hcju/isabelleps huggingface dataset:
evalF_evalC_induct: fixes P :: "((d \<times> d list \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour) discr \<rightarrow> ((nat \<times> (nat \<Rightarrow> contour option)) \<times> d) set) \<Rightarrow> ((call \<times> (nat \<Rightarrow> contour option) \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour) discr \<rightarrow> ((nat \<times> (nat \<Rightarrow> contour option)) \<times> d) set) \<Rightarrow> bool" assumes "adm (\<lambda>(evalF, evalC). P evalF evalC)"   and "P \<bottom> \<bottom>"   and "\<And>a b. P a b \<Longrightarrow> P (\<Lambda> fstate. case undiscr fstate of (DI int, ba) \<Rightarrow> \<bottom> | (DC (Lambda lab vs c, \<beta>), as, ve, ba) \<Rightarrow> if length vs = length as then let \<beta>' = \<beta>(lab \<mapsto> ba); ve' = ve(map (\<lambda>v. (v, ba)) vs [\<mapsto>] as) in b\<cdot>(Discr (c, \<beta>', ve', ba)) else \<bottom> | (DP (Plus c), [], bb) \<Rightarrow> \<bottom> | (DP (Plus c), [DI a1], bb) \<Rightarrow> \<bottom> | (DP (Plus c), [DI a1, DI a2], bb) \<Rightarrow> \<bottom> | (DP (Plus c), [DI a1, DI a2, cnt], ve, b) \<Rightarrow> if isProc cnt then let b' = nb b c; \<beta> = [c \<mapsto> b] in a\<cdot>(Discr (cnt, [DI (a1 + a2)], ve, b')) \<union> {((c, \<beta>), cnt)} else \<bottom> | (DP (Plus c), DI a1 # DI a2 # cnt # aa # lista, bb) \<Rightarrow> \<bottom> | (DP (Plus c), DI a1 # _ # lista, bb) \<Rightarrow> \<bottom> | (DP (Plus c), _ # list, bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), [], bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), [DI v], bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), [DI v, contt], bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), [DI v, contt, contf], ve, b) \<Rightarrow> if isProc contt \<and> isProc contf then if v \<noteq> 0 then let b' = nb b ct; \<beta> = [ct \<mapsto> b] in a\<cdot>(Discr (contt, [], ve, b')) \<union> {((ct, \<beta>), contt)} else let b' = nb b cf; \<beta> = [cf \<mapsto> b] in a\<cdot>(Discr (contf, [], ve, b')) \<union> {((cf, \<beta>), contf)} else \<bottom> | (DP (prim.If ct cf), DI v # contt # contf # ab # lista, bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), _ # list, bb) \<Rightarrow> \<bottom> | (Stop, [], bb) \<Rightarrow> \<bottom> | (Stop, [DI i], x, xa) \<Rightarrow> {} | (Stop, DI i # a # lista, bb) \<Rightarrow> \<bottom> | (Stop, _ # list, bb) \<Rightarrow> \<bottom>) (\<Lambda> cstate. case undiscr cstate of (App lab f vs, \<beta>, ve, b) \<Rightarrow> let f' = \<A> f \<beta> ve; as = map (\<lambda>v. \<A> v \<beta> ve) vs; b' = nb b lab in if isProc f' then a\<cdot>(Discr (f', as, ve, b')) \<union> {((lab, \<beta>), f')} else \<bottom> | (call.Let lab ls c', \<beta>, ve, ba) \<Rightarrow> let b' = nb ba lab; \<beta>' = \<beta>(lab \<mapsto> b'); ve' = ve ++ map_of (map (\<lambda>(v, l). ((v, b'), \<A> (L l) \<beta>' ve)) ls) in b\<cdot>(Discr (c', \<beta>', ve', b')))" shows "P \<F> \<C>"
simps_2: fixes a :: "Arity"   and as :: "Arity list"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "transform_alts (a # as) (Alts e1 e2 # S) = Alts (ccTransform a e1) (ccTransform a e2) # transform_alts as S"
trancl_power: fixes p :: "'b \<times> 'b"   and R :: "('b \<times> 'b) set" shows "(p \<in> R\<^sup>+) = (\<exists>n>0. p \<in> R ^^ n)"
traversed_states_nempty: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "traversed_states s as \<noteq> {}"
