hcju/isabelleps huggingface dataset:
foldl_spmf_append: fixes f :: "'r \<Rightarrow> 's \<Rightarrow> 'r spmf"   and p :: "'r spmf"   and xs :: "'s list"   and ys :: "'s list" shows "foldl_spmf f p (xs @ ys) = foldl_spmf f (foldl_spmf f p xs) ys"
uton_ntou: fixes x :: "nat" shows "uton (ntou x) = x"
hcomp_simps_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" shows "src\<^sub>B (\<nu> \<star>\<^sub>B \<mu>) = src\<^sub>B \<mu>"
edge_det: fixes a :: "'a"   and a' :: "'a" assumes "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a = targetnode a'" shows "a = a'"
tempI: fixes F :: "behavior \<Rightarrow> bool" assumes "\<And>sigma. F sigma" shows "\<turnstile> F"
WP1iD: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (insertDeny p)"
doubleton_eq_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "({a, b} = {c, d}) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
bdia_def: fixes x :: "'a"   and y :: "'a" shows "antirange_semiring.bdia (\<cdot>) ar x y = ar (ar (y \<cdot> x))"
swap_simp: fixes x :: "'b"   and y :: "'a" shows "prod.swap (x, y) = (y, x)"
plus_up_up: fixes x :: "'a"   and y :: "'a" shows "up\<cdot>x + up\<cdot>y = up\<cdot>(x + y)"
add_tvarsT_def: shows "add_tvarsT = fold_atyps (\<lambda>x. case x of Ty literal list \<Rightarrow> id | Tv idn s \<Rightarrow> insert (idn, s))"
visible_def: shows "R.visible \<equiv> {a. \<exists>q. ren a q \<and> int q \<noteq> int (ex a q)}"
share_all_until_volatile_write_thread_local': fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "a \<in> \<O> \<union> all_acquired sb" shows "share (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) (share_all_until_volatile_write ts \<S>) a = share sb \<S> a"
prefixI: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" assumes "ys = xs \<cdot> zs" shows "xs \<le>p ys"
IH1': shows "frechet I t1 (fst \<nu>) (snd \<nu>) = frechet J t1 (fst \<nu>') (snd \<nu>')"
convex_hull_list_eq_set: fixes Vs :: "'b vec list"   and n :: "nat"   and Ws :: "'b vec list" assumes "set Vs \<subseteq> carrier_vec n"   and "set Vs = set Ws" shows "gram_schmidt.convex_hull_list n Vs = gram_schmidt.convex_hull_list n Ws"
removing_attractor_induces_no_deadends: fixes v :: "'a"   and S :: "'a set"   and p :: "Player"   and W :: "'a set"   and w :: "'a" assumes "v \<in> S - attractor p W"   and "v \<rightarrow> w"   and "w \<in> S"   and "\<And>w. \<lbrakk>v \<in> VV p**; v \<rightarrow> w\<rbrakk> \<Longrightarrow> w \<in> S" shows "\<exists>w\<in>S - attractor p W. v \<rightarrow> w"
hseqI': fixes \<mu> :: "'d"   and \<nu> :: "'d" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "D.hseq \<nu> \<mu>"
elims: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind list"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and y :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "transfers x xa = y" obtains s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = []" and "xa = s__" and "y = s__"   | et :: "('var, 'val, 'ret, 'pname) edge_kind" and ets :: "('var, 'val, 'ret, 'pname) edge_kind list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = et # ets" and "xa = s" and "y = transfers ets (transfer et s)"
to_vec_from_vec: fixes M :: "(('c, 'd) vec, 'd) vec" shows "to_vec (from_vec M) = M"
assms: fixes i :: "nat" shows "fresh_floatarith ea i \<or> i < length ys \<and> i < length zs \<and> ys ! i = zs ! i"
