hcju/isabelleps huggingface dataset:
compl_inf: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (inf x y) = sup (uminus x) (uminus y)"
geodesic_segment_param_in_geodesic_spaces_8: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<in> {x--y}" shows "z = geodesic_segment_param {x--y} x (dist x z)"
quotient_eq: shows "{f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // domain_and_range_permutation A B = {F \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B. univ (\<lambda>f. f ` A = B) F}"
forests_bot_2: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and e :: "'b"   and f :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.pd_kleene_allegory star one times sup less_eq less bot inf top conv uminus"   and "(less_eq one e \<and> less_eq (times e e) e) \<and> conv e = e"   and "less_eq (times f (conv f)) one \<and> less_eq (times f (star f)) (uminus one)" shows "inf (times (inf (uminus e) (conv f)) x) (times (inf e (conv f)) y) = bot"
E_LetV: fixes v :: "trm"   and p :: "pat"   and ts :: "trm list"   and t :: "trm" assumes "v \<in> value"   and "\<turnstile> p \<rhd> v \<Rightarrow> ts" shows "(LET p = v IN t) \<longmapsto>\<^sub>c t[0 \<mapsto>\<^sub>s ts]"
avl_size_lowerbound: fixes t :: "('b \<times> nat) tree" assumes "avl t" shows "((1 + sqrt 5) / 2) ^ height t \<le> real (size1 t)"
foldl_map: fixes g :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'b"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
asym_cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "less_eq x y" obtains "asympartp less_eq x y"   | "sympartp less_eq x y"
Cdef: shows "C = (\<Gamma>C, \<Delta>C)"
left_inverse: fixes a :: "'a perm" shows "inverse a * a = 1"
add_pow_rdistr: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "a \<in> carrier P"   and "b \<in> carrier P" shows "a \<otimes>\<^bsub>P\<^esub> [k] \<cdot>\<^bsub>P\<^esub> b = [k] \<cdot>\<^bsub>P\<^esub> (a \<otimes>\<^bsub>P\<^esub> b)"
SpecStuck: fixes s :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and n :: "nat" assumes "\<forall>t. (s, t) \<notin> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> =n\<Rightarrow> Stuck"
union: fixes M :: "'a llist set set" assumes "\<And>t. t \<in> M \<Longrightarrow> t open" shows "\<Union> M open"
single_subset_iff: fixes a :: "'d"   and M :: "'d multiset" shows "({#a#} \<subseteq># M) = (a \<in># M)"
diff_streams_only_if: fixes w :: "'b stream"   and x :: "'b stream" assumes "w \<noteq> x" shows "\<exists>n. w !! n \<noteq> x !! n"
simps_10: shows "T_f64 \<noteq> T_i64"
ds_dom_llp_strong: fixes x :: "'a"   and y :: "'a" shows "(x = d y \<cdot> x) = domain_semigroup.ds_ord d (\<cdot>) (d x) (d y)"
j_2: fixes i :: "nat" assumes "i < length YS" shows "YS ! i = XS ! j i"
elt_set_plus_def: fixes a :: "'b"   and B :: "'b set" shows "a +o B = {c. \<exists>b\<in>B. c = a + b}"
coinduct: fixes R :: "'b stream \<Rightarrow> 'b stream \<Rightarrow> bool"   and stream :: "'b stream"   and stream' :: "'b stream" assumes "R stream stream'"   and "\<And>stream stream'. R stream stream' \<Longrightarrow> shd stream = shd stream' \<and> R (stl stream) (stl stream')" shows "stream = stream'"
sub_kk: fixes i :: "nat"   and r :: "nat \<Rightarrow> 'a" shows "strict_mono (kk i r)"
simps: fixes t :: "('b ivl \<times> 'b) tree" shows "invar t = (inv_max_hi t \<and> Sorted_Less.sorted (Tree2.inorder t))"
