hcju/isabelleps huggingface dataset:
m_assoc: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a"   and z :: "'a \<times> 'a" assumes "x \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"   and "y \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"   and "z \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)" shows "x \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> y \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> z = x \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> (y \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> z)"
simps_12: fixes t :: "Syntax.term"   and f :: "(Syntax.var, Syntax.term) fmap" shows "psubst_term (Prj1 t) f = Prj1 (psubst_term t f)"
min_top2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and x :: "'a" assumes "class.order_top less_eq less top" shows "ord.min less_eq x top = x"
full_abstraction_wrt_preorders_impl_indRelSTEQ_to_SRel_and_TRel_5: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "trans TRel"   and "refl SRel"   and "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q" shows "\<forall>TP TQ. TP \<in>T P \<and> TQ \<in>T Q \<longrightarrow> (TP, TQ) \<in> TRel"
plus_gt_both_mono: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" assumes "x \<succ> y"   and "z \<succ> u" shows "x + z \<succ> y + u"
order_refl: fixes x :: "'c multiset" shows "x \<subseteq># x"
map_scale_eq_0_iff: fixes c :: "'d"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "(c \<cdot> t = 0) = (c = (0::'d) \<or> t = 0)"
update_convs_5: fixes globs' :: "global list \<Rightarrow> global list"   and inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.globs_update globs' \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs' globs, \<dots> = more\<rparr>"
a: shows "steps0 (1, l, r) B stp = (s'', l'', r'')"
prems_1: fixes D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l'', D'\<rangle>"   and "[D']\<^bsub>v,n\<^esub> = Z''"   and "valid_dbm D'" shows "thesis"
sdrop_simps_2: fixes n :: "nat"   and s :: "'a stream" shows "stl (sdrop n s) = sdrop (Suc n) s"
coprime_polynomial_factorization: fixes as :: "'b poly set"   and a1 :: "'b poly"   and a2 :: "'b poly" assumes "as \<subseteq> {q. irreducible q \<and> monic q}"   and "finite as"   and "a1 \<in> as"   and "a2 \<in> as"   and "a1 \<noteq> a2" shows "comm_monoid_mult_class.coprime a1 a2"
max_list_non_empty: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "x \<le> max_list_non_empty xs"
bit_signed_take_bit_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and m :: "nat"   and a :: "'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m a) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> bit a (min m n))"
hoare_triple_init: shows "\<turnstile>\<^sub>p {tensor_P pre (proj_k 0)} hadamard_n n {tensor_P proj_psi (proj_k 0)}"
Gr_domain_2: fixes A :: "'b set"   and R :: "('b \<times> 'c) set" shows "Domain (BNF_Def.Gr A id O R) = A \<inter> Domain R"
induct: fixes x1 :: "'b list"   and x2 :: "'b list"   and R :: "('b \<times> 'b) set"   and P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "(x1, x2) \<in> lprod R"   and "\<And>a b. (a, b) \<in> R \<Longrightarrow> P [a] [b]"   and "\<And>ah at bh bt a b. \<lbrakk>(ah @ at, bh @ bt) \<in> lprod R; P (ah @ at) (bh @ bt); (a, b) \<in> R \<or> a = b\<rbrakk> \<Longrightarrow> P (ah @ a # at) (bh @ b # bt)" shows "P x1 x2"
CondF: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>1 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
Bex_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "Bex A P = (\<exists>x. x \<in> A \<and> P x)"
finite_bind: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "finite S"   and "\<forall>x\<in>S. finite (f x)" shows "finite (S \<bind> f)"