fmrel_rel_fmran: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('e, 'c) fmap"   and y :: "('e, 'd) fmap" assumes "fmrel P x y" shows "rel_fset P (fmran x) (fmran y)"
adjoint_games: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real"   and a :: "char" shows "Games (USubst.adjoint \<sigma> I \<omega>) a = (case SGames \<sigma> a of None \<Rightarrow> Games I a | Some r \<Rightarrow> game_sem I r)"
green_typeII_chain_axioms: shows "green_typeII_chain i j F two_chain s"
strict_arr_char': fixes f :: "'c local.term set" shows "arr f = (\<F>C.arr f \<and> \<F>C.D f = f)"
S_in_MPLS_leq_2_pow_n: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and vs :: "'a set"   and lss :: "('a, bool) fmap list"   and s :: "('a, bool) fmap"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "finite PROB"   and "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"   and "set lss = valid_states (prob_proj PROB vs)"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as' = exec_plan s as \<and> subseq as' as \<and> length as' \<le> Sup {S vs lss PROB s' |s'. s' \<in> valid_states (prob_proj PROB vs)}"
list_all_simps_1: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and xs :: "'d list" shows "list_all P (x # xs) = (P x \<and> list_all P xs)"
filtermap_linear_at_within: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and S :: "'c set" assumes "bij f"   and "isCont f a"   and "\<And>S. open S \<Longrightarrow> open (f ` S)" shows "filtermap f (at a within S) = at (f a) within f ` S"
intro: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" assumes "connection G"   and "galois_bijection_axioms G" shows "galois_bijection G"
fst_splitE: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and s :: "'x \<Rightarrow>\<^sub>0 nat"   and V :: "'x set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "U \<subseteq> X"   and "finite S"   and "0 \<notin> S"   and "(monomial (1::'b) s, V) \<in> set (fst (split t U S))" obtains t' :: "'x \<Rightarrow>\<^sub>0 nat" and x :: "'x" where "t' \<in> .[X]" and "x \<in> X" and "V \<subseteq> U" and "0 \<notin> (\<lambda>s. s - t') ` S" and "s = t' + t + monomial 1 x" and "(monomial (1::'b) s, V) \<in> set (fst (split (t' + t) V ((\<lambda>s. s - t') ` S)))" and "set (snd (split (t' + t) V ((\<lambda>s. s - t') ` S))) \<subseteq> set (snd (split t U S))"
hcomp_assoc: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr"   and \<tau> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>"   and "S.arr \<nu>"   and "S.arr \<tau>"   and "S.src \<mu> = S.trg \<nu>"   and "S.src \<nu> = S.trg \<tau>" shows "(\<mu> \<star>\<^sub>S \<nu>) \<star>\<^sub>S \<tau> = \<mu> \<star>\<^sub>S \<nu> \<star>\<^sub>S \<tau>"
positive_part_def: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.positive_part C = {L. (\<exists>A. L = Pos A) \<and> L \<in> C}"
span_minimal: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "vs1.subspace T" shows "vs1.span S \<subseteq> T"
exp_divide_power_eq: fixes n :: "nat"   and x :: "'b" assumes "0 < n" shows "exp (x / of_nat n) ^ n = exp x"
inverse_matrix_code_rings: fixes A :: "(('b, 'c) vec, 'c) vec" shows "inverse_matrix A = (let d = det A in if is_unit d then Some ((1::'b) div d *k adjugate A) else None)"
path_component_of: fixes X :: "'c topology"   and x :: "'c"   and y :: "'c" shows "path_component_of X x y = (\<exists>T. path_connectedin X T \<and> x \<in> T \<and> y \<in> T)"
