hcju/isabelleps huggingface dataset:
extremeE: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and e :: "'c" assumes "extreme X r e" obtains "e \<in> X" and "\<And>x. x \<in> X \<Longrightarrow> r x e"
join_ge1: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.SL_top le join Top" shows "le x (join x y)"
closed_interval_left: fixes b :: "'c" shows "closed {x. \<forall>i\<in>Basis. x \<bullet> i \<le> b \<bullet> i}"
uminus_closed: fixes u :: "('a \<times> 'a) set" assumes "u \<in> \<ww>" shows "uminus_local_ring_at u \<in> \<ww>"
prefs_wf: fixes i :: "'a" assumes "i \<in> agents'" shows "finite_total_preorder_on alts' (R' i)"
map_comp: fixes g :: "'h \<Rightarrow> 'g"   and f :: "'i \<Rightarrow> 'h"   and v :: "'f \<Rightarrow> 'i" shows "g \<circ> (f \<circ> v) = g \<circ> f \<circ> v"
assms_3: shows "fixespointwise (g \<circ> f) X"
enn2real_plus: fixes a :: "ennreal"   and b :: "ennreal" assumes "a < \<top>"   and "b < \<top>" shows "enn2real (a + b) = enn2real a + enn2real b"
next_not_nil_right: fixes nil :: "'a"   and "next" :: "('a \<times> 'a) set"   and x :: "'a" assumes "graph nil next" shows "(x, nil) \<notin> next"
klift_eta_pres: shows "bd\<^sub>\<F> \<eta> = id"
unity_coeff_ex: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a" shows "\<exists>x. P (l * x) \<equiv> \<exists>x. l dvd x + (0::'a) \<and> P x"
zero_order_4: fixes n :: "'a multiset" shows "({#} \<subset># n) = (n \<noteq> {#})"
assms_2: shows "isSend ev \<longrightarrow> getId ev \<noteq> i"
height_mkt_bal_r2: fixes l :: "'b tree"   and r :: "'b tree"   and n :: "'b" assumes "avl l"   and "avl r"   and "height r \<noteq> height l + 2" shows "height (mkt_bal_r n l r) = 1 + max (height l) (height r)"
BFs_3: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<diamond>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<exists>\<alpha>. \<^bold>\<diamond>\<phi> \<alpha>) in v]"
bigE: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)" obtains c :: "real" where "0 < c" and "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)"
sinvar_all_edges_normal_form_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "sinvar_all_edges_normal_form P \<equiv> \<forall>G nP. sinvar G nP = (\<forall>(e1, e2)\<in>edges G. P (nP e1) (nP e2))"
clop_adj: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "clop (g \<circ> f)"
3: shows "card (\<Union>I\<in>set (subformulae F). atoms (tseytin_toatom F I)) \<le> size F"
DiningPhilosophers_axioms: shows "DiningPhilosophers N"
simps_3: fixes t\<^sub>1 :: "'b tree"   and ts\<^sub>1 :: "'b tree list"   and t\<^sub>2 :: "'b tree"   and ts\<^sub>2 :: "'b tree list" shows "merge (t\<^sub>1 # ts\<^sub>1) (t\<^sub>2 # ts\<^sub>2) = (let h\<^sub>2 = t\<^sub>2 # ts\<^sub>2; h\<^sub>1 = t\<^sub>1 # ts\<^sub>1 in if rank t\<^sub>1 < rank t\<^sub>2 then t\<^sub>1 # merge ts\<^sub>1 h\<^sub>2 else if rank t\<^sub>2 < rank t\<^sub>1 then t\<^sub>2 # merge h\<^sub>1 ts\<^sub>2 else ins_tree (link t\<^sub>1 t\<^sub>2) (merge ts\<^sub>1 ts\<^sub>2))"
nonneg_bounded: shows "\<exists>K\<ge>0. \<forall>a b. norm (blinfun_apply a b) \<le> norm a * norm b * K"
refine: fixes Xi :: "'b list"   and X :: "'c set"   and sctni :: "real list sctn"   and sctn :: "'c sctn" assumes "(Xi, X) \<in> appr_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(RETURN (above_sctn_impl Xi sctni), above_sctn $ X $ sctn) \<in> \<langle>bool_rel\<rangle>nres_rel"
nnfis_def: shows "nnfis \<equiv> \<lambda>f M. {x. nnfisp f M x}"
Bp_if_B: fixes t :: "'b bro"   and h :: "nat" assumes "t \<in> B h" shows "t \<in> Bp h"
fimage_constant: fixes x :: "'a"   and A :: "'a fset"   and c :: "'b" assumes "x |\<in>| A" shows "(\<lambda>x. c) |`| A = {|c|}"