is_NONSAT_PUSH'_def: fixes pr_operation' :: "pr_operation'" shows "is_NONSAT_PUSH' pr_operation' = (pr_operation' = NONSAT_PUSH')"
polyhedron_def: fixes S :: "'b set" shows "polyhedron S \<equiv> \<exists>F. finite F \<and> S = \<Inter> F \<and> (\<forall>h\<in>F. \<exists>a b. a \<noteq> (0::'b) \<and> h = {x. a \<bullet> x \<le> b})"
count_in_diffI: fixes N :: "'a multiset"   and x :: "'a"   and M :: "'a multiset" assumes "\<And>n. count N x = n + count M x \<Longrightarrow> False" shows "x \<in># M - N"
c1: fixes R :: "('a \<times> 'a set) set" shows "R \<cdot> 1\<^sub>\<pi> \<parallel> R = R"
not_Interp_to_Interp_imp_less: fixes A :: "'a"   and C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "A \<notin> Interp C"   and "A \<in> Interp D" shows "C < D"
B: shows "lookup tOps Y (ec, obs) = lookup tOps Y (ec', obs)"
inject_2: fixes x21 :: "'a"   and x22 :: "'a trm list"   and y21 :: "'a"   and y22 :: "'a trm list" shows "(Fn x21 x22 = Fn y21 y22) = (x21 = y21 \<and> x22 = y22)"
set_OAlist_tc_sorted_domain: fixes xs :: "('a, 'b) oalist_tc" shows "set (OAlist_tc_sorted_domain xs) = fst ` set (list_of_oalist_tc xs)"
\<gamma>_def: shows "\<gamma> = hankel_semiannulus r N"
lfp_pulldown: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "\<And>u P. sound P \<Longrightarrow> T u P = t P (u P)"   and "\<And>P Q. \<lbrakk>sound P; sound Q\<rbrakk> \<Longrightarrow> sound (t P Q)"   and "\<And>P. sound P \<Longrightarrow> mono_trans (t P)"   and "\<And>t u. \<lbrakk>le_trans t u; \<And>P. sound P \<Longrightarrow> sound (t P); \<And>P. sound P \<Longrightarrow> sound (u P)\<rbrakk> \<Longrightarrow> le_trans (T t) (T u)"   and "\<And>t P. \<lbrakk>\<And>Q. sound Q \<Longrightarrow> sound (t Q); sound P\<rbrakk> \<Longrightarrow> sound (T t P)"   and "le_trans (T v) v"   and "\<And>P. sound P \<Longrightarrow> sound (v P)"   and "sound P" shows "lfp_trans T P = lfp_exp (t P)"
vconverse_vintersection: fixes r :: "V"   and s :: "V" shows "(r \<inter>\<^sub>\<circ> s)\<inverse>\<^sub>\<circ> = r\<inverse>\<^sub>\<circ> \<inter>\<^sub>\<circ> s\<inverse>\<^sub>\<circ>"
monoid_mult_hom_axioms: shows "monoid_mult_hom (\<lambda>p. coeff p 0)"
is_aligned_add_helper: fixes p :: "'b word"   and n :: "nat"   and d :: "'b word" assumes "is_aligned p n"   and "d < 2 ^ n" shows "p + d AND mask n = d \<and> p + d AND NOT (mask n) = p"
countable: fixes A :: "'b set"   and M :: "'b measure" assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> sets M"   and "countable A" shows "A \<in> sets M"
steps_z_alpha_V: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l', Z'\<rangle>"   and "Z \<subseteq> V" shows "Z' \<subseteq> V"
assms_2: shows "f \<in> borel_measurable (F t)"
transfom_sas_plus_problem_to_strips_problem_operators_valid: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op' :: "('c \<times> 'd) strips_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)" obtains op :: "('c, 'd) sas_plus_operator" where "op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)" and "op' = \<phi>\<^sub>O \<Psi> op" and "is_valid_operator_sas_plus \<Psi> op"
i_cut_max_empty_2: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "Max I \<le> t" shows "I \<down>> t = {}"
enat_iless: fixes n :: "enat"   and m :: "nat" assumes "n < enat m" shows "\<exists>k. n = enat k"
Red1NewArrayNegative: fixes i :: "32 word"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and T :: "ty"   and s :: "'heap \<times> 'addr val list" assumes "i <s 0" shows "uf,P,t \<turnstile>1 \<langle>newA T\<lfloor>Val (Intg i)\<rceil>,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NegativeArraySize),s\<rangle>"
rev_alternating_list: fixes n :: "nat"   and s :: "'b"   and t :: "'b" shows "rev (alternating_list n s t) = (if gcd_nat.greater_eq n 2 then alternating_list n t s else alternating_list n s t)"