same_output_sig: shows "outputs (ioa.asig composition) = outputs (ioa.asig (ioa 0 id2))"
GuardsFlip_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/- F\<^esub> P c UNIV,UNIV" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P c Q,A"
scale_spmf_def: fixes r :: "real"   and p :: "'c spmf" shows "scale_spmf r p = embed_spmf (\<lambda>x. min (inverse (weight_spmf p)) (max 0 r) * spmf p x)"
group_closure_minus_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and s :: "'a"   and S :: "'a set" assumes "class.ab_group_add plus zero minus uminus" shows "(uminus s \<in> ab_group_add.group_closure zero minus S) = (s \<in> ab_group_add.group_closure zero minus S)"
sets: shows "sets (f (SOME x. x \<in> A \<and> x \<in> space M)) = sets N"
null_eqI: fixes n :: "'d" assumes "\<And>f. n \<cdot>\<^sub>D f = n \<and> f \<cdot>\<^sub>D n = n" shows "n = D.null"
xa_def: shows "A2 = mat_of_rows (dim_col A) (map (Matrix.row A) [1..<dim_row A])"
b: shows "local.prefixes (Suc k) j \<down>= b"
resNil_1: fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x>x<y>.P \<sim> \<zero>"
subset_\<P>\<Q>k: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u \<subseteq> LocalLexing.\<Q> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k"
one_monom_def: shows "1 \<equiv> Abs_monom []"
Keys_to_list_eq_pps_to_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "Keys_to_list ps = pps_to_list (Keys (set ps))"
assms: shows "\<forall>v\<in>fset vs. \<phi> v = \<phi>' v"
mat_to_polys_mapping_of: fixes ts :: "('b, 'c) pp list"   and m :: "'d mat" assumes "distinct ts"   and "dim_col m = length ts" shows "mat_to_polys (map pp.mapping_of ts) m = map (Poly_Mapping.map_key PP) (mat_to_polys ts m)"
sound_sum: fixes P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (\<lambda>s. P s + Q s)"
complex_Im_of_nat: fixes n :: "nat" shows "Im (of_nat n) = 0"
simps: fixes i :: "nat"   and u :: "nat"   and c :: "'c \<Rightarrow> bool Heap"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'c Heap"   and s :: "'c" shows "imp_for i u c f s = (if u \<le> i then return s else c s \<bind> (\<lambda>ctn. if ctn then f i s \<bind> imp_for (i + 1) u c f else return s))"
Wpl_def: fixes i :: "int"   and t :: "int tree" shows "Wpl i t = (\<Sum>(k, c)\<in>nodes t. c * b k) + (\<Sum>(k, c)\<in>leaves i t. c * a k)"
integral_empty: fixes f :: "'d \<Rightarrow> 'c" shows "integral {} f = (0::'c)"
Class_revI: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). x \<sim> y}" shows "b \<in> rel.Class a"
pre_splitFace_pre_split_face: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and nvs :: "nat list" assumes "pre_splitFace g ram1 ram2 oldF nvs" shows "pre_split_face oldF ram1 ram2 nvs"
induct: fixes x :: "dbfm"   and P :: "dbfm \<Rightarrow> bool" assumes "wf_dbfm x"   and "\<And>t1 t2. \<lbrakk>wf_dbtm t1; wf_dbtm t2\<rbrakk> \<Longrightarrow> P (DBMem t1 t2)"   and "\<And>t1 t2. \<lbrakk>wf_dbtm t1; wf_dbtm t2\<rbrakk> \<Longrightarrow> P (DBEq t1 t2)"   and "\<And>A1 A2. \<lbrakk>wf_dbfm A1; P A1; wf_dbfm A2; P A2\<rbrakk> \<Longrightarrow> P (DBDisj A1 A2)"   and "\<And>A. \<lbrakk>wf_dbfm A; P A\<rbrakk> \<Longrightarrow> P (DBNeg A)"   and "\<And>A name. \<lbrakk>wf_dbfm A; P A\<rbrakk> \<Longrightarrow> P (DBEx (abst_dbfm name 0 A))" shows "P x"
unitary_intro: fixes a :: "'a" assumes "UNIV = {a}" shows "OFCLASS('a, unitary_class)"
labels_def: fixes ss :: "'b \<times> ('a \<times> 'b) list" shows "labels ss = map fst (snd ss)"
