hcju/isabelleps huggingface dataset:
fst_hyp_edge_at: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b"   and "is" :: "nat list"   and n :: "nat"   and s :: "'c" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "fst (Solved_Task.hyp_edge_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is n s) = Solved_Task.hyp_edge_from freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is n s"
list_all2_Cons1: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and ys :: "'c list" shows "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
Inf_comp_fun: fixes M :: "('d \<Rightarrow> 'c) set"   and f :: "'b \<Rightarrow> 'd" shows "\<Sqinter>M \<circ> f = (\<Sqinter>m\<in>M. m \<circ> f)"
assms_5: shows "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"
Cons_2: shows "a # x = filter (\<lambda>x. \<not> P1 x) l"
lower_minus: fixes A :: "'a interval"   and B :: "'a interval" shows "lower (A - B) = lower A - upper B"
Min_in: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Min less_eq A \<in> A"
secure_implies_c_secure_aux: fixes step :: "'h \<Rightarrow> 'e \<Rightarrow> 'h"   and out :: "'h \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'h"   and I :: "('g \<times> 'g) set"   and D :: "'e \<Rightarrow> 'g"   and xs :: "'e list"   and ys :: "'e list"   and x :: "'e" assumes "secure (c_process step out s\<^sub>0) I (c_dom D)" shows "out (foldl step (foldl step s\<^sub>0 xs) ys) x = out (foldl step (foldl step s\<^sub>0 xs) (c_ipurge I D (D x) ys)) x"
converse_unfold: fixes r :: "('h \<times> 'g) set" shows "r\<inverse> = {(y, x). (x, y) \<in> r}"
wp2WhileTrue': fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "bval b s" shows "wp\<^sub>2 c (wp\<^sub>2 (WHILE b DO c) Q) s + 1 = wp\<^sub>2 (WHILE b DO c) Q s"
emptyLeftProperty: fixes A :: "'c event set"   and t :: "'c event list" shows "\<forall>s. s setinterleaves (([], t), A) \<longrightarrow> s = t"
IH: fixes n :: "nat"   and c :: "'a" assumes "n + 1 < m" shows "(pderiv ^^ m) (Polynomial.monom c n) = 0"
Ext_foo_member_of_Ext: shows "tprg \<turnstile> (Ext, mdecl Ext_foo) member_of Ext"
Abs_ordinal_eq: fixes x :: "ord0"   and y :: "ord0" shows "(Abs_ordinal (ord0rel `` {x}) = Abs_ordinal (ord0rel `` {y})) = (x \<le> y \<and> y \<le> x)"
RETURN_refine: fixes x :: "'f"   and x' :: "'g"   and R :: "('f \<times> 'g) set" assumes "(x, x') \<in> R" shows "RETURN x \<le> \<Down> R (RETURN x')"
bigger_prime: fixes n :: "nat" shows "\<exists>p. prime p \<and> n < p"
PiE_eq: fixes I :: "'b set"   and S :: "'b \<Rightarrow> 'c set"   and T :: "'b \<Rightarrow> 'c set" shows "(Pi\<^sub>E I S = Pi\<^sub>E I T) = (Pi\<^sub>E I S = {} \<and> Pi\<^sub>E I T = {} \<or> (\<forall>i\<in>I. S i = T i))"
rank_mul_left_invertible_mat: fixes A :: "'a mat"   and B :: "'a mat"   and nc :: "nat" assumes "invertible_mat A"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n nc" shows "rank (A * B) = rank B"
span_alt: fixes B :: "'b set" shows "vs1.span B = {\<Sum>x | f x \<noteq> (0::'a). f x *a x |f. {x. f x \<noteq> (0::'a)} \<subseteq> B \<and> finite {x. f x \<noteq> (0::'a)}}"
zmult_div_leq_mono: fixes x :: "int"   and a :: "int"   and b :: "int"   and d :: "int" assumes "0 \<le> x"   and "a \<le> b"   and "0 < d" shows "x * a div d \<le> x * b div d"
2: shows "S' = P2 ** D ** Q2 \<and> invertible P2 \<and> invertible Q2 \<and> Smith_normal_form S'"
word_or_zero: fixes a :: "'b word"   and b :: "'b word" shows "(a OR b = 0) = (a = 0 \<and> b = 0)"
decomp_vars_2: fixes t :: "('fun, 'var) Term.term" shows "vars\<^sub>s\<^sub>t (decomp t) = fv t"
Carmichael_def: fixes n :: "nat" shows "Carmichael n = Lcm (ord n ` totatives n)"
intro: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<And>a x y. scale a (x + y) = scale a x + scale a y"   and "\<And>a b x. scale (a + b) x = scale a x + scale b x"   and "\<And>a b x. scale a (scale b x) = scale (a * b) x"   and "\<And>x. scale (1::'d) x = x" shows "module scale"