cadjoint_univ_prop: fixes G :: "'b \<Rightarrow> 'c" assumes "bounded_clinear G" shows "\<forall>x y. \<langle>(G\<^sup>\<dagger>) x, y\<rangle> = \<langle>x, G y\<rangle>"
sum_slice2: fixes n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'"   and "n' \<in> sum_SDG_slice2 n" shows "n'' \<in> sum_SDG_slice2 n"
Plus_int: fixes b :: "nat"   and a :: "nat" shows "(+) b ` {..<a} = {b..<b + a}"
Red1CondT: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>if (true) e1 else e2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>e1,s\<rangle>"
Units_eq: shows "Units (DirProds G I) = carrier (DirProds G I)"
assms_2: fixes s :: "('v, 'es) state_scheme" assumes "DFS_invar G param s"   and "\<not> cond s" shows "P s"
Obj_Src: fixes t :: "'c local.term" assumes "Arr t" shows "Obj (Src t)"
active_threads_def: shows "active_threads \<equiv> \<lambda>s. {x. active_threadsp s x}"
adjust_idx_def: fixes i :: "nat"   and j :: "nat" shows "adjust_idx i j \<equiv> if j < i then j else Suc j"
that: fixes a :: "V"   and b :: "V" assumes "x = [a, b]\<^sub>\<circ>"   and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "thesis"
prems_6: shows "free_vars e \<subseteq> set (v # vs) \<union> V'"
shift_eq: fixes u :: "'a list"   and v :: "'a stream"   and w :: "'a stream" shows "(u @- v = w) = (u = stake (length u) w \<and> v = sdrop (length u) w)"
FG_advice_nested_prop_atoms\<^sub>\<mu>: fixes \<phi> :: "'b ltln"   and Y :: "'b ltln set" shows "nested_prop_atoms (\<phi>[Y]\<^sub>\<mu>) \<subseteq> nested_prop_atoms\<^sub>\<mu> \<phi> Y"
fcompI: fixes b :: "V"   and c :: "V"   and r :: "V"   and a :: "V"   and s :: "V" assumes "[b, c]\<^sub>\<circ> \<in>\<^sub>\<circ> r"   and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> s" shows "[a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<circ>\<^sub>\<bullet> s"
simps_8: fixes r :: "'a rexp"   and s :: "'a rexp"   and v :: "'a" shows "pnPlus (Plus r s) (Atom v) = pnPlus r (pnPlus s (Atom v))"
IfTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c\<^sub>1 :: "com"   and t :: "char list \<Rightarrow> int"   and c\<^sub>2 :: "com" assumes "bval b s"   and "(c\<^sub>1, s) \<Rightarrow> t" shows "(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t"
eval_power: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and abort :: "bool"   and F :: "'a"   and n :: "nat" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "eval (power_expansion abort F n) = (\<lambda>x. eval F x ^ n)"
convert_TA_initial_inject: fixes ta :: "('c, 'd, 'e, 'f, 'g, 'h) thread_action"   and ta' :: "('c, 'd, 'e, 'f, 'g, 'h) thread_action" shows "(convert_TA_initial ta = convert_TA_initial ta') = (ta = ta')"
unop_f64: fixes c :: "f64"   and fop :: "unop_f" shows "\<lparr>[$C ConstFloat64 c, $Unop_f T_f64 fop]\<rparr> \<leadsto> \<lparr>[$C ConstFloat64 (app_unop_f fop c)]\<rparr>"
positive_is_hermitian: fixes A :: "complex mat" assumes "positive A" shows "hermitian A"
A1: shows "c_assoc_have_key (pr_gr x) key = 0"
tame5: fixes p :: "nat"   and g :: "graph" assumes "Seed\<^bsub>p\<^esub> [next_plane0\<^bsub>p\<^esub>]\<rightarrow>* g"   and "final g"   and "tame g" shows "p \<le> 3"
fc_emptyI: fixes ctx :: "ctx"   and fqn :: "fqn" shows "find_cld [] ctx fqn None"
div_cbox: shows "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q"
shiftr_word_eq: fixes w :: "'a word"   and n :: "nat" shows "w >> n = drop_bit n w"
