hcju/isabelleps huggingface dataset:
path_weightp_distinct_def: fixes l1 :: "'vtx"   and l2 :: "'vtx"   and s :: "'lbl" shows "path_weightp_distinct l1 l2 s \<equiv> \<exists>xs. distinct xs \<and> path l1 l2 xs \<and> s = sum_weights (map (\<lambda>(s, l, t). l) xs)"
induct: fixes P1 :: "(nat \<Rightarrow> classDef option) \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> exp list \<Rightarrow> nat list \<Rightarrow> bool"   and P2 :: "(nat \<Rightarrow> classDef option) \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> exp \<Rightarrow> nat \<Rightarrow> bool"   and x1 :: "nat \<Rightarrow> classDef option"   and x2 :: "nat \<Rightarrow> nat option"   and x3 :: "exp list"   and x4 :: "nat list"   and x5 :: "nat \<Rightarrow> classDef option"   and x6 :: "nat \<Rightarrow> nat option"   and x7 :: "exp"   and x8 :: "nat" assumes "\<And>CT \<Gamma>. P1 CT \<Gamma> [] []"   and "\<And>CT \<Gamma> e0 C0 es Cs. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : C0; P2 CT \<Gamma> e0 C0; CT;\<Gamma> \<turnstile>+ es : Cs; P1 CT \<Gamma> es Cs\<rbrakk> \<Longrightarrow> P1 CT \<Gamma> (e0 # es) (C0 # Cs)"   and "\<And>\<Gamma> x C CT. \<Gamma> x = Some C \<Longrightarrow> P2 CT \<Gamma> (Var x) C"   and "\<And>CT \<Gamma> e0 C0 Cf fi fDef Ci. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : C0; P2 CT \<Gamma> e0 C0; fields(CT,C0) = Cf; lookup Cf (\<lambda>fd. vdName fd = fi) = Some fDef; vdType fDef = Ci\<rbrakk> \<Longrightarrow> P2 CT \<Gamma> (FieldProj e0 fi) Ci"   and "\<And>CT \<Gamma> e0 C0 m Ds C es Cs. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : C0; P2 CT \<Gamma> e0 C0; mtype(CT,m,C0) = Ds \<rightarrow> C; CT;\<Gamma> \<turnstile>+ es : Cs; P1 CT \<Gamma> es Cs; CT \<turnstile>+ Cs <: Ds; length es = length Ds\<rbrakk> \<Longrightarrow> P2 CT \<Gamma> (MethodInvk e0 m es) C"   and "\<And>CT C Df es Ds \<Gamma> Cs. \<lbrakk>fields(CT,C) = Df; length es = length Df; varDefs_types Df = Ds; CT;\<Gamma> \<turnstile>+ es : Cs; P1 CT \<Gamma> es Cs; CT \<turnstile>+ Cs <: Ds\<rbrakk> \<Longrightarrow> P2 CT \<Gamma> (New C es) C"   and "\<And>CT \<Gamma> e0 D C. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P2 CT \<Gamma> e0 D; CT \<turnstile> D <: C\<rbrakk> \<Longrightarrow> P2 CT \<Gamma> (Cast C e0) C"   and "\<And>CT \<Gamma> e0 D C. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P2 CT \<Gamma> e0 D; CT \<turnstile> C <: D; C \<noteq> D\<rbrakk> \<Longrightarrow> P2 CT \<Gamma> (Cast C e0) C"   and "\<And>CT \<Gamma> e0 D C. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P2 CT \<Gamma> e0 D; CT \<turnstile> C \<not><: D; CT \<turnstile> D \<not><: C\<rbrakk> \<Longrightarrow> P2 CT \<Gamma> (Cast C e0) C" shows "(x1;x2 \<turnstile>+ x3 : x4 \<longrightarrow> P1 x1 x2 x3 x4) \<and> (x5;x6 \<turnstile> x7 : x8 \<longrightarrow> P2 x5 x6 x7 x8)"
poly_deg_leI: fixes p :: "('c \<Rightarrow>\<^sub>0 'd) \<Rightarrow>\<^sub>0 'e"   and d :: "'d" assumes "\<And>t. t \<in> keys p \<Longrightarrow> deg_pm t \<le> d" shows "poly_deg p \<le> d"
not_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "(\<not> less_eq x y) = less y x"
simps_2: fixes k :: "'b" shows "cnf (\<^bold>\<not> (Atom k)) = {{k\<inverse>}}"
assms_1: shows "\<forall>x\<in>A. f x = 0"
vors_4: shows "between (vertices f) ram2 ram1 = []"
ht_height: fixes m :: "nat"   and t :: "('b \<times> nat) tree" assumes "HBT m"   and "HBT.hbt m t" shows "ht t = height t"
ltlr_rel_id: fixes R :: "('a \<times> 'a) set" assumes "R = Id" shows "\<langle>R\<rangle>ltlr_rel = Id"