exists_unique_def: shows "exists_unique \<equiv> \<lambda>\<phi>. \<^bold>\<exists>\<alpha>. \<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>)"
msg_zhops_def: fixes m :: "msg" shows "msg_zhops m \<equiv> case m of Rreq hopsc x dipc xa xb oipc xc sipc \<Rightarrow> hopsc = 0 \<longrightarrow> oipc = sipc | Rrep hopsc dipc x xa sipc \<Rightarrow> hopsc = 0 \<longrightarrow> dipc = sipc | _ \<Rightarrow> True"
ex_lit: shows "E\<turnstile>Lit vv\<Colon>-PrimT Boolean"
xor_one_right: fixes x :: "'b" shows "x XOR - (1::'b) = NOT x"
mod_exp_aux_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and m :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "euclidean_semiring_cancel.mod_exp_aux times modulo m = efficient_funpow (\<lambda>x y. modulo (times x y) m)"
wf_Conj: fixes xset :: "('idx, 'pred, 'act) formula set['idx]" assumes "finite (supp xset)"   and "\<And>x. x \<in> set_bset xset \<Longrightarrow> weak_formula x" shows "weak_formula (Conj xset)"
trans: shows "P1 \<rightarrow> \<langle>\<alpha>,P2\<rangle>"
nf_inres_RES: fixes X :: "'c set"   and x :: "'c" shows "nf_inres (RES X) x = (x \<in> X)"
l_new_document_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_document_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_new_document type_wf \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_new_document_get_child_nodes_axioms get_child_nodes get_child_nodes_locs"
B_def: shows "B = (\<lambda>x. x * a) ` A"
derived_set_of_derived_set_subset_gen: fixes X :: "'c topology"   and S :: "'c set" assumes "t1_space X" shows "X derived_set_of (X derived_set_of S) \<subseteq> X derived_set_of S"
oalist_of_list_of_oalist: fixes x :: "'x" shows "oalist_of_list (list_of_oalist x) = x"
deriv_c2: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 I,A"
lnull_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lnull (lappend xs ys) = (lnull xs \<and> lnull ys)"
Diagonal_to_Smith_row_i_PQ': fixes P :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec"   and Q :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P' :: "(('a, 'c) vec, 'c) vec"   and B' :: "(('a, 'b) vec, 'c) vec"   and Q' :: "(('a, 'b) vec, 'b) vec"   and i :: "nat" assumes "P ** A ** Q = B"   and "invertible P"   and "invertible Q"   and "is_bezout_ext bezout"   and "(P', B', Q') = Diagonal_to_Smith_row_i_PQ i bezout (P, B, Q)"   and "isDiagonal B"   and "i < min (nrows A) (ncols A)" shows "B' = P' ** A ** Q' \<and> invertible P' \<and> invertible Q'"
CIDom: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "ZFAxioms T"   and "[A,e,B]\<^bsub>T\<^esub> \<in> mor\<^bsub>CanonicalCat T\<^esub>" shows "dom\<^bsub>CanonicalCat T\<^esub> [A,e,B]\<^bsub>T\<^esub> = A"
that_1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "x = P"   and "xa = E"   and "xb = []"   and "xc = []" shows "thesis"
allowed_insert: fixes i :: "'a"   and e :: "'a \<times> 'b \<times> bool"   and n :: "'a option"   and j :: "nat" assumes "Broadcast (i, Insert e n) \<in> set (history j)" shows "n = None \<or> (\<exists>i' e' n'. n = Some (fst e') \<and> Deliver (i', Insert e' n') \<sqsubset>\<^sup>j Broadcast (i, Insert e n))"
injectivity: fixes x :: "rat"   and y :: "rat" assumes "of_rat x = of_rat y" shows "x = y"
valid_iff_nvalid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A = (\<forall>n. \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A)"
