hcju/isabelleps huggingface dataset:
add_le_imp_le_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and i :: "'a"   and k :: "'a"   and n :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (plus i k) n" shows "less_eq i (minus n k)"
toplevel_summands_not_Plus: fixes x :: "'b rexp" assumes "\<forall>r s. x \<noteq> Plus r s" shows "toplevel_summands x = {x}"
eventually_not_in_closed: fixes P :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and t :: "'c"   and T :: "'c set" assumes "closed P"   and "f t \<notin> P"   and "t \<in> T"   and "continuous_on T f" shows "\<forall>\<^sub>F t in at t within T. f t \<notin> P"
assms_3: fixes i :: "nat" shows "S i = S (Suc i) \<or> (\<exists>R\<in>Rs. pushout_step t R (S i) (S (Suc i)))"
max_node_is_Max_V: fixes la :: "(nat \<times> int \<times> nat) list" assumes "E = set la" shows "max_node la = Max (insert 0 (fst ` E \<union> (snd \<circ> snd) ` E))"
S: fixes x :: "comb"   and y :: "comb"   and z :: "comb" shows "S \<bullet> x \<bullet> y \<bullet> z \<rightarrow>\<^sup>1 x \<bullet> z \<bullet> (y \<bullet> z)"
mult_ffact: fixes a :: "'b"   and n :: "nat" shows "a * ffact n a = ffact (Suc n) a + of_nat n * ffact n a"
finfun_comp_def: fixes g :: "'c \<Rightarrow> 'b"   and f :: "('a, 'c) finfun" shows "finfun_comp g f = finfun_rec (\<lambda>b. finfun_const (g b)) (\<lambda>a b c. finfun_update c a (g b)) f"
frame_rule: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and R :: "assn" assumes "<P> c <Q>" shows "<P * R> c <\<lambda>x. Q x * R>"
not_nb: fixes n :: "nat"   and p :: "fm" assumes "bound n p" shows "bound n (not p)"
fundamental_theorem_eq_4: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2"   and "3 \<le> i"   and "i \<le> k" shows "map_poly to_fract (subresultant_prs_locale.R n \<delta> G1 G2 (n (i - 1) - 1)) = Polynomial.smult (subresultant_prs_locale.\<gamma> n \<delta> f \<beta> i) (F i)"
wprepare_loop_goon_Bk: fixes lm :: "nat list"   and m :: "nat"   and b :: "cell list" assumes "lm \<noteq> []"   and "wprepare_loop_start m lm (b, [Bk])" shows "wprepare_loop_goon m lm (Bk # b, [])"
homeomorphic_imp_surjective_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f" shows "f ` topspace X = topspace Y"
Sup_insert: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and a :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (a \<triangleright> A) = sup a (Sup A)"
cnj_sum: fixes f :: "'a \<Rightarrow> complex"   and s :: "'a set" shows "cnj (sum f s) = (\<Sum>x\<in>s. cnj (f x))"
simps_2: fixes \<mu> :: "'a" shows "\<^bold>\<lfloor>\<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^bold>\<rfloor> = \<^bold>\<langle>\<mu>\<^bold>\<rangle>"
Rep_boolean_subset_inject: fixes x :: "'b boolean_subset"   and y :: "'b boolean_subset" shows "(Rep_boolean_subset x = Rep_boolean_subset y) = (x = y)"
A_insert: fixes h :: "'b tree"   and a :: "'b" assumes "is_root h" shows "real (T_insert a h) + \<Phi> (Pairing_Heap_Tree.insert a h) - \<Phi> h \<le> log 2 (real (size h + 1)) + 1"
characterizationallAllocations: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" shows "(a \<in> allAllocations N G) = (a \<in> injections (Domain a) (Range a) \<and> Range a \<in> all_partitions G \<and> Domain a \<subseteq> N)"
le_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "Listn.le r = list_all2 (\<lambda>x. lesub x r)"
drop_bit_of_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n zero = zero"