simps_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and ys :: "'a list" shows "insort_key_rel f x (y # ys) = (if f x y then x # y # ys else y # insort_key_rel f x ys)"
simps_7: fixes v :: "('a, 'b) form"   and va :: "('a, 'b) form" shows "compl (Impl v va) = Neg (Impl v va)"
hyp: shows "Z + Y \<cdot> X \<subseteq> Y"
resumption_def: shows "resumption \<equiv> {x. resumptionp x}"
case_sum_expand_Inr': fixes f :: "'d + 'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'e" assumes "f \<circ> Inl = g" shows "(h = f \<circ> Inr) = (case_sum g h = f)"
tl_lt_stack_hd_discover: fixes x :: "'v" assumes "stack s \<noteq> []"   and "x \<in> set (tl (stack s))" shows "\<delta> s x < \<delta> s (hd (stack s))"
locateT_def: shows "locateT \<equiv> locate1 brn"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>c a b. \<lbrakk>c \<otimes>\<^bsub>G\<^esub> a = c \<otimes>\<^bsub>G\<^esub> b; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b"   and "\<And>a c b. \<lbrakk>a \<otimes>\<^bsub>G\<^esub> c = b \<otimes>\<^bsub>G\<^esub> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b" shows "monoid_cancel_axioms G"
interiorI: fixes T :: "'b set"   and x :: "'b"   and S :: "'b set" assumes "open T"   and "x \<in> T"   and "T \<subseteq> S" shows "x \<in> interior S"
inst_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and \<phi> :: "'c"   and t :: "'b" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst" shows "Generic_Syntax.inst var subst \<phi> t = subst \<phi> t (Generic_Syntax.Variable var 10)"
one_reducible_lm_2: fixes n :: "nat" shows "one_reducible_to_via (nat_to_ce_set n) univ_ce (c_pair n)"
ghc_sort_def: fixes xs :: "int list" shows "ghc_sort xs = merge_list [] (map (\<lambda>ys. if decr ys then rev ys else ys) (cuts xs))"
af_decompose_1: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "'a set list" shows "af (\<phi> and\<^sub>n \<psi>) w = af \<phi> w and\<^sub>n af \<psi> w"
simps_1: fixes keep :: "('v \<times> 'v) list"   and uu :: "'v graph"   and uv :: "'v \<Rightarrow> 'a" shows "minimalize_offending_overapprox [] keep uu uv = keep"
dgrad_p_set_le_trans: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and H :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "dgrad_p_set_le d F G"   and "dgrad_p_set_le d G H" shows "dgrad_p_set_le d F H"
ideal_1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> I" shows "multA a b \<in> I"
not_isSucc_zero: shows "\<not> s.isSucc s.zero"
fds_scale_zero: fixes c :: "nat" shows "fds_scale c 0 = 0"
I: shows "\<forall>s. initial_state sys s \<longrightarrow> I \<lparr>GST = s, HST = []\<rparr>"
f: fixes z :: "nat" shows "eval f [z, b] \<down>= (if the (eval (Cn 2 r_result1 [Cn 2 r_pdec2 [recf.Id 2 0], Cn 2 r_s [Cn 2 r_append_zeros [recf.Id 2 1, Cn 2 r_pdec1 [recf.Id 2 0]]], Cn 2 r_add [Cn 2 r_length [recf.Id 2 1], Cn 2 r_pdec1 [recf.Id 2 0]]]) [z, b]) = 1 then 0 else 1)"
deriv: fixes x :: "real" assumes "x \<in> {real_of_int a<..} - A" shows "(f has_vector_derivative f' x) (at x)"
sort_by_quicksort_rec: fixes cmp :: "'b comparator"   and xs :: "'b list" shows "Sorting_Algorithms.sort cmp xs = Sorting_Algorithms.sort cmp (filter (\<lambda>x. compare cmp x (xs ! (length xs div 2)) = Less) xs) @ filter (\<lambda>y. compare cmp (xs ! (length xs div 2)) y = Equiv) xs @ Sorting_Algorithms.sort cmp (filter (\<lambda>x. compare cmp x (xs ! (length xs div 2)) = Greater) xs)"
isomorphic_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and a' :: "'b" assumes "category C" shows "category.isomorphic C a a' = (\<exists>f. partial_magma.in_hom C f a a' \<and> category.iso C f)"
