma> :: "'e Gamma"   and ur :: "('s \<times> 's) set" shows "fcrb \<Gamma> ur \<equiv> \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<forall>s\<in>S\<^bsub>SES\<^esub>. \<forall>s''\<in>S\<^bsub>SES\<^esub>. reachable SES s \<and> s v\<longrightarrow>\<^bsub>SES\<^esub> s'' \<longrightarrow> (\<exists>s'\<in>S\<^bsub>SES\<^esub>. \<exists>\<delta>. (\<forall>d\<in>set \<delta>. d \<in> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub>) \<and> s ([c] @ \<delta> @ [v])\<Longrightarrow>\<^bsub>SES\<^esub> s' \<and> (s'', s') \<in> ur)"
union_set_partList4: fixes s :: "'b set"   and ts :: "'b set list" shows "\<Union> (set (partList4 s ts)) = \<Union> (set ts)"
lam_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : hfsetcat.prod a b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>hfsetcat.\<Lambda> a b c f : a \<rightarrow> hfsetcat.exp b c\<guillemotright>"
ftf_syn21: fixes a :: "'b event"   and u :: "'b event list"   and t :: "'b event list"   and A :: "'b event set" shows "(a \<in> set u \<and> a \<notin> set t \<or> a \<in> set t \<and> a \<notin> set u) \<and> a \<in> A \<longrightarrow> setinterleaving (u, A, t) = {}"
gcd_zero_iff': fixes a :: "'a"   and b :: "'a" shows "((0::'a) = some_gcd a b) = (a = (0::'a) \<and> b = (0::'a))"
naturality: fixes f :: "'a" assumes "S.arr f" shows "\<rr>.map (S.cod f) \<cdot>\<^sub>S R f = S.map f \<cdot>\<^sub>S \<rr>.map (S.dom f)"
assms_2: assumes "{#a#} + {#b#} \<subseteq># A" shows "P"
ln_upper_9_neg: fixes x :: "real" assumes "0 < x"   and "x \<le> 1" shows "ln x \<le> ln_upper_9 x"
start_def: shows "start \<equiv> [] # replicate (length (g_V G) - 1) (rev [0..<n])"
THE_default_def: fixes d :: "'b"   and P :: "'b \<Rightarrow> bool" shows "THE_default d P = (if \<exists>!x. P x then THE x. P x else d)"
oldest_token_bounded: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat"   and x :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q n = Some x" shows "x \<le> n"
A: shows "R (P \<turnstile> Q) \<in> {p. is_CSP_process p}"
well_defined_vec_mat_Tensor: fixes M :: "'a list list"   and v :: "'a list" assumes "mat (row_length M) (length M) M" shows "mat (row_length M * length v) (length M) (vec_mat_Tensor v M)"
assms: shows "(t, t') \<in> (\<Union> (range (relations SPR.MC)))\<^sup>*"
modifies_spec: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call p (Modif \<sigma>),{}"
tree_chop_plus: fixes t :: "'a tree"   and t' :: "'a tree" shows "tree_chop (t + t') = tree_chop t + tree_chop t'"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b tree \<Rightarrow> 'b \<Rightarrow> 'b tree \<Rightarrow> 'a"   and tree :: "'b tree" shows "P (case tree of \<langle>\<rangle> \<Rightarrow> f1 | \<langle>x, xa, xb\<rangle> \<Rightarrow> f2 x xa xb) = ((tree = \<langle>\<rangle> \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23. tree = \<langle>x21, x22, x23\<rangle> \<longrightarrow> P (f2 x21 x22 x23)))"
reset_set: fixes cs :: "'e list"   and u :: "'e \<Rightarrow> 'f"   and d :: "'f"   and ts :: "'f list" assumes "\<forall>c\<in>set cs. u c = d" shows "[cs\<rightarrow>d]set_clocks cs ts u = u"
inverse_less_imp_less_neg: fixes a :: "'a"   and b :: "'a" assumes "inverse a < inverse b"   and "b < (0::'a)" shows "b < a"
tensor_prod_distr_right: fixes A :: "'b tensor"   and B :: "'b tensor"   and C :: "'b tensor" assumes "dims A = dims B" shows "C \<otimes> (A + B) = C \<otimes> A + C \<otimes> B"
prod_cases6: fixes y :: "'c \<times> 'd \<times> 'e \<times> 'f \<times> 'g \<times> 'h" obtains a :: "'c" and b :: "'d" and c :: "'e" and d :: "'f" and e :: "'g" and f :: "'h" where "y = (a, b, c, d, e, f)"
Inf_union_distrib: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and A :: "'b::type set"   and B :: "'b::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) ((A::'b::type set) \<union> (B::'b::type set)) = (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (Inf A) (Inf B)"
Ll_length: shows "length Ll = length (prems_of \<iota>)"
M_times_1: fixes x :: "int"   and y :: "int" shows "M (M x * y) = M (x * y)"