closure_of_complement: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of (topspace X - S) = topspace X - X interior_of S"
sgn_abs: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (sgn a) = zero_neq_one.of_bool one zero (a \<noteq> zero)"
Func_elim: fixes g :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and a :: "'d" assumes "g \<in> Func A B"   and "a \<in> A" shows "\<exists>b. b \<in> B \<and> g a = b"
troeger: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y)\<^sup>\<star> \<cdot> z = x\<^sup>\<star> \<cdot> (y \<cdot> (x + y)\<^sup>\<star> \<cdot> z + z)"
ball_maxscope: fixes m :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "bool" assumes "\<forall>x\<in>m. P x \<or> Q" shows "(\<forall>x\<in>m. P x) \<or> Q"
P_def: shows "P \<equiv> compP\<^sub>2 P\<^sub>1"
orbit_swap: fixes s :: "'e"   and f :: "'e \<Rightarrow> 'e"   and t :: "'e" assumes "s \<in> orbit f s"   and "t \<in> orbit f s" shows "s \<in> orbit f t"
induct: fixes P :: "tm \<Rightarrow> bool"   and a0 :: "tm" assumes "\<And>n. P (tm.Bound n)"   and "\<And>a. P a \<Longrightarrow> P (tm.Neg a)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (tm.Add a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (tm.Sub a b)"   and "\<And>c a. P a \<Longrightarrow> P (tm.Mul c a)"   and "\<And>n c a. P a \<Longrightarrow> P (CNP n c a)"   and "\<And>v. P (CP v)" shows "P a0"
simps: fixes G :: "prog"   and a1 :: "state"   and a2 :: "obj_tag"   and a3 :: "loc"   and a4 :: "state" shows "G\<turnstile>a1 \<midarrow>halloc a2\<succ>a3\<rightarrow> a4 = ((\<exists>x s oi. a1 = (Some x, s) \<and> a2 = oi \<and> a3 = undefined \<and> a4 = (Some x, s)) \<or> (\<exists>s a x oi' oi. a1 = Norm s \<and> a2 = oi \<and> a3 = a \<and> a4 = (x, init_obj G oi' (Inl a) s) \<and> new_Addr (heap s) = Some a \<and> (x, oi') = (if atleast_free (heap s) (Suc (Suc 0)) then (None, oi) else (Some (Xcpt (Loc a)), CInst (SXcpt OutOfMemory)))))"
range_from_nat_into: fixes A :: "'d set" assumes "A \<noteq> {}"   and "countable A" shows "range (from_nat_into A) = A"
assms: fixes i :: "nat" assumes "i \<in> pdevs_domain X" shows "P (pdevs_apply X i)"
g_bound_def: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and fs :: "int vec list" shows "LLL.g_bound n m fs_init fs = (\<forall>i<m. \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 \<le> rat_of_nat (LLL.N fs_init))"
flowsto_Union_funE: fixes Gs :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set set"   and X :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and T :: "real set"   and CX :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" assumes "finite Gs"   and "flowsto X T CX (\<Union> Gs)" obtains f :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set \<Rightarrow> ('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" where "\<And>G. G \<in> Gs \<Longrightarrow> flowsto (f G) T CX G" and "X = \<Union> (f ` Gs)"
L5_Controller_One: fixes y :: "nat \<Rightarrow> nat list"   and lIn :: "nat \<Rightarrow> bit"   and lOut :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y lIn lOut z"   and "lOut t = One"   and "z t = []" shows "lIn t = One"
prems_4: shows "nonneg_cexpr (set (v # vs) \<union> V') \<Gamma> e"
interleave_rev_cons2: fixes l :: "'b list"   and a :: "'b list"   and b :: "'b list"   and x :: "'b" assumes "l \<in> a \<otimes> b" shows "l @ [x] \<in> a \<otimes> b @ [x]"
in_set_butlastD: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set (butlast xs)" shows "x \<in> set xs"