rev_finite_subset: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
sorted_insort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.insort_key less_eq (\<lambda>x. x) x xs) = linorder.sorted less_eq xs"
srblit_def: fixes a :: "'b array"   and s :: "nat"   and d :: "nat"   and l :: "nat" shows "srblit a s d l \<equiv> rblit a s a d l"
Union_upper: fixes B :: "'e set"   and A :: "'e set set" assumes "B \<in> A" shows "B \<subseteq> \<Union> A"
PPs_UnE: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and Y :: "'b set" assumes "t \<in> .[X \<union> Y]" obtains tx :: "'b \<Rightarrow>\<^sub>0 nat" and ty :: "'b \<Rightarrow>\<^sub>0 nat" where "tx \<in> .[X]" and "ty \<in> .[Y]" and "t = tx + ty"
dim_invertible: fixes A :: "'a mat"   and n :: "nat" assumes "invertible_mat A"   and "A \<in> carrier_mat n n" shows "matr_option (dim_row A) (mat_inverse A) \<in> carrier_mat n n"
case_prod_beta: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and p :: "'f \<times> 'g" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
CallHigh: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "({HighSec} \<union> G) \<rhd> body : HighSec" shows "G \<rhd> Call : HighSec"
oseq_reachable_in_cterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "((nat \<Rightarrow> 'f) \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'f"   and p :: "('f, 'g, 'e, 'h) seqp"   and S :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> 'g seq_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> bool"   and p' :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "automaton.trans A = oseqp_sos \<Gamma> i"   and "(\<sigma>, p) \<in> oreachable A S U"   and "p' \<in> sterms \<Gamma> p" shows "p' \<in> cterms \<Gamma>"
init_simps_9: fixes i :: "ITypeId" shows "init (InterfaceT i) = nullV"
extr_mono_IK: fixes IK1 :: "msg set"   and IK2 :: "msg set"   and bad :: "agent set"   and H :: "chan set" assumes "IK1 \<subseteq> IK2" shows "extr bad IK1 H \<subseteq> extr bad IK2 H"
c: shows "\<lbrace>p \<cdot> !b\<rbrace>y\<lbrace>q\<rbrace>"
open_empty: fixes "open" :: "'d set \<Rightarrow> bool" assumes "class.topological_space open" shows "open {}"
streams_iff_sset: fixes s :: "'d stream"   and A :: "'d set" shows "(s \<in> streams A) = (sset s \<subseteq> A)"
state_terminates: fixes s1 :: "'state1"   and s2 :: "'state1" assumes "L1.eval s1 s2"   and "finished step1 s2"   and "final1 s2" shows "L1.sem_behaves s1 (Terminates s2)"
semigroup_ow: shows "semigroup = semigroup_ow UNIV"
func_plus: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f + g = (\<lambda>x. f x + g x)"
greedy_path_deadend_v0: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c"   and v0 :: "'c"   and v :: "'c"   and P :: "'c llist" assumes "ParityGame G"   and "ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0 = LCons v P" shows "(P = LNil) = (\<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v0 \<rightarrow>\<^bsub>G\<^esub> w))"
par_strict_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "A B ParStrict D C"
inner_verts_Nil: fixes G :: "('b, 'c) pre_digraph" shows "pre_digraph.inner_verts G [] = []"
dec_unclocked_unchanged: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec"   and r :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_dec ck mn env s d r"   and "ck = False" shows "snd r \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock (fst r)"
rep_eq: fixes x :: "'a"   and xa :: "'b"   and xb :: "('a, 'b) mapping" shows "mapping.rep (Mapping.update x xa xb) = mapping.rep xb(x \<mapsto> xa)"
conv: fixes a :: "'c" assumes "a \<in> A" shows "infsetsum'_converges f (B a)"
function_mult_eval_car: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "f \<in> carrier F"   and "g \<in> carrier F" shows "(f \<otimes>\<^bsub>F\<^esub> g) a = f a \<otimes> g a"