L_sorted: shows "is_top_sorted (adm_edges f l) (L_left @ L_right)"
ensures_weaken_R: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B' :: "'d set" assumes "F \<in> A ensures A'"   and "A' \<subseteq> B'" shows "F \<in> A ensures B'"
segment_mono: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec> b = (segment D a \<subset> segment D b)"
fold_mset_single: fixes s :: "'m"   and x :: "'m" shows "fold_mset munionM s {#x#} = munionM x s"
star_inductr: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and star :: "'a \<Rightarrow> 'a"   and z :: "'a"   and y :: "'a"   and x :: "'a" assumes "class.kleene_algebra_zerol plus times one zero less_eq less star"   and "less_eq (plus z (times y x)) y" shows "less_eq (times z (star x)) y"
graph_homomorphism_concr_graph: fixes G :: "('b, 'c) labeled_graph"   and e :: "('b \<times> 'd \<times> 'd) set"   and v :: "'d set"   and x :: "('d \<times> 'c) set" assumes "graph G"   and "graph (LG e v)" shows "graph_homomorphism (LG e v) G x = (x `` v \<subseteq> vertices G \<and> on_triple x `` e \<subseteq> edges G \<and> univalent x \<and> Domain x = v)"
weak_Sup_distl: fixes Y :: "('c \<times> 'd) set"   and x :: "'c \<times> 'd" assumes "Y \<noteq> {}" shows "times_prod' x (Sup_prod Y) = Sup_prod {times_prod' x y |y. y \<in> Y}"
nth_append_length: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" shows "(xs @ x # ys) ! |xs| = x"
ext: fixes g :: "'c"   and f :: "'c" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
homeomorphic_maps_involution: fixes X :: "'b topology"   and f :: "'b \<Rightarrow> 'b" assumes "continuous_map X X f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f (f x) = x" shows "homeomorphic_maps X X f f"
vvvv: shows "inner_prod v v = vec_norm v * vec_norm v"
B: fixes x :: "'b" assumes "x \<in> alts" shows "x \<in> alts'"
aux_lemma: fixes pa :: "'a list"   and qa :: "'a list" assumes "set pa \<subseteq> carrier R"   and "set qa \<subseteq> carrier R"   and "length qa \<le> length pa" shows "local.eval (poly_add pa qa) a = local.eval pa a \<oplus> local.eval qa a"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>n. P (Bound n)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Add p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Sub p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Mul p q)"   and "\<And>p. P p \<Longrightarrow> P (Neg p)"   and "\<And>p n. P p \<Longrightarrow> P (Pw p n)"   and "\<And>c n p. P (Add c (Mul (Bound n) p)) \<Longrightarrow> P (CN c n p)"   and "\<And>c. P (C c)" shows "P a0"
gb_schema_aux_term_def: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "gb_schema_aux_term d gs = gb_schema_aux_term1 \<inter> gb_schema_aux_term2 d gs"
intersect_segment_xline'_eq_None: fixes prec :: "nat"   and p0 :: "real \<times> real"   and p1 :: "real \<times> real"   and x :: "real" assumes "intersect_segment_xline' prec (p0, p1) x = None"   and "fst p0 \<le> fst p1" shows "closed_segment p0 p1 \<inter> {p. fst p = x} = {}"
is_dghm_op: shows "op_dghm \<GG> : op_dg (op_dg \<BB>) \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> op_dg \<CC>"
f3_def: shows "f3 = (\<lambda>x. C x - g3 x)"
vwellformed_def: shows "vwellformed \<equiv> value_pred.pred (\<lambda>_. wellformed_clauses) (\<lambda>_. True) (\<lambda>_. True)"
assume_prop: fixes p :: "'b Assertion" shows "[\<cdot> p ] * \<bottom> \<squnion> (1::'b) = [\<cdot> p ]"
