hcju/isabelleps huggingface dataset:
pr_strict_irrefl: fixes fn :: "'f \<times> nat" shows "\<not> pr_strict fn fn"
chain3: fixes x :: "'b" shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>f. f x) ` Z)"
ord_le_eq_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a \<le>p b"   and "b = c" shows "a \<le>p c"
min_elt_ex: fixes B :: "'a set" assumes "B \<subseteq> A"   and "B \<noteq> {}" shows "\<exists>x. x \<in> B \<and> (\<forall>y\<in>A. P y x \<longrightarrow> y \<notin> B)"
fls_regpart_deriv: fixes f :: "'a fls" shows "fls_regpart (fls_deriv f) = fps_deriv (fls_regpart f)"
mono_aval': fixes S :: "'a Abs_State.st"   and S' :: "'a Abs_State.st"   and e :: "aexp" assumes "S \<sqsubseteq> S'" shows "aval' e S \<sqsubseteq> aval' e S'"
mult_mult_of: fixes R :: "('a, 'b) ring_scheme" shows "(\<otimes>\<^bsub>Multiplicative_Group.mult_of R\<^esub>) = (\<otimes>\<^bsub>R\<^esub>)"
dgrad_set_le_def: fixes d :: "'b \<Rightarrow> nat"   and S :: "'b set"   and T :: "'b set" shows "dgrad_set_le d S T = (\<forall>s\<in>S. \<exists>t\<in>T. d s \<le> d t)"
int_xor_assoc: fixes x :: "int"   and y :: "int"   and z :: "int" shows "(x XOR y) XOR z = x XOR y XOR z"
N_1: shows "{x \<in> space M. \<not> (\<forall>i. f i x \<le> f (Suc i) x)} \<subseteq> N"
plus_list_ub2: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> A"   and "set ys \<subseteq> A"   and "length xs = length ys" shows "ys [\<sqsubseteq>\<^bsub>r\<^esub>] xs [+f] ys"
order_trans: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
sibling_induct_consistent: fixes t :: "'a tree"   and P :: "'a tree \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "consistent t"   and "\<And>w b a. P (Leaf w b) a"   and "\<And>w w\<^sub>b b w\<^sub>c c a. b \<noteq> c \<Longrightarrow> P (Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; 0 < height t\<^sub>1 \<or> 0 < height t\<^sub>2; a \<in> alphabet t\<^sub>1; sibling t\<^sub>1 a \<in> alphabet t\<^sub>1; a \<notin> alphabet t\<^sub>2; sibling t\<^sub>1 a \<notin> alphabet t\<^sub>2; P t\<^sub>1 a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; 0 < height t\<^sub>1 \<or> 0 < height t\<^sub>2; a \<notin> alphabet t\<^sub>1; sibling t\<^sub>2 a \<notin> alphabet t\<^sub>1; a \<in> alphabet t\<^sub>2; sibling t\<^sub>2 a \<in> alphabet t\<^sub>2; P t\<^sub>2 a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; 0 < height t\<^sub>1 \<or> 0 < height t\<^sub>2; a \<notin> alphabet t\<^sub>1; a \<notin> alphabet t\<^sub>2\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a" shows "P t a"
Pstep: fixes x :: "'a" assumes "P x" shows "\<exists>y. P y \<and> (x, y) \<in> r"
card_PiE: fixes S :: "'c set"   and T :: "'c \<Rightarrow> 'd set" assumes "finite S" shows "card (Pi\<^sub>E S T) = (\<Prod>i\<in>S. card (T i))"
nth_drop: fixes n :: "nat"   and xs :: "'g list"   and i :: "nat" assumes "n \<le> length xs" shows "drop n xs ! i = xs ! (n + i)"
assms_2: fixes x :: "nat" assumes "x dvd n"   and "odd x" shows "x = 1"
reflp_refl_eq: fixes r :: "('a \<times> 'a) set" shows "reflp (\<lambda>x y. (x, y) \<in> r) = refl r"
LCons: fixes xs :: "'a llist"   and ys :: "'b llist"   and x :: "'a"   and y :: "'b" assumes "lfinite xs"   and "lfinite ys"   and "P xs ys" shows "P (LCons x xs) (LCons y ys)"
double_compl: fixes x :: "'b upred" shows "(\<not> \<not> x) = x"
iint_less_eq: fixes a :: "'a idx_interval"   and b :: "'a idx_interval" shows "(a \<le> b) = (idx_interval.int a < idx_interval.int b \<or> idx_interval.int a = idx_interval.int b \<and> idx a \<le> idx b)"