distinct_1: fixes x2 :: "nat" shows "\<bottom>\<^sub>1 \<noteq> [x2]\<^sub>1"
trg_src: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "trg (src \<mu>) = src \<mu>"
adjuster_wit'': fixes mus_acc :: "'a list"   and f :: "'a vec"   and gs :: "'a vec list"   and mus :: "'a list"   and g' :: "'a vec"   and n_gs :: "('a vec \<times> 'a) list"   and acc :: "'a vec" assumes "adjuster_wit mus_acc f gs = (mus, g')"   and "n_gs = map (\<lambda>x. (x, \<parallel>x\<parallel>\<^sup>2)) gs"   and "f \<in> carrier_vec n"   and "acc \<in> carrier_vec n"   and "set gs \<subseteq> carrier_vec n" shows "mus_adjuster f n_gs mus_acc acc = (mus, acc + g')"
wdD_5: assumes "wd TYPE('c)" shows "D = DIM('c)"
assms_1: shows "Deliver (i, Add i e1) \<in> set (history j)"
diff_union_swap: fixes a :: "'d"   and b :: "'d"   and M :: "'d multiset" assumes "a \<noteq> b" shows "add_mset b (M - {#a#}) = add_mset b M - {#a#}"
abs_eq: fixes x :: "'c regular \<times> 'c dense Filters.filter" assumes "eq_onp (\<lambda>x. x \<in> stone_phi.pairs) x x" shows "- Abs_stone_phi_pair x = Abs_stone_phi_pair (stone_phi.pairs_uminus x)"
tup_to_sum_def: shows "tup_to_sum = (\<Lambda> p s. case p of (f, g) \<Rightarrow> case undiscr s of Inl x \<Rightarrow> f\<cdot>(Discr x) | Inr x \<Rightarrow> g\<cdot>(Discr x))"
eqvt_at_apply': fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and p :: "perm" assumes "eqvt_at f x" shows "p \<bullet> f x = f (p \<bullet> x)"
search_empty: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and i :: "nat" assumes "total_cons s"   and "s b \<down>= i"   and "\<And>x. \<phi> i x \<up>" shows "eval (total_cons.r_search s) [b] \<up>"
FOREACHoci_def: fixes R :: "'n \<Rightarrow> 'n \<Rightarrow> bool"   and \<Phi> :: "'n set \<Rightarrow> 'm \<Rightarrow> bool"   and S :: "'n set"   and c :: "'m \<Rightarrow> bool"   and f :: "'n \<Rightarrow> 'm \<Rightarrow> 'm nres"   and \<sigma>0 :: "'m" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> S c f \<sigma>0 \<equiv> ASSERT (finite S) \<bind> (\<lambda>_. SPEC (\<lambda>xs. distinct xs \<and> S = set xs \<and> sorted_wrt R xs) \<bind> (\<lambda>xs. WHILE\<^sub>T\<^bsup>\<lambda>(it, \<sigma>). \<exists>xs'. xs = xs' @ it \<and> \<Phi> (set it) \<sigma>\<^esup> (FOREACH_cond c) (FOREACH_body f) (xs, \<sigma>0) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)))"
monom_mult_zero_right: fixes c :: "'c"   and t :: "'b" shows "MPoly_Type_Class.punit.monom_mult c t 0 = 0"
ZN_def: shows "ZN = (\<lambda>z n. z = int n)"
old_set_rel_def_internal: fixes R :: "('a \<times> 'b) set" shows "old_set_rel R \<equiv> {(S, S'). S' = R `` S \<and> S \<subseteq> Domain R}"
exhaust: fixes y :: "'e + 'f" obtains x1 :: "'e" where "y = pure x1"   | x2 :: "'f" where "y = Inr x2"
compass_non_zero_5: shows "vector [0, 1, 0] \<noteq> 0"
assms_1: shows "x \<in>\<^sub>i range_tm e t"
cont: shows "continuous_on ((\<lambda>x. (x, x)) ` S) (\<lambda>x. dist ((g \<circ> fst) x) (snd x))"
assms_2: shows "circuit_in \<E> C\<^sub>2"
iff: fixes x :: "'a"   and k :: "real" assumes "x \<in> V - {0::'a}" shows "(0 < k \<and> k *\<^sub>R x \<in> S) = (d x = k)"
card_verts_dv: shows "card (verts G) = Suc (card (verts (del_vert u)))"
T1: fixes ds :: "D2 set"   and X :: "X5 set" shows "BSI.stable_on ds X = (\<exists>XD_XH. BSI.stable_pair_on ds XD_XH \<and> X = BSI.match XD_XH)"
connectedD_interval: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and U :: "'c set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.linorder_topology less_eq less open"   and "topological_space.connected open U"   and "x \<in> U"   and "y \<in> U"   and "less_eq x z"   and "less_eq z y" shows "z \<in> U"