independence2: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y = (0::'a)" shows "x \<cdot> y\<^sup>\<star> = x"
decseqD: fixes f :: "nat \<Rightarrow> 'c"   and i :: "nat"   and j :: "nat" assumes "decseq f"   and "i \<le> j" shows "f j \<le> f i"
member_union: fixes xs1 :: "'b set_dlist"   and xs2 :: "'b set_dlist"   and x :: "'b" assumes "ID CEQ('b) \<noteq> None" shows "DList_Set.member (DList_Set.union xs1 xs2) x = (DList_Set.member xs1 x \<or> DList_Set.member xs2 x)"
ivl_inj: fixes p :: "'b ivl"   and q :: "'b ivl" assumes "low p = low q"   and "high p = high q" shows "p = q"
\<pi>n': shows "\<pi>\<^sub>1' n' = local.return"
simps: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and a :: "'a stream" shows "(\<phi> until \<psi>) a = ((\<exists>xs. a = xs \<and> \<psi> xs) \<or> (\<exists>xs. a = xs \<and> \<phi> xs \<and> (\<phi> until \<psi>) (stl xs)))"
in_snd_sym_preprocE: fixes p :: "'t \<Rightarrow>\<^sub>0 'd"   and gs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and fs :: "('t \<Rightarrow>\<^sub>0 'd) list" assumes "p \<in> set (snd (sym_preproc gs fs))" obtains "p \<in> set fs"   | g :: "'t \<Rightarrow>\<^sub>0 'd" and t :: "'a" where "g \<in> set gs" and "p = monom_mult (1::'d) t g"
sum_list_pair_relator_halflist_flip: fixes S :: "'b set"   and s :: "'b"   and t :: "'b" assumes "PreCoxeterSystem S"   and "s \<in> S"   and "t \<in> S" shows "sum_list (pair_relator_halflist s t) = sum_list (pair_relator_halflist t s)"
state_Cfg: fixes s :: "'a"   and d' :: "'a pmf"   and c' :: "'a \<Rightarrow> 'a cfg" shows "state (Cfg s d' c') = s"
length_interact_ge: fixes xs :: "'a list list"   and ys :: "'a list list" assumes "xs \<in> lists (- {[]})"   and "ys \<in> lists (- {[]})" shows "length xs + length ys \<le> length (interact xs ys)"
Hermite_of_row_i_range_res: fixes j :: "'a"   and i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ass :: "'b \<Rightarrow> 'b" assumes "j < i"   and "\<not> is_zero_row i A"   and "res_function res" shows "Hermite_of_row_i ass res A i $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) \<in> range (res (Hermite_of_row_i ass res A i $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))))"
map_graph_fn_graphI: fixes G :: "('a, 'c) labeled_graph"   and f :: "'c \<Rightarrow> 'b" shows "graph (map_graph_fn G f)"
subst_atm_id_subst: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and A :: "'b" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "subst_atm A id_subst = A"
arden_helper: fixes X :: "'b list set"   and A :: "'b list set"   and B :: "'b list set"   and n :: "nat" assumes "X = A @@ X \<union> B" shows "X = A ^^ Suc n @@ X \<union> (\<Union>m\<le>n. A ^^ m @@ B)"
p1: shows "prod_list (map (sq_norm \<circ> row (sylvester_mat f g)) [0..<degree g]) = \<parallel>f\<parallel>\<^sup>2 ^ k"
pair_measure_bind: fixes M1 :: "'b measure"   and M2 :: "'c measure"   and f :: "'b \<times> 'c \<Rightarrow> 'd measure"   and N :: "'d measure" assumes "pair_prob_space M1 M2"   and "f \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M subprob_algebra N" shows "M1 \<Otimes>\<^sub>M M2 \<bind> f = M1 \<bind> (\<lambda>x. M2 \<bind> (\<lambda>y. f (x, y)))"
the_inverse_matrix: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "invertible A" shows "the (inverse_matrix A) = P_Gauss_Jordan A"
subspace_diff_fun_space: shows "subspace manifold_eucl.dest.diff_fun_space"