strict_order_iff: fixes b :: "'b"   and a :: "'b" shows "(b < a) = (a = max a b \<and> a \<noteq> b)"
simps_2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
cexp_mono2': fixes p2 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set"   and q :: "('e \<times> 'e) set" assumes "p2 \<le>o r2"   and "Card_order q"   and "Field p2 = {} \<Longrightarrow> Field r2 = {}" shows "q ^c p2 \<le>o q ^c r2"
extremes_equiv: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and c :: "'b" assumes "extreme X r b"   and "extreme X r c" shows "sympartp r b c"
disc_1: fixes x1 :: "name" shows "is_Sconst (Sconst x1)"
monoid_axioms: shows "monoid (\<^bold>\<squnion>) \<zero>"
nodes_height_upper_bound: fixes k :: "nat"   and t :: "'b btree" assumes "order k t"   and "bal t" shows "nodes t * (2 * k) \<le> bound (2 * k) (height t)"
phiDefNodesI: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and v :: "'var" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "v \<in> uses g m"   and "\<forall>n\<in>set ns. v \<notin> defs g n"   and "length (predecessors g n) \<noteq> 1" shows "n \<in> phiDefNodes g v"
ptensor_mat_dim_row: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_row (st3.ptensor_mat m1 m2) = st3.d0"
harm_remainder_def: fixes N :: "nat"   and n :: "nat" shows "harm_remainder N n = EM_remainder (2 * N + 1) (\<lambda>x. - fact (2 * N + 1) / x ^ (2 * N + 2)) (int n)"
tab_simps_2: shows "src ((\<phi> \<star> f) \<cdot> \<rho>) = src f"
isomorphic_symmetric: fixes f :: "'a"   and g :: "'a" assumes "L.isomorphic f g" shows "L.isomorphic g f"
nth_map_upt: fixes i :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'f" assumes "i < n - m" shows "map f [m..<n] ! i = f (m + i)"
lm083: fixes X :: "'c set"   and Q :: "('c \<times> 'd) set"   and P :: "('c \<times> 'd) set" assumes "X \<inter> Domain Q = {}" shows "(P +< Q) `` X = P `` X"
prems_2: shows "\<Gamma> \<turnstile> Var x : T"
Dag_upd_same_r: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" shows "Dag p l (r(p := p)) t = (p = Null \<and> t = Tip)"
scale_left_diff_distrib: fixes a :: "complex"   and b :: "complex"   and x :: "'a" shows "(a - b) *\<^sub>C x = a *\<^sub>C x - b *\<^sub>C x"
4: shows "(w \<sqinter> - prim_EP w v e) * prim_P w v e\<^sup>T\<^sup>\<star> = w \<sqinter> - prim_EP w v e"
assms_1: shows "A \<in>\<^sub>\<circ> Vset \<alpha>"
nc: shows "\<nexists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l"
prob_space: fixes s :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "T.prob s (space (T s)) = 1"
AboveS_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "Order_Relation.AboveS r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. b \<noteq> a \<and> (a, b) \<in> r}"
simps_4: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<phi> :: "fml"   and \<psi> :: "fml" shows "usubstappf \<sigma> U (\<phi> && \<psi>) = Ando (usubstappf \<sigma> U \<phi>) (usubstappf \<sigma> U \<psi>)"
body': shows "P,E(this \<mapsto> Class (last Cs'), pns [\<mapsto>] Ts) \<turnstile> \<langle>new_body,(h\<^sub>2, l\<^sub>2(this \<mapsto> Ref (a, Cs'), pns [\<mapsto>] vs'))\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2 ++ l\<^sub>3)\<rangle>"
simps_1: fixes t' :: "transition"   and t :: "transition"   and i :: "nat" shows "input_i_stored_in_reg t' t i 0 = (if is_generalisation_of t' t i 0 then Some (i, 0) else None)"
smult_monom: fixes a :: "'b"   and b :: "'b"   and n :: "nat" shows "Polynomial.smult a (monom b n) = monom (a * b) n"
convex_differences: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "convex T" shows "convex (\<Union>x\<in>S. \<Union>y\<in>T. {x - y})"