assms_1: shows "P \<Longrightarrow>\<^sub>\<tau> P''"
Cons: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and E :: "assigned"   and es :: "expr list"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E"   and "Env\<turnstile> nrm E \<guillemotright>\<langle>es\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>e # es\<rangle>\<guillemotright> A"
wf_syscls_initialization_list_is_class: fixes P :: "'a prog"   and C :: "String.literal" assumes "wf_syscls P"   and "C \<in> set initialization_list" shows "is_class P C"
shared_le_def: fixes m\<^sub>1 :: "nat \<Rightarrow> bool option"   and m\<^sub>2 :: "nat \<Rightarrow> bool option" shows "m\<^sub>1 \<subseteq>\<^sub>s m\<^sub>2 \<equiv> m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2 \<and> read_only m\<^sub>1 \<subseteq> read_only m\<^sub>2"
hom_uminus: fixes x :: "'a poly poly"   and xa :: "'a"   and y :: "'a" shows "poly2 (- x) xa y = - poly2 x xa y"
pj_zero: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set"   and x :: "'d" assumes "Ring R"   and "ideal R I"   and "x \<in> carrier R" shows "(pj R I x = \<zero>\<^bsub>R /\<^sub>r I\<^esub>) = (x \<in> I)"
P_eq: fixes x :: "'b" assumes "getRel S_Idt G `` {x} \<noteq> {}" shows "Eps (P (Eps (P x))) = Eps (P x)"
assms_2: shows "\<infinity>\<^sub>h \<notin> ocircline_set H"
in_star_iff_concat: fixes w :: "'b list"   and A :: "'b list set" shows "(w \<in> star A) = (\<exists>ws. set ws \<subseteq> A \<and> w = concat ws \<and> [] \<notin> set ws)"
cinner_sum_right: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (comm_monoid_add.sum plus zero f A) = (\<Sum>y\<in>A. cinner x (f y))"
simps_2: fixes x2 :: "'a" shows "Look \<noteq> ForwardOrEdit x2"
assms: fixes a :: "'a"   and b :: "'a" shows "g (f a b) = f (g a) (g b)"
inv_comp_left_1: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.iso (CC.comp g f)"   and "CC.iso g" shows "CC.inv (CC.comp g f) = CC.comp (CC.inv f) (CC.inv g)"
Collect_mem_eq: fixes A :: "'k set" shows "{x. x \<in> A} = A"
directed_extend: fixes X :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "'b set" assumes "directed X less_eq"   and "directed Y less_eq"   and "\<forall>x\<in>X. \<forall>y\<in>Y. less_eq x y" shows "directed (X \<union> Y) less_eq"
simps_2: fixes n :: "'c" shows "pos_arctic_delta (Num_arc_delta n) = ((0::'c) \<le> n)"
lem: fixes f :: "'n \<Rightarrow> 'a"   and a :: "'n"   and b :: "'n"   and g :: "'n \<Rightarrow> real" assumes "f integrable_on cbox a b"   and "g integrable_on cbox a b"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> norm (f x) \<le> g x" shows "norm (integral (cbox a b) f) \<le> integral (cbox a b) g"
steps: shows "(ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>')"
cnt_rev: fixes x :: "'b"   and xs :: "'b list" shows "cnt x (rev xs) = cnt x xs"
simps_14: fixes a :: "floatarith" shows "fold_const_fa (Ln a) = Ln (fold_const_fa a)"
hom_one: shows "coeff 1 0 = (1::'a)"
distance_attains_inf: fixes s :: "'b set"   and a :: "'b" assumes "closed s"   and "s \<noteq> {}" obtains x :: "'b" where "x \<in> s" and "\<And>y. y \<in> s \<Longrightarrow> dist a x \<le> dist a y"
points_index_0_right_imp: fixes B :: "'b set multiset"   and ps :: "'b set" assumes "\<And>b. b \<in># B \<Longrightarrow> \<not> ps \<subseteq> b" shows "B index ps = 0"
rep_simps_4: fixes r :: "'a" assumes "ide r" shows "local.dom (rep r) = tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>*"
commute: fixes a :: "'i"   and b :: "'i" shows "max a b = max b a"
vimage_inter_cong: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and y :: "'c set" assumes "\<And>w. w \<in> S \<Longrightarrow> f w = g w" shows "f -` y \<inter> S = g -` y \<inter> S"
cod_dom: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "J.cod (J.dom f) = J.dom f"
partial_evaluation_is_saturated: fixes S :: "'at Literal set set"   and C :: "'at Literal set" assumes "saturated_binary_rule resolvent S" shows "saturated_binary_rule ordered_resolvent (partial_evaluation S C)"
f'_def: shows "f' \<equiv> \<lambda>z. c * f z"