modulo_finite_3_def: fixes x :: "finite_3"   and y :: "finite_3" shows "x mod y = (case y of a\<^sub>1 \<Rightarrow> x | _ \<Rightarrow> a\<^sub>1)"
simps_3: fixes t\<^sub>1 :: "term"   and t\<^sub>2 :: "term" shows "linear (t\<^sub>1 $ t\<^sub>2) = (linear t\<^sub>1 \<and> linear t\<^sub>2 \<and> \<not> is_free t\<^sub>1 \<and> fdisjnt (frees t\<^sub>1) (frees t\<^sub>2))"
prems_5: shows "a \<notin> \<O> \<union> all_acquired (x # sb)"
jozsa_transform_is_gate: shows "gate (n + 1) U\<^sub>f"
lex_pm_antisym: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" assumes "lex_pm s t"   and "lex_pm t s" shows "s = t"
BoundOutputR_def: fixes x3 :: "name"   and x1 :: "name"   and x2 :: "pi" shows "x3<\<nu>x1> \<prec> x2 \<equiv> Abs_residual (BoundOutputR_Rep x3 ([x1].x2))"
list_to_option_SomeE: fixes xs :: "'a list"   and x :: "'a" assumes "list_to_option xs = Some x" obtains "xs = [x]"
Markov_Decision_Process_axioms: shows "Markov_Decision_Process \<K>"
inf_period_4: fixes d :: "'a"   and D :: "'a"   and t :: "'a" assumes "d dvd D" shows "\<forall>x k. (\<not> d dvd x + t) = (\<not> d dvd x - k * D + t)"
eval\<^sub>G_subst: fixes S :: "'c ltl set"   and \<phi> :: "'c ltl" shows "eval\<^sub>G S \<phi> = subst \<phi> (\<lambda>\<chi>. Some (eval\<^sub>G S \<chi>))"
content_subset: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "cbox a b \<subseteq> box c d" shows "content (cbox a b) \<le> content (box c d)"
g_max_def: fixes m :: "'s"   and P :: "'k \<times> 'v \<Rightarrow> bool" shows "g_max m P \<equiv> reverse_iterateoi m (\<lambda>\<sigma>. \<sigma> = None) (\<lambda>x \<sigma>. if P x then Some x else None) None"
category_with_finite_products_axioms: shows "category_with_finite_products (\<cdot>)"
Wax_satPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "TE.satPB \<xi> Wax"
Min_insert: fixes A :: "'b word set"   and x :: "'b word" assumes "finite A"   and "A \<noteq> {}" shows "signed.Min (insert x A) = signed.min x (signed.Min A)"
right_inverse_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 b :: "'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "(divide a b = one) = (a = b)"
scp1: shows "singular_chain (p - Suc 0) (subtopology X T) (chain_boundary p c)"
usumE: fixes u :: "('a, 'b) node set"   and A :: "('a, 'b) node set set"   and B :: "('a, 'b) node set set" assumes "u \<in> usum A B" obtains x :: "('a, 'b) node set" where "x \<in> A" and "u = In0 x"   | y :: "('a, 'b) node set" where "y \<in> B" and "u = In1 y"
eval_fds_norm_power_le: fixes f :: "'a fds"   and s :: "real"   and n :: "nat" assumes "fds_abs_converges (fds_norm f) s" shows "eval_fds (fds_norm (f ^ n)) s \<le> eval_fds (fds_norm f) s ^ n"
n_in_obs: fixes n :: "'node"   and S :: "'node set" assumes "valid_node n"   and "n \<in> S" shows "obs n S = {n}"
asymmetric_var: fixes x :: "'a" shows "asymmetric x = asymmetric_var x"
assms_4: fixes s :: "'b set" assumes "s \<in> set ss" shows "local.subspace s"
uniformly_continuous_on_uniformity: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "uniformly_continuous_on s f = (LIM (x, y) uniformity_on s. (f x, f y) :> uniformity)"
len: shows "length (map ((*\<^sub>R) 2) (ccw.selsort 0 (inl (snd X)))) \<noteq> 1"
singleton_ball: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>{a}. P x) = P a"