that: fixes ks :: "'ord stream" assumes "alw (\<lambda>ks. (shd (stl ks), shd ks) \<in> Ord) ks"   and "alw (ev (\<lambda>ks. shd (stl ks) \<noteq> shd ks)) ks" shows "thesis"
wf_disc_sel_common_primitive_2: shows "wf_disc_sel (is_Dst_Ports, dst_ports_sel) Dst_Ports"
simps_1: fixes x :: "'a"   and uu :: "'a list" shows "safe_hd (x # uu) = Some x"
stutter_reduced_equivalent: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "\<natural>\<sigma> \<approx> \<sigma>"
square_free_factorization_def: fixes p :: "'a poly"   and cbs :: "'a \<times> ('a poly \<times> nat) list" shows "local.square_free_factorization p cbs \<equiv> case cbs of (c, bs) \<Rightarrow> p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i) \<and> (p = 0 \<longrightarrow> c = (0::'a) \<and> bs = []) \<and> (\<forall>a i. (a, i) \<in> set bs \<longrightarrow> square_free a \<and> 0 < degree a) \<and> (\<forall>a i b j. (a, i) \<in> set bs \<longrightarrow> (b, j) \<in> set bs \<longrightarrow> (a, i) \<noteq> (b, j) \<longrightarrow> coprime a b) \<and> distinct bs"
simps: fixes i :: "nat" shows "h i = (if i \<le> 1 then 1 else if i = 2 then f 2 ^ \<delta> 1 else f i ^ \<delta> (i - 1) / h (i - 1) ^ (\<delta> (i - 1) - 1))"
adjugate_transpose: fixes A :: "('c, 'd) sq_matrix" shows "adjugate (Square_Matrix.transpose A) = Square_Matrix.transpose (adjugate A)"
If_def: fixes P :: "bool"   and x :: "'d"   and y :: "'d" shows "if P then x else y \<equiv> THE z. (P = True \<longrightarrow> z = x) \<and> (P = False \<longrightarrow> z = y)"
subsums_def: fixes c\<^sub>2 :: "('a, 'b) conf"   and c\<^sub>1 :: "('a, 'b) conf" shows "c\<^sub>2 \<sqsubseteq> c\<^sub>1 \<equiv> states c\<^sub>2 \<subseteq> states c\<^sub>1"
moebius_mb_cmat_def: fixes I :: "complex \<times> complex \<times> complex \<times> complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" shows "moebius_mb_cmat I M = similarity I M"
S5nI: fixes M :: "('b, 'd, 'c) KripkeStructure" assumes "\<And>a. equiv (worlds M) (relations M a)" shows "S5n M"
simps_6: fixes V :: "'c set"   and t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and S :: "('d, 'c) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (insert\<langle>t,s\<rangle> # S) = (fv t \<subseteq> V \<and> fv s \<subseteq> V \<and> wf'\<^sub>s\<^sub>s\<^sub>t V S)"
nop_onto_n: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_onto A n f" shows "n \<in>\<^sub>\<circ> \<omega>"
prm_set_id: fixes S :: "'b set" shows "\<epsilon> {$} S = S"
inf_not_lt: fixes x :: "'d DBMEntry" shows "(\<infinity> \<prec> x) = False"
sup: fixes x :: "nat" shows "(SUP i\<in>Y. f i x) = Sup (range (g x))"
entailment_implies_validity: fixes atom_ordering :: "('a \<times> 'a) set"   and S1 :: "'a Literal set set"   and S2 :: "'a Literal set set"   and I :: "'a set" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.entails_formula S1 S2"   and "propositional_atoms.validate_formula I S1" shows "propositional_atoms.validate_formula I S2"
chamber_in_subcomplex: fixes Y :: "'b set set"   and C :: "'b set" assumes "codomain.ChamberSubcomplex Y"   and "C \<in> Y"   and "codomain.chamber C" shows "SimplicialComplex.maxsimp Y C"
translation_galois: fixes T :: "'c set"   and a :: "'c"   and S :: "'c set" shows "(T = (+) a ` S) = (S = (+) (- a) ` T)"
f_Exec_Stream_causal: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "xs \<down> n = ys \<down> n" shows "f_Exec_Comp_Stream trans_fun xs c \<down> n = f_Exec_Comp_Stream trans_fun ys c \<down> n"