subgroupE_2: fixes H :: "nat set" assumes "subgroup H G" shows "H \<noteq> {}"
less_eq_insert1_iff: fixes x :: "hf"   and y :: "hf"   and z :: "hf" shows "(y \<triangleleft> x \<le> z) = (x \<^bold>\<in> z \<and> y \<le> z)"
oa_facts_8: fixes v :: "i"   and x :: "\<nu>" shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<A>\<lparr>A!,x\<^sup>P\<rparr> in v]"
rep_eq: fixes x :: "rat vec" shows "Rep_linear_poly (poly_of_vec x) = (\<lambda>x. if x < dim_vec x then x $ x else 0)"
Top_top: fixes L :: "(alt \<times> alt) set"   and b :: "alt" assumes "L \<in> Lin"   and "\<And>a. a \<noteq> b \<Longrightarrow> a <\<^bsub>L\<^esub> b" shows "Top {b} L = L"
assms_3: shows "q \<cdot> b = (0::'a)"
trg_simps_1: fixes F :: "('a E.term, 'a) arr" assumes "arr F" shows "Dom (trg F) = E.Trg (Dom F)"
assms_4: shows "e * f\<^sup>\<star> * e = bot"
replicate_set_eq: fixes xs :: "'c list"   and y :: "'c" assumes "\<forall>x\<in>set xs. x = y" shows "xs = replicate (length xs) y"
h6: assumes "0 \<le> Av"   and "0 < Bv" shows "(Av + Bv * sqrt Cv < 0) = (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
from_nat_not_eq: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and a :: "nat"   and b :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "a \<noteq> mod_type.to_nat Rep b"   and "a < CARD('a)" shows "mod_type.from_nat Abs a \<noteq> b"
bij: shows "bij_betw \<pi> \<V> \<V>'"
impI2: fixes A :: "'a set"   and q :: "'a"   and p :: "'a" assumes "A \<turnstile> q" shows "A \<turnstile> p .\<longrightarrow>. q"
Sup_distr: fixes X :: "'b set"   and y :: "'b" shows "\<Squnion> X \<cdot> y = (\<Squnion>x\<in>X. x \<cdot> y)"
injection_defined: fixes f :: "'b \<rightarrow> 'a"   and g :: "'a \<rightarrow> 'b"   and z :: "'a" assumes "\<forall>x. f\<cdot>(g\<cdot>x) = x"   and "z \<noteq> \<bottom>" shows "g\<cdot>z \<noteq> \<bottom>"
lexord_eq_fusion_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and g1 :: "('c, 'd) generator"   and g2 :: "('c, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "ord.lexord_eq_fusion less g1 g2 s1 s2 = ord.lexordp_eq less (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
is_type_simps: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list" shows "is_type P Void \<and> is_type P Boolean \<and> is_type P Integer \<and> is_type P NT \<and> is_type P (Class C) = is_class P C"
eventually_at_top_linorder: fixes P :: "'a \<Rightarrow> bool" shows "eventually P at_top = (\<exists>N. \<forall>n\<ge>N. P n)"
msort_distinct_sorted: fixes xs :: "'a list" shows "distinct (msort xs) \<and> sorted (msort xs)"
judge_imp_agree: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> e : \<tau>" shows "\<Gamma> \<turnstile> e, e, e : \<tau>"
dcharacters_def: fixes n :: "nat" shows "dcharacters n = {\<chi>. dcharacter n \<chi>}"
simps_1: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "successively P [] = True"
weight_less_def: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "weight_less s t \<equiv> vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t) \<and> weight s < weight t"
pick_swap_qSwap: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "good X" shows "pick (X #[x1 \<and> x2]_xs) #= pick X #[[x1 \<and> x2]]_xs"
simps_2: fixes xs_ys :: "(int \<times> int) list" shows "interpolation_poly_int Lagrange xs_ys = (let rxs_ys = map (\<lambda>(x, y). (rat_of_int x, rat_of_int y)) xs_ys; rp = interpolation_poly Lagrange rxs_ys in if Ball (set (coeffs rp)) is_int_rat then Some (map_poly int_of_rat rp) else None)"