additive_sum: fixes f :: "'a set \<Rightarrow> ennreal"   and S :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "positive M f"   and "Measure_Space.additive M f"   and "finite S"   and "A ` S \<subseteq> M"   and "disjoint_family_on A S" shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union> (A ` S))"
invpst_baldL: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invpst l"   and "invpst r" shows "invpst (baldL l a r)"
eval_models: fixes ec :: "'s odlist"   and t :: "'s Trace"   and \<phi> :: "('a, 'p) Kform"   and s :: "'s" assumes "spr_simAbs ec = spr_jkbpCSt t"   and "subjective agent \<phi>"   and "s \<in> toSet ec" shows "(toSet (eval envVal ec \<phi>) \<noteq> {}) = spr_repMC (toSet ec), s \<Turnstile> \<phi>"
assms: shows "\<forall>t\<in>M. analyzed_in t M"
permutes_image_mset: fixes p :: "'d \<Rightarrow> 'd"   and A :: "'d set" assumes "p permutes A" shows "image_mset p (mset_set A) = mset_set A"
antisymmetric_axioms: shows "antisymmetric A (\<sqsubseteq>)"
rt_fresh_asD1: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
arr_char: fixes Obj :: "'b set"   and Null :: "'b"   and f :: "'b" assumes "discrete_category Obj Null" shows "partial_magma.arr (discrete_category.comp Obj Null) f = (f \<in> Obj)"
intro: fixes T :: "'b set set"   and S :: "'c set set"   and f :: "'c \<Rightarrow> 'b" assumes "\<And>m. m open\<^bsub>T\<^esub> \<Longrightarrow> carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>" shows "continuous_axioms f S T"
simps_4: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "quadratic_sub var a b c d (Neq p) = (let p1 = quadratic_part_1 var a b d (Neq p); p2 = quadratic_part_2 var c p1; A = isolate_variable_sparse p2 var 0; B = isolate_variable_sparse p2 var 1 in Or (fm.Atom (Less (- A * B))) (fm.Atom (Neq (A\<^sup>2 - B\<^sup>2 * c))))"
HC_intros_3: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (F::'b formula) \<^bold>\<rightarrow> (G::'b formula) \<^bold>\<rightarrow> F"
VcompNml_Ide_Nml: fixes t :: "'a local.term"   and a :: "'a local.term" assumes "Nml t"   and "Ide a"   and "Dom a = Cod t" shows "a \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"
max_ary_def: shows "max_ary = Max (size ` (CC \<union> {C}))"
bin_clr_conv_NAND: fixes n :: "nat"   and i :: "int" shows "bin_sc n False i = i AND NOT (1 << n)"
inorder_split_min: fixes t :: "'b bro"   and h :: "nat"   and a :: "'b"   and t' :: "'b bro" assumes "t \<in> T h" shows "(split_min t = None) = (inorder t = []) \<and> (split_min t = Some (a, t') \<longrightarrow> inorder t = a # inorder t')"
that: fixes i :: "nat"   and i' :: "nat" assumes "i < k"   and "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'"   and "\<pi> (Suc i) \<noteq> \<pi>' (Suc i')"   and "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<or> (\<forall>j'>i'. j' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> i')" shows "thesis"
assms_3: shows "DD ars r (\<tau>, \<sigma>, \<sigma>', \<tau>')"
mset_pass\<^sub>2: fixes h :: "'b tree" shows "mset_tree (pass\<^sub>2 h) = mset_tree h"
one_uint8_def: shows "1 \<equiv> Abs_uint8 1"
underS_Under_trans: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and C :: "'a set" assumes "trans r"   and "antisym r"   and "a \<in> underS r b"   and "b \<in> Under r C" shows "a \<in> UnderS r C"
right_associated_product: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "\<exists>w. P w x \<and> P w y \<and> P w z" shows "\<forall>w. P w (x \<otimes> (y \<otimes> z)) = (P w x \<and> P w y \<and> P w z)"