select_convs_3: fixes iiface :: "iface"   and oiface :: "iface"   and src :: "'a word \<times> nat"   and dst :: "'a word \<times> nat"   and proto :: "protocol"   and sports :: "16 word \<times> 16 word"   and dports :: "16 word \<times> 16 word"   and more :: "'b" shows "src \<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports, \<dots> = more\<rparr> = src"
h: shows "h \<in> lspace (lebesgue_on S) (ennreal p)"
cases: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" assumes "a1 \<R>\<lbrakk>\<cdot>\<rbrakk> a2" obtains S :: "'procS" where "S__ \<in>S a1" and "\<lbrakk>S__\<rbrakk> \<in>T a2"   | S :: "'procS" where "\<lbrakk>S\<rbrakk> \<in>T a1" and "S \<in>S a2"
less_complex_def: fixes x :: "complex"   and y :: "complex" shows "x < y \<equiv> Im x = Im y \<and> Re x < Re y"
eval_final_same: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "final e" shows "e = e' \<and> s = s'"
bound_on_all_plans_bounds_problem_plan_bound_: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and P :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> bool"   and f :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> nat" assumes "\<forall>PROB' as s. finite PROB \<and> P PROB' \<and> s \<in> valid_states PROB' \<and> as \<in> valid_plans PROB' \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB')"   and "P PROB"   and "finite PROB" shows "problem_plan_bound PROB < f PROB"
bound_child_parent_lemma_s_2: fixes P :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> bool"   and f :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> ('c, bool) fmap \<Rightarrow> nat"   and PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap" assumes "\<forall>PROB as s. finite PROB \<and> P PROB \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB s)" shows "finite PROB \<and> P PROB \<and> s \<in> valid_states PROB \<longrightarrow> problem_plan_bound_s PROB s < f PROB s"
test_distrib: fixes p :: "'a"   and q :: "'a"   and y :: "'a"   and x :: "'a" assumes "test p"   and "test q" shows "(p + q) \<cdot> (q \<cdot> y + !q \<cdot> x) = q \<cdot> y + !q \<cdot> p \<cdot> x"
valid_index_weave_1: fixes is1 :: "nat list"   and ds :: "nat list"   and A :: "nat set"   and is2 :: "nat list" assumes "is1 \<lhd> nths ds A"   and "is2 \<lhd> nths ds (- A)" shows "weave A is1 is2 \<lhd> ds"
sublist_drop: fixes n :: "nat"   and xs :: "'b list" shows "drop n xs \<le>f xs"
currentLevelPrefixToLevelEq: fixes level :: "nat"   and M :: "('b \<times> bool) list" assumes "level \<le> currentLevel M" shows "currentLevel (prefixToLevel level M) = level"
postpreList_inv: fixes S :: "(char list \<Rightarrow> int) \<Rightarrow> 'a"   and s :: "char list \<Rightarrow> int"   and C :: "acom"   and upds :: "('b \<times> ('a \<Rightarrow> nat)) list"   and l :: "'b \<Rightarrow> nat" assumes "S s = S (postQ C s)" shows "postList (map (\<lambda>(x, e). (x, \<lambda>s. e (S s))) upds) l s = preList (map (\<lambda>(x, e). (x, \<lambda>s. e (S s))) upds) C l s"
finite_fv_regex: fixes r :: "'a Regex.regex"   and fv :: "'a \<Rightarrow> 'b set" assumes "\<And>z. z \<in> regex.atms r \<Longrightarrow> finite (fv z)" shows "finite (Regex.fv_regex fv r)"
reflexive: fixes a :: "'a" assumes "a \<in> S" shows "(a, a) \<in> E"
coretract_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "coretract G \<equiv> galois_connection G \<and> coretract_axioms G"
obtain_v1_v2_2: shows "v2 \<in> vs"