bl_size: shows "\<forall>bl\<in>#filter_mset ((\<in>) x) \<B>. int (card {p \<in> \<V>. p \<in> bl}) = \<k>"
summable_LIMSEQ_zero: fixes f :: "nat \<Rightarrow> 'e" assumes "summable f" shows "f \<longlonglongrightarrow> (0::'e)"
Rep_cases: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'b" assumes "type_definition Rep Abs A"   and "y \<in> A" obtains x :: "'a" where "y = Rep x"
N2_enabled_at_b: shows "\<turnstile> pc2 = #b \<longrightarrow> Enabled (<N2>_(x, y, sem, pc1, pc2))"
closure_same_affine_hull: fixes S :: "'a set" shows "affine hull closure S = affine hull S"
trans_SeqC: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and c1' :: "('test, 'atom) com"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>c (c1', s')" shows "(c1 ;; c2, s) \<rightarrow>c (c1' ;; c2, s')"
distinct_13: fixes x2 :: "'c"   and x41 :: "('b, 'c) fm"   and x42 :: "('b, 'c) fm" shows "Nom x2 \<noteq> (x41 \<^bold>\<or> x42)"
factors_of_same_degree_def: fixes i :: "nat"   and f :: "'b poly" shows "factors_of_same_degree i f = (i \<noteq> 0 \<and> degree f \<noteq> 0 \<and> monic f \<and> (\<forall>g. irreducible g \<longrightarrow> g dvd f \<longrightarrow> degree g = i))"
\<gamma>: shows "\<gamma> = \<beta> @ \<delta>'"
zero_less_nat_eq: fixes z :: "int" shows "(0 < nat z) = (0 < z)"
inf_commute: fixes x :: "'a upred"   and y :: "'a upred" shows "(x \<and> y) = (y \<and> x)"
division_points_def: fixes k :: "'a set"   and d :: "'a set set" shows "division_points k d = {(j, x). j \<in> Basis \<and> interval_lowerbound k \<bullet> j < x \<and> x < interval_upperbound k \<bullet> j \<and> (\<exists>i\<in>d. interval_lowerbound i \<bullet> j = x \<or> interval_upperbound i \<bullet> j = x)}"
compress_normalize_input_interfaces_nnf: fixes m :: "'c common_primitive match_expr"   and m' :: "'c common_primitive match_expr" assumes "normalized_nnf_match m"   and "compress_normalize_input_interfaces m = Some m'" shows "normalized_nnf_match m'"
fresh_for_def: fixes V :: "nat set" shows "fresh_for V = (SOME n. n \<notin> V)"
Collect_conj_eq: fixes P :: "'q \<Rightarrow> bool"   and Q :: "'q \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
INT_E: fixes b :: "'b"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set"   and a :: "'c" assumes "b \<in> \<Inter> (B ` A)" obtains "b \<in> B a"   | "a \<notin> A"
predicate2I: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
soldAllocationsEquivalenceVariant: shows "soldAllocations = soldAllocations' \<and> soldAllocations' = soldAllocations'' \<and> soldAllocations = soldAllocations''"
step: shows "(s, s') \<in> obsv_round r S v d_f Ob"
simulation2: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and tl2 :: "'j"   and s2' :: "'i" assumes "bisimulation trsys1 trsys2 bisim tlsim"   and "bisim s1 s2"   and "trsys2 s2 tl2 s2'" shows "\<exists>s1' tl1. trsys1 s1 tl1 s1' \<and> bisim s1' s2' \<and> tlsim tl1 tl2"
genideal_self': fixes R :: "('a, 'b) ring_scheme"   and i :: "'a" assumes "ring R"   and "i \<in> carrier R" shows "i \<in> Idl\<^bsub>R\<^esub> {i}"
seqE: fixes g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.seq g f" obtains "Cop_S.arr f" and "Cop_S.arr g" and "Cop_S.dom g = Cop_S.cod f"