flush_step_preserves_write_sops: fixes m :: "nat \<Rightarrow> nat"   and sb :: "'a memref list"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and m' :: "nat \<Rightarrow> nat"   and sb' :: "'a memref list"   and \<O>' :: "nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and t :: "nat" assumes "(m, sb, \<O>) \<rightarrow>\<^sub>f (m', sb', \<O>')"   and "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t" shows "\<forall>i\<in>\<Union> (fst ` write_sops sb'). i < t"
vars_mult: fixes p :: "'a mpoly"   and q :: "'a mpoly" shows "vars (p * q) \<subseteq> vars p \<union> vars q"
assms: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "\<not> bl \<subset> b"
cov: fixes x :: "'a" assumes "x \<in> S" shows "x \<in> U x"
lincomb_Nil: fixes rs :: "'f list"   and ms :: "'v list" assumes "rs = [] \<or> ms = []" shows "rs \<bullet>\<cdot> ms = (0::'v)"
H_increasing_Suc: fixes k :: "nat" shows "(case H (Suc k) of (n, N, j') \<Rightarrow> insert n N) \<subseteq> (case H k of (n, N, j') \<Rightarrow> N)"
better_NotR_substc: fixes d :: "coname"   and M :: "trm"   and x :: "name"   and z :: "name"   and P :: "trm" assumes "d \<sharp> M" shows "NotR x.M d{d:=z.P} = fresh_fun (\<lambda>a'. Cut <a'>.NotR x.M a' z.P)"
empty_replicate: fixes n :: "nat"   and x :: "'a" shows "([] = replicate n x) = (n = 0)"
sound_def: fixes P :: "'c \<Rightarrow> real" shows "sound P \<equiv> Expectations.bounded P \<and> nneg P"
Cast: 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 exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l)\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l)\<rangle>"
dispatch_fail: fixes s :: "'b sparc_state"   and v :: "word32"   and e :: "Exception list" assumes "snd (execute_instruction () s) = False \<and> get_trap_set s = {} \<and> exe_mode_val s \<and> fetch_instruction (delayed_pool_write s) = Inr v \<and> decode_instruction v = Inl e" shows "False"
split_cong: fixes q :: "'b \<times> 'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and p :: "'b \<times> 'c" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
closedin_closed_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "closedin X S" shows "closedin (subtopology X S) T = (closedin X T \<and> T \<subseteq> S)"
\<Delta>_max_degree: fixes u :: "'a" assumes "u \<in> V" shows "degree_vertex u \<le> \<Delta>"
birkhoff_sum_integral_2: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "integrable M f" shows "integral\<^sup>L M (birkhoff_sum f n) = real n *\<^sub>R integral\<^sup>L M f"
simps_4: fixes f1 :: "ename \<Rightarrow> 'a"   and f2 :: "'a"   and x1 :: "ename" shows "(case EName x1 of EName x \<Rightarrow> f1 x | This \<Rightarrow> f2) = f1 x1"
closed_cdi_Disj: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and A :: "nat \<Rightarrow> 'c set" assumes "closed_cdi \<Omega> M"   and "range A \<subseteq> M"   and "disjoint_family A" shows "\<Union> (range A) \<in> M"
ProjInd_aHom: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and j :: "'c"   and f :: "('c \<Rightarrow> 'd) \<Rightarrow> 'f"   and B :: "'f set" assumes "\<forall>j\<in>I. aGroup (A j)"   and "j \<in> I"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to f (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B"   and "j \<in> I" shows "ProjInd I A f j \<in> aHom (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f) (A j)"