has_bochner_integral_I0i_power_exp_m': fixes k :: "nat" shows "has_bochner_integral lborel (\<lambda>x. x ^ k * exp (- x) * indicat_real {0..} x) (fact k)"
valofd_def: fixes x :: "('c, 'd) IEEE.float" shows "valofd x = 2 / 2 ^ bias TYPE(('c, 'd) IEEE.float) * (real (fraction x) / 2 ^ LENGTH('d))"
document_ptr_kinds_eq_h2: shows "document_ptr_kinds h2 = document_ptr_kinds h'"
Sup_sup_pred: fixes x :: "'c \<times> 'd"   and P :: "'c \<times> 'd \<Rightarrow> bool" shows "sup_prod x (Sup_prod {y. P y}) = Sup_prod {y. y = x \<or> P y}"
inducts: fixes P :: "'c list \<Rightarrow> bool"   and list :: "'c list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
eventually_at_infinityI: fixes c :: "real"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. c \<le> norm x \<Longrightarrow> P x" shows "eventually P at_infinity"
projection_on_intersection: fixes l :: "'b list"   and X :: "'b set"   and Y :: "'b set" assumes "l \<upharpoonleft> X = []" shows "l \<upharpoonleft> (X \<inter> Y) = []"
set_spmf_sampl_uni_units: fixes q :: "nat" shows "set_spmf (sample_uniform_units q) = {..<q} - {0}"
is_none_code_1: shows "Option.is_none \<bottom> = True"
fls_inverse_deriv: fixes a :: "'a fls" shows "fls_deriv (inverse a) = - fls_deriv a * (inverse a)\<^sup>2"
prems: fixes vs :: "'val list" assumes "liveVal' g (v' # vs)" shows "thesis"
ceiling_less_iff: fixes x :: "'a"   and z :: "int" shows "(\<lceil>x\<rceil> < z) = (x \<le> of_int z - (1::'a))"
trimming: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "current \<Gamma> g"   and "wave \<Gamma> g"   and "g \<le> f"   and "\<And>x. \<lbrakk>x \<in> RF\<^sup>\<circ> (TER f); x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> KIR g x"   and "\<E> (TER g) - A \<Gamma> = \<E> (TER f) - A \<Gamma>" shows "local.trimming g"
disc_12: fixes x4 :: "nat" shows "is_REL (REL x4)"
minus_int_def: fixes z :: "Quotient_Int.int"   and w :: "Quotient_Int.int" shows "z - w = z + - w"
mirror_l: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "iszlfm p (a # bs)" shows "iszlfm (mirror p) (a # bs)"
b_len: shows "length (base (ds' \<union> ds) b) = dm"
floor_eq: fixes n :: "int"   and x :: "real" assumes "real_of_int n < x"   and "x < real_of_int n + 1" shows "\<lfloor>x\<rfloor> = n"
domI: fixes m :: "'b \<Rightarrow> 'a option"   and a :: "'b"   and b :: "'a" assumes "m a = Some b" shows "a \<in> dom m"
distinct_87: fixes x81 :: "'d"   and x82 :: "'a \<Rightarrow> nat"   and x83 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "{x81}deliver(x82) . x83 \<noteq> call(x10)"
alpha_abs_set_stronger1: fixes as :: "atom set"   and x :: "'b"   and p' :: "perm"   and as' :: "atom set"   and x' :: "'b" assumes "(as, x) \<approx>set (=) supp p' (as', x')" shows "\<exists>p. (as, x) \<approx>set (=) supp p (as', x') \<and> supp p \<subseteq> as \<union> as'"
component_mult_def: fixes v :: "'a vec"   and w :: "'a vec" shows "component_mult v w = Matrix.vec (min (dim_vec v) (dim_vec w)) (\<lambda>i. v $ i * w $ i)"
mon_n_def: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and n :: "'g" shows "mon_n fg n \<equiv> mon fg (proc_of fg n)"
setops\<^sub>s\<^sub>s\<^sub>t_unlabel_subset_if_no_label: fixes l :: "'a"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and l' :: "'a" assumes "\<not> list_ex (is_LabelN l) A" shows "setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l A) \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l' A)"
subst_lift_cond: fixes \<sigma> :: "'c \<Rightarrow> 'c"   and s :: "('b, 'c) uexpr" shows "\<lceil>\<sigma>\<rceil>\<^sub>s \<dagger> \<lceil>s\<rceil>\<^sub>\<leftarrow> = \<lceil>\<sigma> \<dagger> s\<rceil>\<^sub>\<leftarrow>"