n2m_ctor_rec_G_def: fixes s1 :: "('e, 'e G F \<times> 'g) G0 \<Rightarrow> 'f"   and s2 :: "('e G \<times> 'f, 'e G F \<times> 'g) F0 \<Rightarrow> 'g" shows "n2m_ctor_rec_G s1 s2 = ctor_rec_G (s1 \<circ> map_pre_G\<^sub>M id id <map_F fst, ctor_rec_F (s2 \<circ> map_pre_GF\<^sub>M id id (map_prod (map_F fst) id) \<circ> BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf)> \<circ> BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf)"
wfP_lstrict_prefix: shows "wfP lstrict_prefix"
simple_function_finite_support: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "simple_function M f"   and "integral\<^sup>N M f < \<infinity>"   and "\<And>x. 0 \<le> f x" shows "emeasure M {x \<in> space M. f x \<noteq> 0} \<noteq> \<infinity>"
cSup_eq_maximum: fixes z :: "'b multiset"   and X :: "'b multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X = z"
finite_gpv_rel''D2: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and gpv :: "('g, 'i, 'k) gpv"   and gpv' :: "('h, 'j, 'l) gpv"   and \<I> :: "('j, 'l) \<I>"   and \<I>' :: "('i, 'k) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "finite_gpv \<I> gpv'"   and "rel_\<I> C R \<I>' \<I>" shows "finite_gpv \<I>' gpv"
SMP_fun_map_snd_subset: fixes X :: "('fun, 'var) Term.term list"   and f :: "'fun" shows "SMP (trms\<^sub>s\<^sub>t (map Send X)) \<subseteq> SMP (trms\<^sub>s\<^sub>t [send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t])"
classical_operator_ket_finite: fixes \<pi> :: "'c \<Rightarrow> 'b option"   and x :: "'c" shows "classical_operator \<pi> *\<^sub>V ket x = (case \<pi> x of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
zero_pp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat_pp :: "'b \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'b"   and lex_comp' :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and deg' :: "'b \<Rightarrow> nat" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'" shows "rep_nat_pp zero = 0"
Abs_fset_inverse: fixes y :: "'b set" assumes "y \<in> {A. finite A}" shows "\<langle>Abs_fset y\<rangle>\<^sub>f = y"
max_y_xy: shows "maximum j = maximum (i \<squnion> j)"
intros_3: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and Ts :: "'c set"   and t' :: "'c"   and t :: "'c" assumes "final_thread.all_final_except final s Ts"   and "t' \<notin> interrupts s" shows "final_thread.must_wait final s t (Inr (Inr t')) Ts"
conv_plus_commute: fixes x :: "'a" shows "x\<^sup>+\<^sup>T = x\<^sup>T\<^sup>+"
is_point_def: fixes x :: "'a" shows "is_point x \<equiv> is_vector x \<and> is_inj x \<and> x \<noteq> 0"
facesAt_def: fixes g :: "graph"   and v :: "nat" shows "facesAt g v \<equiv> faceListAt g ! v"
ACI_decidable: fixes r :: "'a rexp"   and s :: "'a rexp" shows "ACI r s = (\<guillemotleft>r\<guillemotright> = \<guillemotleft>s\<guillemotright>)"
wt_elim_cases_1: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and vn :: "lname"   and T :: "tys" assumes "E,dt\<Turnstile>In2 (LVar vn)\<Colon>T" obtains T :: "ty" where "Tb__ = Inl T" and "lcl E vn = Some T" and "is_acc_type (prg E) (pkg E) T"