notinset_notinEdge1: fixes x :: "nat"   and xs :: "nat list"   and y :: "nat" assumes "x \<notin> set xs" shows "(x, y) \<notin> Edges xs"
irreducible_def: fixes p :: "'a" shows "irreducible p = (p \<noteq> (0::'a) \<and> \<not> p dvd (1::'a) \<and> (\<forall>a b. p = a * b \<longrightarrow> a dvd (1::'a) \<or> b dvd (1::'a)))"
lcdh_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a set spmf" shows "local.lcdh.lcdh \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) \<bind> (\<lambda>zs. return_spmf (\<^bold>g [^] (x * y) \<in> zs))))"
YZ: shows "random_variable (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x))"
SUP_union: fixes M :: "'f \<Rightarrow> 'd \<times> 'e"   and A :: "'f set"   and B :: "'f set" shows "Sup_prod (M ` (A \<union> B)) = sup_prod (Sup_prod (M ` A)) (Sup_prod (M ` B))"
symrun_interp_idem: fixes \<Gamma> :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<Gamma> @ \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
mult_zero_right: fixes a :: "'a" shows "a \<sqinter> bot = bot"
assms: shows "low_equal s1 s2 \<and> Some t1 = memory_write 10 addr bm data s1 \<and> Some t2 = memory_write 10 addr bm data s2"
Pair_bottom_iff: fixes x :: "'c"   and y :: "'d" shows "((x, y) = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)"
proj_arcs_G_simps_3: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map"   and a :: "'d" assumes "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM"   and "a \<notin> {uv, rev_G uv}" shows "subdiv1_contr.proj_arcs_G rev_G uv uw vw a = a"
ideD_1: fixes a :: "'b" assumes "Adj.D.ide a" shows "Adj.D.arr a"
collapse_1: fixes "term" :: "('e, 'f) Term.term" assumes "is_Var term" shows "Var (the_Var term) = term"
bind_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c set" shows "Set.bind A f = {x. \<exists>B\<in>f ` A. x \<in> B}"
sets_image_in_sets: fixes N :: "'b measure"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and M :: "'c measure" assumes "space N = X"   and "f \<in> N \<rightarrow>\<^sub>M M" shows "sets (vimage_algebra X f M) \<subseteq> sets N"
isoI: fixes f :: "'c"   and g :: "'c" assumes "C.inverse_arrows f g" shows "C.iso f"
length_drop2: shows "length (nths (drop 2 xs) (Collect even)) = length (nths (drop 2 xs) {a. odd a})"
SubstTermP_imp_VarP: fixes v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm" shows "{SubstTermP v i t u} \<turnstile> VarP v"
icomp_fcomp: fixes \<theta> :: "char list \<Rightarrow> (char list, char list) Term.term"   and i :: "char list \<Rightarrow> (char list, char list) Term.term" shows "\<theta> \<circ>\<^sub>s i = (\<lambda>x. fterm_to_iterm (((\<lambda>x. iterm_to_fterm (\<theta> x)) \<cdot> (\<lambda>x. iterm_to_fterm (i x))) x))"
f5: shows "(r'' \<otimes> r', s'' \<otimes> s') \<in> carrier rel"
C_eq_All_untilSorted_withSimps: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l)) = C p"
not_kD_sqn: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<notin> kD rt" shows "sqn rt dip = 0"
union_zmset_add_zmset_right: fixes A :: "'b zmultiset"   and a :: "'b"   and B :: "'b zmultiset" shows "A + add_zmset a B = add_zmset a (A + B)"
bij_lens_weak: fixes x :: "'d \<Longrightarrow> 'e" assumes "bij_lens x" shows "weak_lens x"
continuous_snd: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "continuous F f" shows "continuous F (\<lambda>x. snd (f x))"