lcoset_eq_reps_subset: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "order.greater_eq (a +o B) (a +o A)" shows "order.greater_eq B A"
Inf_synlr_def: fixes A :: "('a, 'b) synlr set" shows "Inf A \<equiv> mksynlr (\<Inter> (unsynlr ` A))"
self_finance_updated: fixes Mkt :: "('c, 'd) discrete_market"   and asset :: "'d"   and n :: "nat"   and w :: "'c"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and v :: "real" assumes "prices Mkt asset (Suc n) w \<noteq> 0"   and "portfolio pf" shows "cls_val_process Mkt (self_finance Mkt v pf asset) (Suc n) w = cls_val_process Mkt pf (Suc n) w + prices Mkt asset (Suc n) w * remaining_qty Mkt v pf asset (Suc n) w"
leftderives_rule_step: fixes a :: "symbol list"   and u :: "symbol list"   and N :: "symbol"   and v :: "symbol list"   and w :: "symbol list" assumes "leftderives a (u @ [N] @ v)"   and "(N, w) \<in> \<RR>"   and "is_word u"   and "is_sentence v" shows "leftderives a (u @ w @ v)"
ContraAssume: fixes B :: "fmla"   and H :: "fmla set"   and A :: "fmla" assumes "Robinson_Arithmetic.neg B \<in> H" shows "insert B H \<turnstile> A"
exec_step_ind_Return_Init: fixes D :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "jvm_method"   and P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and h :: "heap"   and stk\<^sub>0 :: "val list"   and loc\<^sub>0 :: "val list"   and pc :: "nat"   and ics :: "init_call_status"   and stk' :: "val list"   and loc' :: "val list"   and C' :: "char list"   and m' :: "char list"   and pc' :: "nat"   and ics' :: "init_call_status"   and frs' :: "frame list"   and sh :: "sheap" assumes "(D, b, Ts, T, m) = method P C\<^sub>0 clinit" shows "exec_step_ind (StepI Return) P h stk\<^sub>0 loc\<^sub>0 C\<^sub>0 clinit pc ics ((stk', loc', C', m', pc', ics') # frs') sh (None, h, (stk', loc', C', m', pc', ics') # frs', sh(C\<^sub>0 \<mapsto> (fst (the (sh C\<^sub>0)), Done)))"
F2_get_f: shows "Finca_get F2 f = Some fd2"
In0_eq: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" shows "(In0 M = In0 N) = (M = N)"
in_encrypt_oracle: fixes x :: "'a" shows "callee_invariant hash_oracle'' (\<lambda>(s, uu_). x \<in> s)"
assms: shows "low_equal s1 s2 \<and> fst (memory_read 8 addra s1) = fst (memory_read 8 addra s2)"
ide_img: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f" shows "partial_magma.ide S (set_category.img S \<AA> f)"
iterates_above_triangle: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b"   and y :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"   and "y \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"   and "a \<in> Field leq" shows "x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f y \<or> y \<in> bourbaki_witt_fixpoint.iterates_above lub leq f x"
bot_similar'_step: fixes s :: "Value \<Rightarrow> CValue \<Rightarrow> bool" shows "similar'_step s \<bottom> \<bottom>"
prj_simps_6: shows "C.dom \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0 = HHH\<mu>\<nu>\<pi>\<rho>.chine"
term_ok'_betapply: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and u :: "Core.term" assumes "term_ok' (sig \<Theta>) f"   and "term_ok' (sig \<Theta>) u" shows "term_ok' (sig \<Theta>) (f \<bullet> u)"
ntrs_replace_context_s: fixes s :: "'a list"   and c :: "'a list multiset"   and cr :: "'a list multiset"   and ee :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and crn :: "'a list multiset" assumes "((s, c + cr), ee, s', c' + cr) \<in> ntrs fg"   and "mon_c fg crn \<subseteq> mon_c fg cr" shows "((s, c + crn), ee, s', c' + crn) \<in> ntrs fg"