summable_Re: fixes f :: "nat \<Rightarrow> complex" assumes "summable f" shows "summable (\<lambda>x. Re (f x))"
integral_mult_right: fixes S :: "'c set"   and c :: "'b"   and f :: "'c \<Rightarrow> 'b" shows "integral S (\<lambda>x. c * f x) = c * integral S f"
induced_graph_imp_graph: fixes H :: "('a, 'b) pre_digraph" assumes "induced_subgraph H G" shows "sym_digraph H"
ext_inject: fixes Obj :: "'d set"   and Mor :: "'e set"   and Dom :: "'e \<Rightarrow> 'd"   and Cod :: "'e \<Rightarrow> 'd"   and Id :: "'d \<Rightarrow> 'e"   and Comp :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and more :: "'f"   and Obj' :: "'d set"   and Mor' :: "'e set"   and Dom' :: "'e \<Rightarrow> 'd"   and Cod' :: "'e \<Rightarrow> 'd"   and Id' :: "'d \<Rightarrow> 'e"   and Comp' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and more' :: "'f" shows "(\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = \<lparr>Obj = Obj', Mor = Mor', Dom = Dom', Cod = Cod', Id = Id', Comp = Comp', \<dots> = more'\<rparr>) = (Obj = Obj' \<and> Mor = Mor' \<and> Dom = Dom' \<and> Cod = Cod' \<and> Id = Id' \<and> Comp = Comp' \<and> more = more')"
conjugate_comp: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "residuated f"   and "residuated g" shows "conjugate (f \<circ> g) = conjugate g \<circ> conjugate f"
map\<^sub>0_simp: fixes a :: "'c" assumes "C.obj a" shows "I\<^sub>C.map\<^sub>0 a = a"
comp_tfr\<^sub>s\<^sub>e\<^sub>t_M: shows "comp_tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> M"
valid_path_join_D2: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "valid_path (g1 +++ g2)"   and "pathfinish g1 = pathstart g2" shows "valid_path g2"
circ_mult_star: fixes x :: "'a" shows "x\<^sup>\<circ> * x\<^sup>\<star> = x\<^sup>\<circ>"
evalRR_sub: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "x \<in> carrier R" shows "eval R R id x (p \<ominus>\<^bsub>UP R\<^esub> q) = eval R R id x p \<ominus> eval R R id x q"
finite_lists_length_eq: fixes A :: "'a set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
assms: shows "\<nexists>t. local.prefixes t j \<up>"
eq_iff: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (a \<le> b \<and> b \<le> a)"
resultant_0_gcd: fixes f :: "'a poly"   and g :: "'a poly" shows "(resultant f g = (0::'a)) = (degree (gcd f g) \<noteq> 0)"
subst_notIn: fixes \<phi> :: "fm"   and t :: "tm"   and x :: "name" assumes "\<phi> \<in> UNIV"   and "t \<in> UNIV"   and "x \<in> UNIV"   and "x \<notin> Fvars \<phi>" shows "\<phi>(x::=t) = \<phi>"
pow2_le_abs: fixes a :: "real"   and b :: "real" shows "(a\<^sup>2 \<le> b\<^sup>2) = (\<bar>a\<bar> \<le> \<bar>b\<bar>)"
Theorem_5: fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "stable_on ds X"   and "x \<in> X" shows "\<exists>y\<in>cop ds. (x, y) \<in> Pd (Xd x)"
containsI: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "contains A x"
indicator_le_1: fixes S :: "'d set"   and x :: "'d" shows "indicator S x \<le> (1::'c)"
finite_sorted_edge_distinct: fixes S :: "'a set" assumes "finite S" obtains L :: "'a list" where "distinct L" and "wsorted L" and "S = set L"
invariantStateSpace_thm_5: fixes ss :: "('d, 'e) fmap set"   and domain :: "'d set"   and vs :: "'d set" assumes "stateSpace ss domain" shows "stateSpace (ss_proj ss vs) (domain \<inter> vs)"
int_sdiv_negated_is_minus1: fixes a :: "int"   and b :: "int" assumes "a \<noteq> 0" shows "(a sdiv b = - a) = (b = - 1)"
s_def: shows "s = punit.lt (rep_list p) - punit.lt (rep_list g)"