arg_Im_sgn: fixes z :: "complex" assumes "Im z \<noteq> 0" shows "sgn (arg z) = sgn (Im z)"
moebius_comp: fixes M1 :: "moebius"   and M2 :: "moebius" shows "moebius_pt (moebius_comp M1 M2) = moebius_pt M1 \<circ> moebius_pt M2"
autoref_FAIL: fixes R :: "('a \<times> 'b) set" shows "(FAIL, FAIL) \<in> \<langle>R\<rangle>nres_rel"
le_iff_sup: fixes x :: "'b"   and y :: "'b" shows "(x \<le> y) = (x \<squnion> y = y)"
atm_of_atms_subst: fixes As :: "'a list"   and \<sigma> :: "'s"   and Bs :: "'a list" shows "(atm_of_atms As \<cdot>a \<sigma> = atm_of_atms Bs) = (map (\<lambda>A. A \<cdot>a \<sigma>) As = Bs)"
cring_ltrm_sub: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "0 < deg R g"   and "f (deg R f) \<otimes> g (deg R g) [^] deg R f \<noteq> \<zero>" shows "monom P (Cring_Poly.compose R f g (deg R (Cring_Poly.compose R f g))) (deg R (Cring_Poly.compose R f g)) = monom P (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g (deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g))) (deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g))"
ex_graph_def: shows "ex_graph \<equiv> Lgraph {(home, cloud), (sphone, cloud), (cloud, hospital)} ex_loc_ass ex_creds ex_locs"
semiring_simprules_3: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
add_sheaf_spec_extensional: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set"   and a :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and b :: "'c set \<Rightarrow> ('c \<times> 'c) set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.add_sheaf_spec R addition multiplication zero U a b \<in> extensional U"
unit_in_hom_2: shows "\<guillemotleft>\<eta>\<^sub>0 : src e\<^sub>0 \<Rightarrow> d\<^sub>0 \<star> e\<^sub>0\<guillemotright>"
ereal_mono_minus_cancel: fixes c :: "ereal"   and a :: "ereal"   and b :: "ereal" assumes "c - a \<le> c - b"   and "0 \<le> c"   and "c < \<infinity>" shows "b \<le> a"
monoD: fixes f :: "'b \<times> 'c \<Rightarrow> 'd"   and x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "prod.mono f"   and "less_eq_prod x y" shows "f x \<le> f y"
multiplicity_characterization_nat: fixes S :: "nat set"   and f :: "nat \<Rightarrow> nat"   and p :: "nat"   and n :: "nat" assumes "S = {p. 0 < f p}"   and "finite S"   and "\<forall>p\<in>S. prime p"   and "prime p"   and "n = (\<Prod>p\<in>S. p ^ f p)" shows "multiplicity p n = f p"
cases: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c list" assumes "Graph_Defs.steps E a" obtains x :: "'c" where "a = [x__]"   | x :: "'c" and y :: "'c" and xs :: "'c list" where "a = x # y # xs" and "E x y" and "Graph_Defs.steps E (y # xs)"
LeftDerivationFix_splits_at_symbol: fixes a :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and b :: "symbol list" assumes "LeftDerivationFix a i D j b" shows "\<exists>U a1 a2 b1 b2 n. splits_at a i a1 U a2 \<and> splits_at b j b1 U b2 \<and> n \<le> length D \<and> LeftDerivation a1 (take n D) b1 \<and> derivation_ge (drop n D) (Suc (length b1)) \<and> LeftDerivation a2 (derivation_shift (drop n D) (Suc (length b1)) 0) b2 \<and> (n = length D \<or> n < length D \<and> is_word (b1 @ [U]))"
extremum_unique: fixes a :: "'c" shows "(top \<le> a) = (a = top)"
snd_map_prod: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'b"   and x :: "'d \<times> 'e" shows "snd (map_prod f g x) = g (snd x)"
hyps_20: shows "A\<^sub>Q \<sharp>* \<Psi>"
isLam_graph_aux_def: shows "isLam_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>isLam_aux x. x1 = Var x \<and> x2 = False) \<or> (\<exists>isLam_aux x e. x1 = Lam [x]. e \<and> x2 = True) \<or> (\<exists>isLam_aux e x. x1 = App e x \<and> x2 = False) \<or> (\<exists>isLam_aux as e. x1 = Let as e \<and> x2 = False) \<or> (\<exists>isLam_aux b. x1 = Bool b \<and> x2 = False) \<or> (\<exists>isLam_aux scrut e1 e2. x1 = (scrut ? e1 : e2) \<and> x2 = False))"