assign\<^sub>1: fixes x :: "'Var"   and \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'AExp"   and t :: "Sec" assumes "x \<notin> dom \<Gamma>"   and "\<Gamma> \<turnstile>\<^sub>a e \<in> t"   and "t \<sqsubseteq> dma x" shows "\<turnstile> \<Gamma> {x \<leftarrow> e} \<Gamma>"
mult_le_cancel_left1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq c (times c b) = ((less zero c \<longrightarrow> less_eq one b) \<and> (less c zero \<longrightarrow> less_eq b one))"
norm_ge_zero: fixes x :: "'a" shows "0 \<le> norm x"
g: fixes z :: "complex" assumes "z \<in> cball 0 1" shows "h z = cos (complex_of_real pi * g z)"
primitive_extension_def: shows "primitive_extension = (SOME \<Phi>. primitive_dchar conductor \<Phi> \<and> (\<forall>k. \<chi> k = \<Phi> k * principal_dchar n k))"
c_fresh1: shows "atom c \<sharp> trans_fm [i] A"
vseq_implies_hpar_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "seq \<nu> \<mu>" shows "src \<nu> = src \<mu>"
span_base: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and S :: "'d set" assumes "module scale"   and "a \<in> S" shows "a \<in> module.span scale S"
get_owner_document_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok Shadow_DOM.get_owner_document ptr"
less_Suc_induct: fixes i :: "nat"   and j :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> bool" assumes "i < j"   and "\<And>i. P i (Suc i)"   and "\<And>i j k. \<lbrakk>i < j; j < k; P i j; P j k\<rbrakk> \<Longrightarrow> P i k" shows "P i j"
gcd_mult_distrib: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "times k (gcd a b) = times (gcd (times k a) (times k b)) (unit_factor k)"
S_lemma: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and ec :: "bool"   and cc :: "bool"   and rc :: "bool"   and cs :: "mClkState"   and rs :: "rpcState"   and hs1 :: "histState"   and hs2 :: "histState" shows "\<turnstile> unchanged (e p, c p, r p, m p, rmhist!p) \<longrightarrow> unchanged S rmhist ec cc rc cs rs hs1 hs2 p"
lookup_pair_map_pair: fixes xs :: "('a \<times> 'e) list"   and f :: "'a \<times> 'e \<Rightarrow> 'a \<times> 'f"   and k :: "'a"   and v :: "'e" assumes "oalist_inv_raw xs"   and "snd (f (k, 0::'e)) = (0::'f)"   and "\<And>a b. comp (fst (f a)) (fst (f b)) = comp (fst a) (fst b)" shows "lookup_pair (map_pair f xs) (fst (f (k, v))) = snd (f (k, lookup_pair xs k))"
axioms_1: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah" assumes "diagram J C D" shows "category C"
certificate_valid: fixes B :: "agent"   and KB :: "nat"   and evs :: "event list" assumes "certificate B KB \<in> parts (knows Spy evs)"   and "evs \<in> tls" shows "KB = pubK B"
order_iff: fixes a :: "'g"   and b :: "'g" shows "(a \<le> b) = (a = min a b)"
max_norm_set_proptys: fixes A :: "(('b, 'd) vec, 'c) vec" shows "finite {\<bar>A $ i $ j\<bar> |i j. i \<in> UNIV \<and> j \<in> UNIV}"
abc_Hoare_unhalt_def: fixes P :: "nat list \<Rightarrow> bool"   and p :: "abc_inst list" shows "{P} p \<up> \<equiv> \<forall>args. P args \<longrightarrow> (\<forall>n. abc_notfinal (abc_steps_l (0, args) p n) p)"
starfun: fixes f :: "'d \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'd" shows "(*f* f) (star_n X) = star_n (\<lambda>n. f (X n))"
that: fixes m :: "'c multiset" assumes "count m i = length Qs + k"   and "indexed_eval_aux (Qs @ [P]) i m \<noteq> \<zero>" shows "thesis"
update_i_def: fixes ba :: "'b list \<times> 'b list"   and x :: "'b" shows "update_i ba x = (fst ba, x # tl (snd ba))"
maxsimp_map_into_image: fixes x :: "'a set" assumes "domain.chamber x" shows "SimplicialComplex.maxsimp (f \<turnstile> X) (f ` x)"