ipt_iprange_compress_dst_matching: fixes \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and ml :: "'c ipt_iprange negation_type list"   and a :: "action"   and p :: "('c, 'd) tagged_packet_scheme" shows "match_list (common_matcher, \<alpha>) (map (Match \<circ> Dst) (ipt_iprange_compress ml)) a p = matches (common_matcher, \<alpha>) (alist_and (NegPos_map Dst ml)) a p"
chine_eq_apex: shows "H\<mu>H\<nu>\<pi>.chine = H\<mu>H\<nu>\<pi>.apex"
equation_3: fixes x1 :: "com \<times> (char list \<Rightarrow> int)"   and x2 :: "char list \<Rightarrow> int" shows "x1 \<Rightarrow> x2 = Predicate.holds (big_step_Pii_i x1 x2)"
run_asset_ii_unused: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "\<exists>C. PriKey C \<notin> used (fst (run_asset_ii n s))"
S: shows "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"
insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A" shows "mset_set (insert x A) = add_mset x (mset_set A)"
prj1_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "category_with_pullbacks C" shows "category_with_pullbacks.prj1 C f g \<equiv> if partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g then fst (SOME x. limit_cone cospan_shape.comp C (cospan_diagram.map C f g) (partial_magma.dom C (fst x)) (cospan_diagram.mkCone C f g (fst x) (snd x))) else partial_magma.null C"
distinct_binop: fixes xs :: "('d \<times> 'e) list"   and ys :: "('d \<times> 'f) list"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and z1 :: "'e"   and z2 :: "'f" assumes "distinct (map fst xs)"   and "distinct (map fst ys)"   and "rsorted (map fst xs)"   and "rsorted (map fst ys)" shows "distinct (map fst (binop f z1 z2 xs ys))"
hn_merge3_2: fixes XX :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R1 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and x :: "'c"   and x' :: "'d"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and Rm :: "'c \<Rightarrow> 'd \<Rightarrow> assn" assumes "NO_MATCH (hn_invalid XX) R1"   and "hn_ctxt R1 x x' \<or>\<^sub>A hn_ctxt R2 x x' \<Longrightarrow>\<^sub>t hn_ctxt Rm x x'" shows "hn_ctxt R1 x x' \<or>\<^sub>A hn_invalid R2 x x' \<Longrightarrow>\<^sub>t hn_invalid Rm x x'"
that: fixes ns :: "'node SDG_node list" assumes "CFG_node (_Entry_) ics-ns\<rightarrow>\<^sub>d* n'"   and "n \<in> set ns" shows "thesis"
bit_unsigned_iff: 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 w :: "'c word"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (semiring_1.unsigned one plus zero w) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> bit w n)"
Quotient3_rel_abs: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and r :: "'b"   and s :: "'b" assumes "Quotient3 R Abs Rep"   and "R r s" shows "Abs r = Abs s"
intTF_def: fixes \<sigma> :: "'a"   and a :: "'b" shows "intTF \<sigma> a \<equiv> True"
Ferio: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A E B"   and "B I C" shows "A Z C"
Bleast_def: fixes S :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "Bleast S P = (LEAST x. x \<in> S \<and> P x)"
hyperpow_def: fixes R :: "'a star"   and N :: "nat star" shows "R pow N = (*f2* (^)) R N"
point_loop_not_acyclic: fixes p :: "'a"   and x :: "'a"   and n :: "nat" assumes "is_point p"   and "p \<le> power.power 1' (;) x (Suc n) ; p" shows "\<not> is_acyclic x"
less_imp_neq: fixes x :: "'b list"   and y :: "'b list" assumes "strict_suffix x y" shows "x \<noteq> y"
assms: shows "\<forall>i\<in>A. periodic_arithmetic (h i) k"
distinct_map_of_rev: fixes xs :: "('d \<times> 'e) list" assumes "distinct (map fst xs)" shows "map_of (rev xs) = map_of xs"
starset_n_diff: fixes A :: "nat \<Rightarrow> 'a set"   and B :: "nat \<Rightarrow> 'a set" shows "*sn* (\<lambda>n. A n - B n) = *sn* A - *sn* B"