hyps_4: shows "\<forall>n w. (\<Sum>i\<in>F. discounted_value r (f i) n w) = discounted_value r (\<lambda>m x. \<Sum>i\<in>F. f i m x) n w"
vunionI1: fixes a :: "V"   and A :: "V"   and B :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> A \<union>\<^sub>\<circ> B"
awalk_Nil_iff: fixes u :: "'a"   and v :: "'a" shows "awalk u [] v = (u = v \<and> u \<in> pverts G)"
Nneg: fixes x :: "int \<times> int" shows "INum (~\<^sub>N x) = - INum x"
swapEnvIm_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and rho :: "'f \<Rightarrow> 'g \<Rightarrow> ('h, 'i, 'varSort, 'var, 'j) qTerm set option" shows "swapEnvIm xs x y rho \<equiv> \<lambda>zs. lift (swap xs x y) (rho zs)"
circ_plus_same: fixes x :: "'a" shows "x\<^sup>\<star> * x = x\<^sup>+"
pairwise_insert: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and s :: "'c set" shows "pairwise r (insert x s) = ((\<forall>y. y \<in> s \<and> y \<noteq> x \<longrightarrow> r x y \<and> r y x) \<and> pairwise r s)"
distinct_11: shows "UnlockFail \<noteq> ReleaseAcquire"
iter_widen_termination: fixes P :: "'a acom \<Rightarrow> bool"   and f :: "'a acom \<Rightarrow> 'a acom"   and c0 :: "'a acom" assumes "\<And>c. P c \<Longrightarrow> P (f c)"   and "\<And>c c'. \<lbrakk>P c; P c'\<rbrakk> \<Longrightarrow> P (c \<nabla>\<^sub>c c')"   and "wf ({(c, c \<nabla>\<^sub>c c') |c c'. P c \<and> P c' \<and> \<not> c' \<sqsubseteq> c}\<inverse>)"   and "P c0"   and "c0 \<sqsubseteq> f c0" shows "\<exists>c. iter_widen f c0 = Some c"
unique_NB: fixes B :: "agent"   and A :: "agent"   and NA :: "nat"   and nb :: "msg"   and evs :: "event list"   and B' :: "agent"   and A' :: "agent"   and NA' :: "nat" assumes "Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B') \<lbrace>Agent A', Nonce NA', nb\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> yahalom"   and "B \<notin> bad"   and "B' \<notin> bad" shows "NA' = NA \<and> A' = A \<and> B' = B"
special_def: fixes t :: "triangle" shows "special t = (isosceles t \<or> right_angled t)"
ceovisidiffimi_ffie_simp: fixes p :: "'b"   and i :: "'b"   and q :: "'b" assumes "(p, i) \<in> e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>"   and "(i, q) \<in> f \<union> f\<inverse> \<union> e" shows "(p, q) \<in> e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>"
sum_rel_simp_1: fixes a :: "'c"   and a' :: "'e"   and Rl :: "('c \<times> 'e) set"   and Rr :: "('d \<times> 'f) set" shows "((Inl a, Inl a') \<in> \<langle>Rl, Rr\<rangle>sum_rel) = ((a, a') \<in> Rl)"
uve: shows "a\<^sup>2 + 3 * b\<^sup>2 = (u\<^sup>2 + 3 * v\<^sup>2) * (c\<^sup>2 + 3 * d\<^sup>2) \<and> coprime u v \<and> a = c * u + e * 3 * d * v \<and> b = c * v - e * d * u \<and> \<bar>e\<bar> = 1"
onel: fixes x :: "'a" shows "(1::'a) \<oplus> x = x"
SesKey_is_session_key: fixes Tgs_B :: "agent"   and A :: "agent"   and SesKey :: "nat"   and T :: "nat"   and evs :: "event list" assumes "Crypt (shrK Tgs_B) \<lbrace>Agent A, Agent Tgs_B, Key SesKey, Number T\<rbrace> \<in> parts (knows Spy evs)"   and "Tgs_B \<notin> bad"   and "evs \<in> kerbIV" shows "SesKey \<notin> range shrK"
CSP1_R1_commute2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "CSP1 (R1 P) = R1 (CSP1 P)"
simps: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" shows "Rangep r a = (\<exists>aa b. a = b \<and> r aa b)"
bit_exp_iff: fixes m :: "nat"   and n :: "nat" shows "bit ((2::'a) ^ m) n = ((2::'a) ^ m \<noteq> (0::'a) \<and> m = n)"