dist_fs_cvec_iff: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex" assumes "z \<noteq> vec_zero"   and "w \<noteq> vec_zero" shows "dist_fs_cvec z w = 2 * sqrt (1 - (cmod \<langle>z,w\<rangle>)\<^sup>2 / (\<langle>z\<rangle>\<^sup>2 * \<langle>w\<rangle>\<^sup>2))"
square_free_m_def: fixes f :: "int poly" shows "p.square_free_m f = (p.Mp f \<noteq> p.Mp 0 \<and> (\<forall>g. p.degree_m g \<noteq> 0 \<longrightarrow> \<not> p.dvdm (g * g) f))"
eq_inc_same: fixes a :: "Arity"   and v1 :: "Value"   and v2 :: "Value" assumes "eq a v1 v2" shows "eq (inc\<cdot>a) v1 v2"
assms_2: shows "consume (Trans s1 a1 ou1 s1') vl1 vl1'"
u1_3: shows "u1 = k1 \<otimes> a \<oplus> v1"
ex_power_ivl1: fixes b :: "nat"   and k :: "nat" assumes "2 \<le> b"   and "1 \<le> k" shows "\<exists>n. b ^ n \<le> k \<and> k < b ^ (n + 1)"
part_equivp_transp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "part_equivp R"   and "R x y"   and "R y z" shows "R x z"
mod_self: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a a = zero"
simps_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "('c \<times> nat) list" assumes "class.linorder less_eq less" shows "linorder.monom_mult_list less [] n = n"
fmadd_empty_2: fixes m :: "('a, 'b) fmap" shows "m ++\<^sub>f fmempty = m"
assms_1: shows "ns1 O s1 \<subseteq> s1"
pow_int_divide: fixes a :: "'b"   and x :: "'b"   and b :: "int" shows "a / pow_int x b = a * pow_int x (- b)"
qtable2: shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne \<psi>) rel2"
simps_2: fixes s :: "'b set"   and ss :: "'b set list"   and ts :: "'b set list" shows "partitioning1 (s # ss) ts = partList4 s (partitioning1 ss ts)"
Collect_case_prod_mono: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "A \<le> B" shows "{(x, y). A x y} \<subseteq> {(x, y). B x y}"
multiplicity_gt_zero_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "(0 < factorial_semiring.multiplicity times one p x) = dvd.dvd times p x"
cases: fixes a :: "hf" assumes "a \<in> Special_ax" obtains v :: "hf" and x :: "hf" and ax :: "hf" and y :: "hf" and sx :: "hf" where "a = q_Imp sx (q_Ex ax)" and "AbstForm v 0 x ax" and "SubstForm v y x sx" and "Form x" and "is_Var v" and "Term y"
OclInt2_def: shows "\<two> = (\<lambda>_. \<lfloor>\<lfloor>2\<rfloor>\<rfloor>)"
simps_8: fixes v :: "'c match_expr"   and vb :: "char list"   and rs :: "'c rule list" shows "process_ret (Rule v (Goto vb) # rs) = Rule v (Goto vb) # process_ret rs"
h0: shows "\<forall>x\<in>set (characterize_consistent_signs_at_roots p (qs1 @ qs2)). \<exists>x1\<in>set (characterize_consistent_signs_at_roots p qs1). \<exists>x2\<in>set (characterize_consistent_signs_at_roots p qs2). x = x1 @ x2"
verit_sum_simplify: fixes a :: "'a" shows "a + (0::'a) = a"
measure_space: fixes M :: "'c measure" shows "measure_space (space M) (sets M) (emeasure M)"
axioms_3: fixes G :: "('c, 'd) pre_digraph" assumes "digraph G" shows "nomulti_digraph G"
init_simps_7: fixes c :: "CTypeId" shows "init (CClassT c) = nullV"
prec_dom_independent: shows "\<forall>s d a a'. rcurrent s \<noteq> d \<and> rprecondition s d a \<longrightarrow> rprecondition (rstep s a') d a"
that: fixes P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> \<tau>.P \<Longrightarrow>\<^sub>\<tau> P''"   and "\<Psi> \<rhd> P' \<sim> P''" shows "thesis"
invertible_left_inverse: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "monoid M composition unit"   and "monoid.invertible M composition unit u"   and "u \<in> M" shows "composition (monoid.inverse M composition unit u) u = unit"
\<delta>_1: shows "subst_domain \<delta> = insert x {y \<in> S. \<Gamma> (Var y) = Var a}"