f10: shows "\<forall>s sa w wa. \<not> low_equal s sa \<or> user_reg_val w wa s = user_reg_val w wa sa"
compP\<^sub>1_def: shows "compP\<^sub>1 \<equiv> compP (\<lambda>(pns, body). compE\<^sub>1 (this # pns) body)"
nat_mult_less_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "(k * m < k * n) = (m < n)"
Abs_ccs_inject: fixes x :: "ccs_Rep"   and y :: "ccs_Rep" assumes "x \<in> Collect ccs_Rep_set"   and "y \<in> Collect ccs_Rep_set" shows "(Abs_ccs x = Abs_ccs y) = (x = y)"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b Regex.regex" shows "regex.atms (regex.map_regex f v) = f ` regex.atms v"
g: shows "monom 1 CARD('a) - monom 1 1 = (\<Prod>x\<in>UNIV. [:0, 1:] - [:x:]) * g"
fr_out_1: shows "frontier (outside (path_image c1 \<union> path_image c2)) = path_image c1 \<union> path_image c2"
p_supdist_inf: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - y \<le> - (x \<sqinter> y)"
induced_arrowI_2: fixes \<chi>' :: "(bool, unit) J.arr \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "D.cones_map (induced_arrow a' \<chi>') (D.mkCone p0 p1) = \<chi>'"
prems_2: shows "snd (cs c cid) \<noteq> NotStarted"
assms_2: shows "\<forall>n. \<exists>k. \<forall>m. k \<le> int m \<longrightarrow> eint (int n) \<le> val_Zp (s (Suc m) \<ominus> s m)"
simps_6: shows "ReleaseAcquire \<noteq> Lock"
mset_remove1: fixes a :: "'b"   and xs :: "'b list" shows "mset (remove1 a xs) = remove1_mset a (mset xs)"
eval_eval_exec_max: fixes s1 :: "state"   and e1 :: "expr"   and v1 :: "val"   and n1 :: "nat"   and t1 :: "state"   and s2 :: "state"   and e2 :: "expr"   and v2 :: "val"   and n2 :: "nat"   and t2 :: "state"   and s3 :: "state"   and c :: "stmt"   and n3 :: "nat"   and t3 :: "state" assumes "s1 -e1\<succ>v1-n1\<rightarrow> t1"   and "s2 -e2\<succ>v2-n2\<rightarrow> t2"   and "s3 -c-n3\<rightarrow> t3" shows "s1 -e1\<succ>v1-max (max n1 n2) n3\<rightarrow> t1 \<and> s2 -e2\<succ>v2-max (max n1 n2) n3\<rightarrow> t2 \<and> s3 -c-max (max n1 n2) n3\<rightarrow> t3"
Cinfinite_limit: fixes x :: "'a"   and r :: "('a \<times> 'a) set" assumes "x \<in> Field r"   and "Cinfinite r" shows "\<exists>y\<in>Field r. x \<noteq> y \<and> (x, y) \<in> r"
map_ann_hoare_def: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and Ps :: "(('b, 'a, 'c) ann \<times> 'b set \<times> 'b set) list"   and Ts :: "('b, 'a, 'c) com list" shows "\<Gamma>,\<Theta>[\<tturnstile>]\<^bsub>/F \<^esub>Ps Ts \<equiv> \<forall>i<length Ts. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (fst (Ps ! i)) (Ts ! i) (fst (snd (Ps ! i))), (snd (snd (Ps ! i)))"
domains_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "partial_magma.domains C f = (if partial_magma.arr C f then {partial_magma.dom C f} else {})"
fls_eq_conv_fps_eqI: fixes f :: "'a fls"   and g :: "'a fls" assumes "0 \<le> fls_subdegree f"   and "0 \<le> fls_subdegree g"   and "fls_regpart f = fls_regpart g" shows "f = g"
\<Gamma>_Var_fst: fixes \<tau> :: "('fun, 'atom) Term.term"   and n :: "nat"   and m :: "nat" shows "\<Gamma> (Var (\<tau>, n)) = \<Gamma> (Var (\<tau>, m))"
ide_def: fixes a :: "'a \<times> 'a" shows "CC.ide a \<equiv> CC.comp a a \<noteq> CC.null \<and> (\<forall>f. (CC.comp f a \<noteq> CC.null \<longrightarrow> CC.comp f a = f) \<and> (CC.comp a f \<noteq> CC.null \<longrightarrow> CC.comp a f = f))"
map_spmf_conv_bind_spmf: fixes f :: "'i \<Rightarrow> 'h"   and p :: "'i spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
divide_cong: shows "\<forall>\<^sub>F x in at_bot. poly q x / poly p x = qq x / pp x * dd x"