assms: shows "seqll i P ((\<sigma>, p), a, \<sigma>', p')"
infsetsum_prod_PiE: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)"   and "\<And>x. x \<in> A \<Longrightarrow> f x abs_summable_on B x" shows "(\<Sum>\<^sub>ag\<in>Pi\<^sub>E A B. \<Prod>x\<in>A. f x (g x)) = (\<Prod>x\<in>A. infsetsum (f x) (B x))"
const_onD: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and x :: "'c"   and y :: "'b" assumes "const_on f S x"   and "y \<in> S" shows "f y = x"
at_within_ball: fixes e :: "real"   and x :: "'a"   and y :: "'a" assumes "0 < e"   and "dist x y < e" shows "at y within ball x e = at y"
smalleraux2: fixes a :: "'b"   and b :: "'b"   and l :: "'b set list"   and c :: "'b"   and d :: "'b" assumes "{a, b} \<in> set l"   and "{c, d} \<in> set l"   and "{a, b} \<noteq> {c, d}"   and "smaller (DenyAllFromTo a b) (DenyAllFromTo c d) l" shows "\<not> smaller (DenyAllFromTo c d) (DenyAllFromTo a b) l"
split_set_1: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and ts :: "('b btree \<times> 'b) list"   and z :: "'b"   and ls :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and b :: "'b"   and rs :: "('b btree \<times> 'b) list" assumes "split split"   and "split ts z = (ls, (a, b) # rs)" shows "(a, b) \<in> set ts"
other_claim: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and j :: "nat" assumes "is_path \<pi>"   and "\<pi> i = \<pi> j"   and "\<pi> i \<noteq> local.return"   and "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"   and "j icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0" shows "i = j"
f_3: fixes epsilon :: "real" assumes "0 < epsilon" shows "\<not> \<bar>set_lebesgue_integral M (f epsilon) h\<bar> < delta"
HF2_holds: fixes e :: "(name, hf) finfun" shows "eval_fm e HF2"
prod_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<forall>a\<in>A. less_eq zero (f a)" shows "less_eq zero (comm_monoid_mult.prod times one f A)"
step: shows "lnth TNs i \<leadsto>LGC lnth TNs (Suc i)"
WT_plus_oracleD1: fixes \<I>l :: "('a, 'c) \<I>"   and \<I>r :: "('b, 'd) \<I>"   and left :: "'e \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'e) spmf"   and right :: "'e \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) spmf"   and s :: "'e" assumes "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd>" shows "\<I>l \<turnstile>c left s \<surd>"
sequence_infinite_lemma: fixes f :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "\<forall>n. f n \<noteq> l"   and "f \<longlonglongrightarrow> l" shows "infinite (range f)"
Matrix_col_is_Legacy_col: fixes j :: "nat"   and A :: "complex Matrix.mat" assumes "j < dim_col A" shows "Matrix.col A j = vec_of_list (Matrix_Legacy.col (mat_to_cols_list A) j)"
derived_set_of_def: fixes X :: "'c topology"   and S :: "'c set" shows "X derived_set_of S \<equiv> {x \<in> topspace X. \<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<noteq> x \<and> y \<in> S \<and> y \<in> T)}"
complete: fixes x :: "'a" assumes "x \<notin> S" shows "f x = (0::'b)"
removeAll_multiset: fixes a :: "'b list"   and x :: "'b" assumes "distinct a"   and "x \<in> set a" shows "mset a = {#x#} + mset (removeAll x a)"
matrix_inv_def: fixes A :: "(('a, 'c) vec, 'b) vec" shows "matrix_inv A = (SOME A'. A ** A' = mat (1::'a) \<and> A' ** A = mat (1::'a))"
C_def: shows "C = (\<lambda>n. UNIV - B ((1 / 2) ^ n))"
virtual_simulates_direct_steps: fixes ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "(ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>')" shows "(remove_rels ts, m, \<S>) \<Rightarrow>\<^sub>v\<^sup>* (remove_rels ts', m', \<S>')"