inf_progress: fixes f :: "nat \<Rightarrow> ('s, 'p, 'f) com list \<times> (('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list \<times> ('s, 'f) xstate"   and c :: "('s, 'p, 'f) com"   and cs :: "('s, 'p, 'f) com list"   and css :: "(('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list"   and s :: "('s, 'f) xstate"   and \<Gamma> :: "'p \<Rightarrow> ('s, 'p, 'f) com option"   and i :: "nat" assumes "f 0 = (c # cs, css, s)"   and "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"   and "\<forall>i. \<not> (CS (f i) = cs \<and> CSS (f i) = css)" shows "\<exists>pcs pcss. if pcss = [] then CSS (f i) = css \<and> CS (f i) = pcs @ cs else CS (f i) = pcs \<and> CSS (f i) = butlast pcss @ [(fst (last pcss) @ cs, snd (last pcss) @ cs)] @ css"
diagram_determines_ide_2: fixes D :: "'j \<Rightarrow> 'c" assumes "diagram (\<cdot>\<^sub>J) (\<cdot>) D" shows "\<chi>.\<Phi>.Ya.Cop_S.Map (J_C.MkIde D) = D"
Inv2a_inner_def: fixes s :: "state"   and p :: "Proc" shows "Inv2a_inner s p = (\<forall>bk\<in>blocksOf s p. Inv2a_innermost s p bk)"
single_eq_add_mset: fixes a :: "'b"   and b :: "'b"   and M :: "'b multiset" shows "(\<LM> a  \<RM> = add_mset b M) = (b = a \<and> M = \<Empt>)"
Glog: fixes t :: "'b tree" shows "real (lrh t) \<le> log 2 (real (size1 t))"
pmf_of_alist_def: fixes xs :: "('a \<times> real) list" shows "pmf_of_alist xs = embed_pmf (\<lambda>x. case map_of xs x of None \<Rightarrow> 0 | Some p \<Rightarrow> p)"
pol_coeff_mem: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and j :: "nat" assumes "pol_coeff S c"   and "j \<le> fst c" shows "snd c j \<in> carrier S"
matrix_sum_cong: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and f' :: "nat \<Rightarrow> 'a mat"   and d :: "nat" assumes "\<And>k. k < n \<Longrightarrow> f k = f' k" shows "matrix_sum d f n = matrix_sum d f' n"
senior_le_token: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "semi_mojmir_def.senior \<delta> q\<^sub>0 w x n \<le> x"
assms_4: shows "r \<cdot> b \<le> b \<cdot> r"
in_set_conv_decomp_last: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
rel_interior_aff_dim: fixes S :: "'b set" assumes "convex S" shows "aff_dim (rel_interior S) = aff_dim S"
cases: fixes a1 :: "state"   and a2 :: "com"   and a3 :: "nat"   and a4 :: "state" assumes "a1 -a2-a3\<rightarrow> a4" obtains t :: "state" and f :: "state \<Rightarrow> state set" and s :: "state" and n :: "nat" where "a1 = sa__" and "a2 = Do f__" and "a3 = na__" and "a4 = ta__" and "ta__ \<in> f__ sa__"   | s0 :: "state" and c0 :: "com" and n :: "nat" and s1 :: "state" and c1 :: "com" and s2 :: "state" where "a1 = s0" and "a2 = (c0__; c1__)" and "a3 = nb__" and "a4 = s2" and "s0 -c0__-nb__\<rightarrow> s1" and "s1 -c1__-nb__\<rightarrow> s2"   | b :: "state \<Rightarrow> bool" and s :: "state" and c0 :: "com" and n :: "nat" and t :: "state" and c1 :: "com" where "a1 = sb__" and "a2 = IF ba__ THEN c0a__ ELSE c1a__" and "a3 = nc__" and "a4 = tb__" and "ba__ sb__" and "sb__ -c0a__-nc__\<rightarrow> tb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and n :: "nat" and t :: "state" and c0 :: "com" where "a1 = sc__" and "a2 = IF bb__ THEN c0 ELSE c1" and "a3 = nd__" and "a4 = tc__" and "\<not> bb__ sc__" and "sc__ -c1-nd__\<rightarrow> tc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and n :: "nat" where "a1 = sd__" and "a2 = WHILE bc__ DO ca__" and "a3 = ne__" and "a4 = sd__" and "\<not> bc__ sd__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and n :: "nat" and t :: "state" and u :: "state" where "a1 = se__" and "a2 = WHILE b DO cb__" and "a3 = nf__" and "a4 = u" and "b se__" and "se__ -cb__-nf__\<rightarrow> td__" and "td__ -WHILE b DO cb__-nf__\<rightarrow> u"   | s :: "state" and p :: "pname" and n :: "nat" and t :: "state" where "a1 = sf__" and "a2 = CALL p" and "a3 = Suc ng__" and "a4 = te__" and "sf__ -body p-ng__\<rightarrow> te__"   | f :: "state \<Rightarrow> state" and s :: "state" and c :: "com" and n :: "nat" and t :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = s" and "a2 = LOCAL f; c; g" and "a3 = n" and "a4 = g s t" and "f s -c-n\<rightarrow> t"