dvd_lcm1: fixes a :: "'a"   and b :: "'a" shows "a dvd lcm a b"
sum_Pinfty: fixes f :: "'b \<Rightarrow> ereal"   and P :: "'b set" shows "(sum f P = \<infinity>) = (finite P \<and> (\<exists>i\<in>P. f i = \<infinity>))"
mod_div_mult_eq: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (modulo a b) (times (divide a b) b) = a"
image_image: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
assms_1: fixes \<sigma> :: "nat \<Rightarrow> val" assumes "\<sigma> \<in> space (state_measure (V \<union> V') \<Gamma>)" shows "\<delta> \<sigma> = \<delta>' \<sigma>"
False: shows "\<nexists>d. \<psi> d \<in> \<R>"
sel_1: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "un_Node1 \<langle>x21, x22, x23\<rangle> = x21"
upt_rw: shows "[0..<Suc (Suc k)] = [0..<Suc k] @ [Suc k]"
fv_list\<^sub>s\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>s\<^sub>t: fixes S :: "('d, 'c) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t S = set (fv_list\<^sub>s\<^sub>s\<^sub>t S)"
conga_left_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.CongA Bet Cong C B A D E F"
intro: fixes B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b" assumes "partial_magma.ide B b" shows "constant_functor_axioms B b"
typing_map_typeof: fixes ves :: "e list"   and vs :: "v list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and tvs :: "t list" assumes "ves = $$* vs"   and "\<S>\<bullet>\<C> \<turnstile> ves : [] _> tvs" shows "tvs = map typeof vs"
twiddle_def: fixes \<beta> :: "(Location \<times> Location) set"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" shows "(s \<equiv>\<^sub>\<beta> t) = (noLowDPs s \<and> noLowDPs t \<and> fst s  \<approx>\<^sub>\<beta> fst t \<and> twiddleHeap \<beta> (snd s) (snd t))"
subst_type_graph_aux_def: shows "subst_type_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>subst_type_aux t' x. x1 = (One, t', x) \<and> x2 = One) \<or> (\<exists>subst_type_aux t1 t2 t' x. x1 = (Fun t1 t2, t', x) \<and> x2 = Fun (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t1, t', x)) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t2, t', x)) \<and> p (t1, t', x) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t1, t', x)) \<and> p (t2, t', x) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t2, t', x))) \<or> (\<exists>subst_type_aux t1 t2 t' x. x1 = (Syntax.Sum t1 t2, t', x) \<and> x2 = Syntax.Sum (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t1, t', x)) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t2, t', x)) \<and> p (t1, t', x) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t1, t', x)) \<and> p (t2, t', x) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t2, t', x))) \<or> (\<exists>subst_type_aux t1 t2 t' x. x1 = (Syntax.Prod t1 t2, t', x) \<and> x2 = Syntax.Prod (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t1, t', x)) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t2, t', x)) \<and> p (t1, t', x) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t1, t', x)) \<and> p (t2, t', x) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t2, t', x))) \<or> (\<exists>subst_type_aux y t' x t. x1 = (Mu y t, t', x) \<and> x2 = Mu y (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t, t', x)) \<and> atom y \<sharp> (t', x) \<and> p (t, t', x) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t, t', x))) \<or> (\<exists>subst_type_aux y t' x. x1 = (Alpha y, t', x) \<and> x2 = (if y = x then t' else Alpha y)) \<or> (\<exists>subst_type_aux t t' x. x1 = (AuthT t, t', x) \<and> x2 = AuthT (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t, t', x)) \<and> p (t, t', x) (((\<lambda>uu_. uu_) \<circ> subst_type_aux) (t, t', x))))"