Rep_induct: fixes y :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "y \<in> A"   and "\<And>x. P (Rep x)" shows "P y"
less_eq_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "(x \<le> y) = (fst x \<le> fst y \<and> snd x \<le> snd y)"
IH: assumes "0 < m"   and "\<And>d. d \<in> set ds \<Longrightarrow> m \<le> d" shows "card {i. i < prod_list ds \<and> (\<forall>i0\<in>set (digit_encode ds i). i0 < m)} = m ^ length ds"
keys_monomial: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_monomial p" shows "keys p = {lt p}"
eigenvector_mat_kernel_char_matrix: fixes A :: "'b mat"   and n :: "nat"   and v :: "'b Matrix.vec"   and lam :: "'b" assumes "A \<in> carrier_mat n n"   and "eigenvector A v lam" shows "v \<in> mat_kernel (char_matrix A lam)"
1: shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<phi>"
wt_App_\<delta>: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "wt (App s t) = wt t" shows "wt s = \<delta>"
simps_9: fixes p :: "('a, 'b) form" shows "params (Exists p) = params p"
Max_in_image: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "A \<noteq> {}" obtains x :: "'c" where "x \<in> A" and "Max (f ` A) = f x"
assert_bound_nolhs_atoms_equiv_bounds: fixes s :: "('i, 'a) state"   and ats :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "(snd ` ats) \<doteq> \<B> s" shows "(snd ` (ats \<union> {a})) \<doteq> \<B> (assert_bound a s)"
insert_disjoint_1: fixes a :: "'g"   and A :: "'g set"   and B :: "'g set" shows "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
norm1_nonzero: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" assumes "v \<noteq> 0" shows "0 < norm1 v"
eq_neutr_iff: fixes a :: "'c"   and b :: "'c" shows "(sup a b = bot) = (a = bot \<and> b = bot)"
hom_power: fixes x :: "real poly"   and n :: "nat" shows "map_poly of_real (x ^ n) = map_poly of_real x ^ n"
iT: shows "isTrue (PPf n \<langle>\<phi>\<rangle>)"
A: shows "((s, c), e, s', c') \<in> ntrs fg"
simps_7: fixes a :: "('d, 'e, 'f) hp"   and b :: "('d, 'e, 'f) hp" shows "SIGP (a ;; b) = SIGP a \<union> SIGP b"
order_consistentD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and a' :: "'b" assumes "order_consistent r s"   and "r a a'"   and "s a' a" shows "a = a'"
lifted_CFG_fund_prop: fixes prog :: "cmd"   and n :: "w_node LDCFG_node"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option" assumes "labels_LDCFG_nodes prog n c"   and "\<langle>c,s\<rangle> \<rightarrow>* \<langle>c',s'\<rangle>" shows "\<exists>n' as. CFG.path src trg (lift_valid_edge (valid_edge prog) sourcenode targetnode kind (_Entry_) (_Exit_)) n as n' \<and> transfers (kinds as) s = s' \<and> preds (kinds as) s \<and> labels_LDCFG_nodes prog n' c'"
bre'_invar_final: fixes Qi :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set" shows "\<forall>sc. sc \<in> wa_invar (bre'_algo Qi \<delta>) \<and> sc \<notin> wa_cond (bre'_algo Qi \<delta>) \<longrightarrow> (Qi \<inter> fst sc = {}) = (Qi \<inter> b_accessible \<delta> = {})"
sconst_cycle: fixes x :: "'b" shows "sconst x = cycle [x]"
to: shows "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"
list_all3_mapD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'j \<Rightarrow> 'g"   and xs :: "'j list"   and g :: "'k \<Rightarrow> 'h"   and ys :: "'k list"   and h :: "'l \<Rightarrow> 'i"   and zs :: "'l list" assumes "list_all3 P (map f xs) (map g ys) (map h zs)" shows "list_all3 (\<lambda>x y z. P (f x) (g y) (h z)) xs ys zs"
cotangent_field_def: fixes f :: "'a \<Rightarrow> real"   and p :: "'a" shows "cotangent_field f p = restrict0 (tangent_space p) (\<lambda>X. X f)"