THEN_refine_2: fixes wl :: "'V \<Rightarrow> 'W infty option"   and res :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and it :: "('W \<times> 'V) set"   and wl' :: "'V \<Rightarrow> 'W infty option"   and res' :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and w' :: "'W"   and v'a :: "'V" assumes "uinvarm v (\<alpha>w wl) (\<alpha>r res) it (wl', res')"   and "dinvarm (wl, res)"   and "mpath_weight' (res v) + Num w' < mpath_weight' (res' v'a)"   and "path_weight' (\<alpha>r res v) + Num w' < path_weight' (\<alpha>r res' v'a)"   and "v'a \<in> \<alpha>w wl'"   and "\<alpha>r res v \<noteq> None" shows "mpath' (res v) \<noteq> None"
interleaves_equal_fst: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and ys' :: "'a list" assumes "xs \<simeq> {ys, zs, P}"   and "xs \<simeq> {ys', zs, P}" shows "ys = ys'"
FactGroup_onto: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h"   and "h ` carrier G = carrier H" shows "(\<lambda>X. the_elem (h ` X)) ` carrier (G Mod kernel G H h) = carrier H"
conservative_mpt_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "conservative_mpt M T \<equiv> mpt M T \<and> conservative M T"
glob_def: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "\<box>\<^sub>c\<gamma> \<equiv> \<lambda>t n. \<forall>n'\<ge>n. \<gamma> t n'"
\<iota>_in_hom: shows "\<guillemotleft>\<iota>\<^sub>D : D.tensor D.unity D.unity \<rightarrow>\<^sub>D D.unity\<guillemotright>"
poly_y_x_o_poly_lift: shows "poly_y_x \<circ> poly_lift = (\<lambda>x. [:x:])"
sorted_less_sorted_list_of_set: fixes S :: "'a set" shows "Sorted_Less.sorted (sorted_list_of_set S)"
card_order_on_ordIso: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and r' :: "('a \<times> 'a) set" assumes "card_order_on A r"   and "card_order_on A r'" shows "r =o r'"
dl_iso_inv_lifted_pair: fixes xf :: "('c, 'd) lifted_distrib_lattice_top"   and f :: "('c, 'd) phi" shows "(top, up_filter (Rep_lifted_distrib_lattice_top xf f)) \<in> triple.pairs (Rep_phi f)"
prolong_betw3: fixes Q :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "a \<noteq> b" shows "\<exists>c\<in>Q. \<exists>d\<in>Q. [[a b c]] \<and> [[a b d]] \<and> c \<noteq> d"
M2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ F \<longrightarrow> G" shows "\<turnstile> \<box>[F]_v \<longrightarrow> \<box>[G]_v"
sup_idem: fixes x :: "'c multiset" shows "x \<union># x = x"
limuls_eq: shows "Liminf_llist (lmap lclss_of_state Sts) = Ql"
minus_Mp_1: fixes f :: "int poly"   and g :: "int poly" shows "r.eq_m (r.Mp f - g) (f - g)"
pt_set_bij1: fixes pi :: "('b \<times> 'b) list"   and x :: "'a"   and X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "(pi \<bullet> x \<in> X) = (x \<in> rev pi \<bullet> X)"
set_in_convex_hull_list: fixes Vs :: "'b vec list"   and n :: "nat" assumes "set Vs \<subseteq> carrier_vec n" shows "set Vs \<subseteq> gram_schmidt.convex_hull_list n Vs"
lunit_coherence: fixes f :: "'c" assumes "C.ide f" shows "D'.lunit (F f) = F (C'.lunit f) \<cdot>\<^sub>D \<Phi> (snd (C'.trg f, f), fst (C'.trg f, f)) \<cdot>\<^sub>D (F'.unit (C'.trg f) \<star>\<^sub>D\<^sup>o\<^sup>p F f)"