\<Gamma>_wf': fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<Gamma> t)"
same_gen_iapath_by_common_arc: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and w :: "'a"   and q :: "('a \<times> 'a) list"   and x :: "'a"   and e :: "'a \<times> 'a" assumes "verts3 (with_proj slim) \<subseteq> V"   and "V \<subseteq> pverts slim"   and "S.gen_iapath V u p v"   and "S.gen_iapath V w q x"   and "e \<in> set p"   and "e \<in> set q" shows "p = q"
finite_product_sigma_finite_axioms_def: fixes I :: "'c set" shows "finite_product_sigma_finite_axioms I \<equiv> finite I"
not_arr_null: shows "\<not> \<chi>.J.arr \<chi>.J.null"
less_float_def: shows "(<) \<equiv> map_fun real_of_float (map_fun real_of_float id) (<)"
no_before_inI: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" assumes "x < y in xs" shows "(\<not> y < x in xs) = True"
iter_induct_nil: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and x :: "'b::refinement_lattice" assumes "finite_or_infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"   and "(nil::'b::refinement_lattice) \<sqinter> (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) (x::'b::refinement_lattice) \<sqsubseteq> x" shows "finite_or_infinite_iteration.iter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice) \<sqsubseteq> (x::'b::refinement_lattice)"
full_pmdlI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and K :: "nat set" assumes "\<And>v. v \<in> keys p \<Longrightarrow> component_of_term v \<in> K" shows "p \<in> full_pmdl K"
rel_pmf_1: fixes x :: "'a option"   and y :: "'b option" assumes "(x, y) \<in> set_pmf pq" shows "rel_option P x y"
second_summands_in_sums: fixes y :: "i"   and x :: "i"   and v :: "i"   and z :: "i" assumes "CEM" shows "P y (x \<oplus> y) \<and> P v (z \<oplus> v)"
P11_a: fixes a :: "'a"   and b :: "'a" shows "a \<le> b l\<rightarrow> a"
rep_eq: fixes x :: "'a dual"   and xa :: "'a dual" shows "undual (inf x xa) = sup (undual x) (undual xa)"
unfold_filterStep: fixes s :: "'b"   and p :: "'c \<rightarrow> tr"   and h :: "'b \<rightarrow> ('c, 'b) Step" assumes "s \<noteq> \<bottom>" shows "unfold\<cdot>(filterStep\<cdot>p\<cdot>h)\<cdot>s = filterL\<cdot>p\<cdot>(unfold\<cdot>h\<cdot>s)"
simps_2: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and e :: "'d"   and es :: "'d list" shows "pre_digraph.awalk_verts G u (e # es) = tail G e # pre_digraph.awalk_verts G (head G e) es"
A': fixes kv :: "'c \<times> 'd"   and x :: "'e" shows "(case kv of (x, xa) \<Rightarrow> f x xa) x \<le> (case kv of (x, xa) \<Rightarrow> f' x xa) x"
zero_uint16_def: shows "0 \<equiv> Abs_uint16 0"
r: shows "r = (- (3::'a) * b ^ 4 + (256::'a) * e - (64::'a) * b * d + (16::'a) * b\<^sup>2 * c) / (256::'a)"
simps_2: fixes b :: "bool"   and l1 :: "'a tree"   and x :: "'a"   and r1 :: "'a tree"   and t2 :: "'a tree" shows "mrbst_inter_diff b \<langle>l1, x, r1\<rangle> t2 = (case split_bst' x t2 of (sep, l2, r2) \<Rightarrow> mrbst_inter_diff b l1 l2 \<bind> (\<lambda>l. mrbst_inter_diff b r1 r2 \<bind> (\<lambda>r. if sep = b then return_pmf \<langle>l, x, r\<rangle> else mrbst_join l r)))"
atLeast0_atMost_Suc_eq_insert_0: fixes n :: "nat" shows "{0..Suc n} = insert 0 (Suc ` {0..n})"
indexed_weak_nominal_ts_def: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool" shows "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition \<equiv> weak_nominal_ts satisfies transition \<tau> \<and> indexed_weak_nominal_ts_axioms TYPE('a) TYPE('c)"
rt_fresh_as_sym: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "rt2 \<approx>\<^bsub>dip\<^esub> rt1"
field_tOp_commute: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier K"   and "y \<in> carrier K" shows "x \<cdot>\<^sub>r y = y \<cdot>\<^sub>r x"