cInf_eq_minimum: fixes z :: "'c"   and X :: "'c set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" shows "Inf X = z"
Int_UN_distrib: fixes B :: "'c set"   and A :: "'d \<Rightarrow> 'c set"   and I :: "'d set" shows "B \<inter> \<Union> (A ` I) = (\<Union>i\<in>I. B \<inter> A i)"
norm_comb': fixes x :: "'basis euclidean_space" shows "d * norm x \<le> norm (comb' x)"
sc_Ring: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme" assumes "R module M" shows "Ring R"
assms: shows "(((a, b), ba), (aa, bb), bc) \<in> edges"
Sigma_fm_OR: fixes A :: "fm"   and B :: "fm" assumes "Sigma_fm A"   and "Sigma_fm B" shows "Sigma_fm (A OR B)"
that: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set" assumes "valid_decomp X ps"   and "standard_decomp (poly_deg f0) ps"   and "cone_decomp T ps"   and "(\<And>f. f \<in> F \<Longrightarrow> homogeneous f) \<Longrightarrow> hom_decomp ps"   and "direct_decomp (ideal (insert f0 F) \<inter> P[X]) [ideal {f0} \<inter> P[X], T]" shows "thesis"
disj: shows "A \<inter> B \<inter> (u - s) = {}"
ran_map_graph: fixes f :: "'d \<Rightarrow> 'c option" shows "ran f = Range (map_graph f)"
row_space_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "row_space A = vec.span (rows A)"
in_residues_Hermite_of: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "'b"   and A :: "(('a, 'c) vec, 'b) vec"   and j :: "'b" assumes "ass_function ass"   and "res_function res"   and "is_bezout_ext bezout"   and "\<not> is_zero_row i (Hermite_of A ass res bezout)"   and "j < i" shows "Hermite_of A ass res bezout $ j $ (LEAST n. Hermite_of A ass res bezout $ i $ n \<noteq> (0::'a)) \<in> range (res (Hermite_of A ass res bezout $ i $ (LEAST n. Hermite_of A ass res bezout $ i $ n \<noteq> (0::'a))))"
Fract_quot_of_fract: fixes x :: "'a fract" shows "Fraction_Field.Fract (fst (quot_of_fract x)) (snd (quot_of_fract x)) = x"
sandwich_natural: fixes A :: "'o"   and a :: "'a" assumes "A \<in> Ob"   and "a \<in> F \<^bsub>\<o>\<^esub> A" shows "\<sigma>(A,a) : Hom(A,_) \<Rightarrow> F in Func (AA , Set )"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "('b \<Rightarrow> 'c \<Rightarrow> 'b spmf) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'b) spmf) \<Rightarrow> ('b \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'b) spmf) \<Rightarrow> 'a"   and core :: "('b, 'c, 'd, 'f, 'e, 'g) core" shows "P (case core of Core x xa xb \<Rightarrow> f x xa xb) = (\<forall>x1 x2 x3. core = Core x1 x2 x3 \<longrightarrow> P (f x1 x2 x3))"
augment_empty: fixes S :: "nat \<Rightarrow> bool option"   and x :: "nat set" shows "S \<oplus>\<^bsub>x\<^esub> {} = S"
Quotient_ltln\<^sub>P: shows "Quotient (\<sim>\<^sub>P) abs_ltln\<^sub>P rep_ltln\<^sub>P cr_ltln\<^sub>P"
rem_comps_spec_def: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "rem_comps_spec bs data = (card (component_of_term ` Keys (fst ` set bs)) = fst data + card (const_lt_component ` (fst ` set bs - {0}) - {None}))"
winding_number_exp_2pi: fixes p :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path p"   and "z \<notin> path_image p" shows "pathfinish p - z = exp (complex_of_real (2 * pi) * \<i> * winding_number p z) * (pathstart p - z)"
gState_progress_relI: fixes prog :: "program"   and g :: "'b gState_scheme"   and g' :: "'b gState_scheme" assumes "gState_inv prog g"   and "gState_inv prog g'"   and "length (gState.channels g) \<le> length (gState.channels g')"   and "dom (lm.\<alpha> (gState.vars g)) = dom (lm.\<alpha> (gState.vars g'))" shows "(g, g') \<in> gState_progress_rel prog"
remove1_tail_invar: fixes bq :: "('c, 'd) SkewBinomialTree list"   and t :: "('c, 'd) SkewBinomialTree" assumes "tail_invar bq" shows "tail_invar (remove1 t bq)"