assms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F G \<tau>"
of_int_eq_0_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus z = zero) = (z = 0)"
plane_of_def: fixes sctn :: "'e sctn" shows "plane_of sctn = plane (normal sctn) (pstn sctn)"
inj_comm_semiring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_comm_semiring_hom hom \<equiv> comm_semiring_hom hom \<and> inj_semiring_hom hom"
analz_insert_freshK3_bis: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list"   and authK' :: "nat" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "authK \<noteq> authK'"   and "authK' \<notin> range shrK"   and "evs \<in> kerbIV" shows "(Key servK \<in> analz (insert (Key authK') (knows Spy evs))) = (servK = authK' \<or> Key servK \<in> analz (knows Spy evs))"
alphaAbs_ex_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_ex_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
notTerminalNotSATAxiom: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "\<not> terminal subs gamma" shows "\<not> SATAxiom (sequent gamma)"
f_xss: shows "perm_restrict f (\<Union> (sset xss)) = lists_succ xss"
encode_operator_precondition_valid_1: fixes t :: "nat"   and op :: "(nat \<times> nat) strips_operator"   and sv :: "sat_plan_variable"   and f :: "'a \<Rightarrow> nat"   and asses :: "'a list" assumes "t < h"   and "op \<in> set ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>)"   and "sv \<in> atoms \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>) op))) \<^bold>\<or> Atom (State t (f v))) asses" shows "valid_state_var sv"
T_pass1': fixes hs :: "'b heap list" assumes "no_Emptys hs" shows "T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>1' hs = Pairing_Heap_Tree_Analysis.T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>1 (homs hs)"
spmf_conv_measure_spmf: fixes p :: "'b spmf"   and x :: "'b" shows "spmf p x = measure (measure_spmf p) {x}"
times_\<omega>_minus_numeral: fixes m :: "hmultiset"   and n :: "num" shows "m * \<omega> - numeral n = m * \<omega>"
l2_inv3I: fixes x :: "l2_state" assumes "\<And>Rb A B gnx. \<lbrakk>Auth A B \<langle>Number 1, gnx, Exp Gen (NonceF (Rb $ ny))\<rangle> \<in> chan x; A \<notin> bad x \<and> B \<notin> bad x\<rbrakk> \<Longrightarrow> \<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress x Ra) {xnx, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Ra xgnx = Some gnx \<and> guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny)))" shows "x \<in> l2_inv3"
num_sylow_dvd_remainder: shows "card (subgroups_of_size (q ^ 1)) dvd p"
wt_sym_gt_0: fixes f :: "'s" shows "0 < wt_sym f"
a_b_4: shows "q \<approx>[\<R>] mix_pmf \<beta> p r"
"2_3": fixes i :: "nat" assumes "i < m2" shows "(sel2 i \<longrightarrow> (as2 i, as2 (Suc i)) \<in> A) \<and> (\<not> sel2 i \<longrightarrow> (as2 i, as2 (Suc i)) \<in> B)"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and f3 :: "'e \<Rightarrow> 'd \<Rightarrow> 'c"   and step :: "('e, 'd) step" shows "P (case step of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = ((step = Done \<longrightarrow> P f1) \<and> (\<forall>x2. step = Skip x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x31 x32. step = Yield x31 x32 \<longrightarrow> P (f3 x31 x32)))"