SUP_mult_right_ennreal: fixes f :: "'a \<Rightarrow> ennreal"   and I :: "'a set"   and c :: "ennreal" shows "\<Squnion> (f ` I) * c = (\<Squnion>i\<in>I. f i * c)"
vars_def: fixes X :: "exp" shows "vars X = \<Union> (freevars ` Rep_Exp X)"
mon_s_uncons: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and a :: "'b"   and as :: "'b list" shows "mon_s fg (a # as) = mon_n fg a \<union> mon_s fg as"
strict_implies_order: fixes b :: "nat list"   and a :: "nat list" assumes "b <\<^sub>v a" shows "b \<le>\<^sub>v a"
INT_simps_10: fixes B :: "'b \<Rightarrow> 'a set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "\<Inter> (B ` f ` A) = (\<Inter>a\<in>A. B (f a))"
tau_step: fixes P :: "'state"   and P' :: "'state"   and P'' :: "'state" assumes "P \<rightarrow>\<^sub>S \<langle>S_action.Act \<tau>,P'\<rangle>"   and "P' \<Rightarrow>\<^sub>S P''" shows "P \<Rightarrow>\<^sub>S P''"
RG_proj_mult_rightdelta': fixes g' :: "'g"   and r :: "'a"   and g :: "'g"   and s :: "'a" assumes "g' \<in> H" shows "Supgroup.RG_proj (r \<delta>\<delta> g * (s \<delta>\<delta> g')) = Supgroup.RG_proj (r \<delta>\<delta> g) * (s \<delta>\<delta> g')"
elims_3: fixes x :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xa :: "'g list" assumes "\<not> sorted_wrt x xa" obtains P :: "'g \<Rightarrow> 'g \<Rightarrow> bool" and xb :: "'g" and ys :: "'g list" where "x = P" and "xa = xb # ys" and "\<not> (Ball (set ys) (P xb) \<and> sorted_wrt P ys)"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "'a vec" assumes "a x xa = y" obtains i :: "nat" where "x = ia__" and "xa = 0" and "y = fs ! ia__"   | i :: "nat" and l :: "nat" where "x = i" and "xa = Suc l" and "y = (1::'a) / d l \<cdot>\<^sub>v (d (Suc l) \<cdot>\<^sub>v a i l - \<mu>' i l \<cdot>\<^sub>v gso' l)"
left_neutral: fixes a :: "'c" shows "(0::'c) XOR a = a"
g'_def: shows "g' = rec_nat x0' (\<lambda>i x. SOME x'. (x, x') \<in> S' \<and> (f' (Suc i), x') \<in> R \<and> (x0', x') \<in> S'\<^sup>*)"
eventually_nonneg_bigtheta_pow_realpow: fixes f :: "primfun \<times> real"   and e :: "nat" shows "\<Theta>(\<lambda>x. eval_primfun f x ^ e) = \<Theta>(\<lambda>x. eval_primfun f x powr real e)"
unfold_simps_2: fixes A :: "'y set"   and B :: "'y set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
interchange: shows "(c::'a::refinement_lattice) \<iinter> (d::'a::refinement_lattice) \<sqsubseteq> c // (i::'a::refinement_lattice) \<iinter> (j::'a::refinement_lattice) \<parallel> i \<iinter> d // j"
hyps_3: assumes "Q root'"   and "\<forall>a\<in>set xs. P a" shows "Q root''"
fv_not_fresh: fixes x :: "'c"   and e :: "'b" shows "atom x \<sharp> e = (x \<notin> fv e)"
sorted_filter: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'f"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map f xs)" shows "linorder.sorted less_eq (map f (filter P xs))"
c_tr_map: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s :: "'g"   and xs :: "'e list" shows "map fst (c_tr step out s xs) = xs"
is_fair_run_def: fixes s :: "nat \<Rightarrow> 'a" shows "A.is_fair_run s \<equiv> \<exists>l. A.is_lrun l s \<and> A.is_lfair l s"
notin_fset: fixes S :: "'b fset"   and x :: "'b" shows "(x |\<notin>| S) = (x \<notin> fset S)"
add_mset_inter_add_mset: fixes a :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" shows "add_mset a A \<inter># add_mset a B = add_mset a (A \<inter># B)"
sel_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "hd (x21 # x22) = x21"