ctr_1: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "lnull xs \<or> \<not> P (lhd xs)" shows "ltakeWhile P xs = LNil"
dg_Obj_vsubset_Vset: shows "\<CC>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
rel_compp: fixes R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "lrel (R OO S) = lrel R OO lrel S"
VarB_def: fixes x2 :: "vrs"   and x1 :: "ty" shows "VarB x2 x1 \<equiv> Abs_binding (VarB_Rep x2 x1)"
sub_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (sub_instr instr s1)) \<and> t2 = snd (fst (sub_instr instr s2))" shows "low_equal t1 t2"
hom_a_inv: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "ring_hom_cring R S h"   and "x \<in> carrier R" shows "h (\<ominus>\<^bsub>R\<^esub> x) = \<ominus>\<^bsub>S\<^esub> h x"
cod_cod: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.cod (B.VV.cod f) = B.VV.cod f"
runWriterT_fmapWT: fixes f :: "'h \<rightarrow> 'e"   and m :: "'h\<cdot>('g, 'f) writerT" shows "runWriterT\<cdot>(fmapWT\<cdot>f\<cdot>m) = runWriterT\<cdot>m \<bind> (\<Lambda> (Writer\<cdot>w\<cdot>x). return\<cdot>(Writer\<cdot>w\<cdot>(f\<cdot>x)))"
simps_1: fixes c :: "'d \<Rightarrow> 'd \<Rightarrow> order"   and k :: "'d" shows "rbt_comp_lookup c rbt.Empty k = None"
conditional_mutual_information_generic_nonneg: fixes S :: "'e measure"   and T :: "'f measure"   and P :: "'g measure"   and X :: "'a \<Rightarrow> 'e"   and Px :: "'e \<Rightarrow> real"   and Z :: "'a \<Rightarrow> 'g"   and Pz :: "'g \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'f"   and Pyz :: "'f \<times> 'g \<Rightarrow> real"   and Pxz :: "'e \<times> 'g \<Rightarrow> real"   and Pxyz :: "'e \<times> 'f \<times> 'g \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "sigma_finite_measure P"   and "distributed M S X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space S \<Longrightarrow> 0 \<le> Px x"   and "distributed M P Z (\<lambda>x. ennreal (Pz x))"   and "\<And>z. z \<in> space P \<Longrightarrow> 0 \<le> Pz z"   and "distributed M (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x)) (\<lambda>x. ennreal (Pyz x))"   and "\<And>y z. \<lbrakk>y \<in> space T; z \<in> space P\<rbrakk> \<Longrightarrow> 0 \<le> Pyz (y, z)"   and "distributed M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) (\<lambda>x. ennreal (Pxz x))"   and "\<And>x z. \<lbrakk>x \<in> space S; z \<in> space P\<rbrakk> \<Longrightarrow> 0 \<le> Pxz (x, z)"   and "distributed M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x)) (\<lambda>x. ennreal (Pxyz x))"   and "\<And>x y z. \<lbrakk>x \<in> space S; y \<in> space T; z \<in> space P\<rbrakk> \<Longrightarrow> 0 \<le> Pxyz (x, y, z)"   and "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"   and "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))" shows "0 \<le> conditional_mutual_information b S T P X Y Z"
map_def: fixes f :: "'a" shows "hom'.map f = (if hom'.arr f then f else hom'.null)"
SINK_Sup: fixes Y :: "('a \<times> 'a \<Rightarrow> ennreal) set" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "countable (support_flow (\<Squnion> Y))" shows "SINK (\<Squnion> Y) = \<Inter> (SINK ` Y)"
pair_in_pair_image_iff: fixes s :: "('fun, 'b) Term.term"   and t :: "('fun, 'b) Term.term"   and P :: "(('fun, 'b) Term.term \<times> ('fun, 'b) Term.term) set" shows "(pair (s, t) \<in> pair ` P) = ((s, t) \<in> P)"
f_def: shows "f = (\<lambda>x. 2 * x - 2 - ln x * ln x)"
compile_rules: shows "srules C_info (Rewriting_Sterm.compile rs)"
Inl_def: fixes a :: "hf" shows "HF.Inl a \<equiv> \<langle>0, a\<rangle>"
cod_eqI': fixes a :: "'a"   and f :: "'a" assumes "a \<in> S.codomains f" shows "a = S.cod f"
invrules_5: assumes "j \<le> m" shows "makespan T = Max\<^sub>0 (t ` {1..j})"