automg_def: fixes G :: "('e, 'f) Group_scheme" shows "automg G = \<lparr>carrier = {f. gbij\<^bsub>G,G\<^esub> f}, Group.top = \<lambda>g\<in>{f. gbij\<^bsub>G,G\<^esub> f}. restrict (GCOMP g G) {f. gbij\<^bsub>G,G\<^esub> f}, iop = restrict (Algebra3.INVFUN G G) {f. gbij\<^bsub>G,G\<^esub> f}, one = I\<^bsub>G\<^esub>\<rparr>"
main_part: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets M" shows "main_part M S = S"
frontier_closures: fixes S :: "'a set" shows "frontier S = closure S \<inter> closure (- S)"
hcomp_def: fixes g :: "'a"   and f :: "'a" shows "S.hcomp g f = (if S.arr f \<and> S.arr g \<and> trg f = src g then g \<star> f else S.null)"
has_domain_iff_has_codomain: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "(A_B.domains f \<noteq> {}) = (A_B.codomains f \<noteq> {})"
lnth_ltake: fixes m :: "nat"   and n :: "enat"   and xs :: "'c llist" assumes "enat m < n" shows "lnth (ltake n xs) m = lnth xs m"
intros_4: fixes \<phi> :: "'a ltln"   and \<chi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<turnstile>\<^sub>s \<chi>" shows "\<phi> and\<^sub>n \<psi> \<turnstile>\<^sub>s \<chi>"
progress: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty"   and l :: "char list \<Rightarrow> val option" assumes "wwf_prog P"   and "P,E,h \<turnstile> e : T"   and "P \<turnstile> h \<surd>"   and "P \<turnstile> E \<surd>"   and "\<D> e \<lfloor>dom l\<rfloor>"   and "\<not> final e" shows "\<exists>e' s'. P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"
3: shows "[k] \<langle>(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms)\<rangle> ms = [k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms"
rep_eq: fixes x :: "circline_mat"   and xa :: "circline_mat" shows "ocircline_eq_clmat x xa = ocircline_eq_cmat (Rep_circline_mat x) (Rep_circline_mat xa)"
l_eq_r_in_eqs: fixes X :: "'b list set"   and rhs :: "'b trm set"   and A :: "'b list set" assumes "(X, rhs) \<in> Init (UNIV // \<approx>A)" shows "X = lang_rhs rhs"
Hash_Sign: fixes X :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Hash X \<noteq> Sign X' Y'"
map_simp: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.map f = f"
inf_add_K_right: fixes a :: "'c"   and b :: "'c" shows "a \<le>\<^sub>\<K> a + b"
qSwap_preserves_qGood2: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood (X #[[x \<and> y]]_zs)" shows "qGood X"
sum_msum_pdevs_cases: fixes f :: "'b pdevs"   and n :: "nat"   and e :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'b pdevs" assumes "degree f \<le> n"   and "\<And>i. e i (0::'b) = (0::'c)" shows "(\<Sum>i<degree (msum_pdevs n f g). e i (if i < n then pdevs_apply f i else pdevs_apply g (i - n))) = (\<Sum>i<degree f. e i (pdevs_apply f i)) + (\<Sum>i<degree g. e (i + n) (pdevs_apply g i))"
sign_simps_17: fixes aa :: "'c"   and ba :: "'c"   and ca :: "'c" shows "(aa + ba) * ca = aa * ca + ba * ca"
of_nat_ge_1_iff: fixes x :: "nat" shows "((1::'b) \<le> of_nat x) = (1 \<le> x)"
is_kur: shows "K\<^bsub>3,3\<^esub> K \<or> K\<^bsub>5\<^esub> K"
F_def: shows "F \<equiv> res_freeword_funlift f S"
singletonI: fixes a :: "'a" shows "s2p (\<eta> a) a"
rbt_map_entry_is_rbt: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and t :: "('c, 'd) rbt" shows "ord.is_rbt less (ord.rbt_map_entry less k f t) = ord.is_rbt less t"