gcd_greatest: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c dvd a"   and "c dvd b" shows "c dvd normalization_euclidean_semiring_class.gcd a b"
cases: fixes a1 :: "'b llist"   and a2 :: "'b llist" assumes "emb a1 a2" obtains xs :: "'b llist" where "a1 = LNil" and "a2 = xsa__" and "lfinite xsa__"   | xs :: "'b llist" and ys :: "'b llist" and x :: "'b" and zs :: "'b list" where "a1 = LCons x xs" and "a2 = prepend zs (LCons x ys)" and "emb xs ys"
INF_lower2: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and i :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and u :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq (f i) u" shows "less_eq (Inf (f ` A)) u"
Up_To_Technique: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and R :: "('a list \<times> 'a list) set" assumes "Strong_Security.d_Bisimulation_Up_To_USdB SR DA d R" shows "R \<subseteq> Strong_Security.USdB SR DA d"
safe_hd_None: fixes xs :: "'a list" shows "(safe_hd xs = None) = (xs = [])"
parCasesOutputFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and T :: "('a, 'b, 'c) psi"   and A\<^sub>P\<^sub>Q :: "name list"   and \<Psi>\<^sub>P\<^sub>Q :: "'b"   and Prop :: "('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> T"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* M"   and "extractFrame (P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>"   and "distinct A\<^sub>P\<^sub>Q"   and "A\<^sub>P\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>P\<^sub>Q \<sharp>* P"   and "A\<^sub>P\<^sub>Q \<sharp>* Q"   and "A\<^sub>P\<^sub>Q \<sharp>* M"   and "\<And>P' A\<^sub>P \<Psi>\<^sub>P A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>P; distinct A\<^sub>Q; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>P \<sharp>* \<Psi>\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>\<^sub>P; A\<^sub>P \<sharp>* A\<^sub>Q; A\<^sub>P\<^sub>Q = A\<^sub>P @ A\<^sub>Q; \<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)"   and "\<And>Q' A\<^sub>P \<Psi>\<^sub>P A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>P; distinct A\<^sub>Q; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>P \<sharp>* \<Psi>\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>\<^sub>P; A\<^sub>P \<sharp>* A\<^sub>Q; A\<^sub>P\<^sub>Q = A\<^sub>P @ A\<^sub>Q; \<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')" shows "Prop T"
simps_3: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "merge_wrt rel (x # xs) (y # ys) = (if x = y then y # merge_wrt rel xs ys else if rel x y then x # merge_wrt rel xs (y # ys) else y # merge_wrt rel (x # xs) ys)"
map_tpOfV_getVars: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and \<sigma>l :: "'e list" assumes "Signature TYPE('e) wtFsym wtPsym" shows "map Signature.tpOfV (Signature.getVars \<sigma>l) = \<sigma>l"
norm_pths_1: fixes x :: "'c"   and y :: "'c" shows "(x = y) = (norm (x - y) \<le> 0)"
apartment_face_distance: fixes A :: "'a set set"   and C :: "'a set"   and F :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "C \<in> A"   and "F \<in> A" shows "ChamberComplex.face_distance A F C = face_distance F C"
enc: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "\<langle>\<phi>\<rangle> \<in> num"
assms_4: shows "prv (eqv \<phi>1 \<phi>2)"
abs_eq: fixes x :: "'a set"   and xa :: "'a \<Rightarrow> 'c" assumes "eq_onp (\<lambda>A. |A| <o natLeq +c |UNIV|) x x" shows "map_bset xa (Abs_bset x) = Abs_bset (xa ` x)"
length_pos_if_in_set: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set xs" shows "0 < length xs"
GrD2: fixes x :: "'a"   and fx :: "'b"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "(x, fx) \<in> BNF_Def.Gr A f" shows "f x = fx"
Abinds_env_restr_cong: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "ae f|` domA \<Delta> = ae' f|` domA \<Delta>" shows "ABinds \<Delta>\<cdot>ae = ABinds \<Delta>\<cdot>ae'"
a: shows "xs \<noteq> [] \<and> 0 < x \<and> 0 < mean xs"
sep_combine2_1: fixes P :: "bool"   and Q :: "bool"   and F :: "'c \<Rightarrow> bool" shows "(\<up> P \<and>* \<up> Q \<and>* F) = (\<up> (P \<and> Q) \<and>* F)"
normal_moment_abs_odd: fixes \<sigma> :: "real"   and \<mu> :: "real"   and k :: "nat" assumes "0 < \<sigma>" shows "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * \<bar>x - \<mu>\<bar> ^ (2 * k + 1)) (2 ^ k * \<sigma> ^ (2 * k + 1) * fact k * sqrt (2 / pi))"
verts3_in_slim_G: fixes G :: "'b pair_pre_digraph"   and x :: "'b" assumes "pair_sym_digraph G"   and "x \<in> verts3 (with_proj G)" shows "x \<in> pverts (pair_pre_digraph.slim G)"
inv_f_f: fixes x :: "int poly" shows "inv (map_poly ((*) c)) (map_poly ((*) c) x) = x"
sqSub_mem_lem: fixes A :: "'f set option"   and A' :: "'f set option"   and a :: "'f" assumes "A \<sqsubseteq> A'"   and "a \<in>\<in> A" shows "a \<in>\<in> A'"
z1: shows "\<forall>x. z1 \<sqsubset> x \<longrightarrow> P1 x = P1'"
L: fixes s :: "'state" assumes "s \<in> S" shows "L s \<subseteq> AP"
cancel1_semiconfluent: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "cancel1 xs ys"   and "cancel zs ys" shows "\<exists>us. cancel us xs \<and> cancel us zs"
R2: fixes p :: "'b fm"   and i :: "'b" assumes "\<turnstile> p" shows "\<turnstile> K i p"
lbound_ofD2: fixes x :: "'a"   and y :: "'a"   and b :: "'a" assumes "lbound_of x y b" shows "y \<succeq> b"
prop_nth_sdrop_pair: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<omega> :: "'a stream"   and \<omega>' :: "'b stream"   and j :: "nat" assumes "\<forall>i. P (\<omega> !! i) (\<omega>' !! i)" shows "\<forall>i. P (sdrop j \<omega> !! i) (sdrop j \<omega>' !! i)"
empty_iff: fixes c :: "'a" shows "s2p {} c = False"
s1': shows "[s1\<^sup>2 + 1 = 0] (mod int (4 * k + 1))"
outs_plus_\<I>: fixes \<I>l :: "('e, 'g) \<I>"   and \<I>r :: "('f, 'h) \<I>" shows "outs_\<I> (\<I>l \<oplus>\<^sub>\<I> \<I>r) = outs_\<I> \<I>l <+> outs_\<I> \<I>r"
finite_vimageI: fixes F :: "'b set"   and h :: "'c \<Rightarrow> 'b" assumes "finite F"   and "inj h" shows "finite (h -` F)"
conv_radius_cmult_left: fixes c :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "conv_radius (\<lambda>n. c * f n) = conv_radius f"
Derivation_implies_derives: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "Derivation a D b" shows "derives a b"
procs_subset_\<Gamma>: shows "\<forall>p\<in>dom \<Gamma>. procs (the (\<Gamma> p)) \<subseteq> dom \<Gamma>"
CLS_in_hom: fixes f :: "'a" assumes "B.is_left_adjoint f" shows "\<guillemotleft>\<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> : \<lbrakk>\<lbrakk>src f\<rbrakk>\<rbrakk> \<rightarrow> \<lbrakk>\<lbrakk>trg f\<rbrakk>\<rbrakk>\<guillemotright>"
card_psubset: fixes B :: "'d set"   and A :: "'d set" assumes "finite B"   and "A \<subseteq> B"   and "card A < card B" shows "A \<subset> B"
ball_trivial: fixes x :: "'b" shows "ball x 0 = {}"
abs_eq_iff: fixes x :: "cmsg"   and y :: "cmsg" shows "(Ab x = Ab y) = eq x y"
finite_lists_length_eq: fixes A :: "'b set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
stirling_integral_complex_of_real: fixes x :: "real"   and n :: "nat" assumes "0 < x"   and "0 < n" shows "stirling_integral n (complex_of_real x) = complex_of_real (stirling_integral n x)"
row_space_def: fixes B :: "rat mat" shows "gs.row_space B = gs.span (set (rows B))"
map_of_Some_None_split: fixes t :: "(('a \<times> 'b) \<times> 'c \<times> 'd) list"   and C :: "'b"   and fs :: "('a \<times> 'c \<times> 'd) list"   and t' :: "(('a \<times> 'b) \<times> 'c \<times> 'd) list"   and F :: "'a"   and y :: "'c \<times> 'd" assumes "t = map (\<lambda>(F, y). ((F, C), y)) fs @ t'"   and "map_of t' (F, C) = None"   and "map_of t (F, C) = \<lfloor>y\<rfloor>" shows "map_of (map (\<lambda>((F, D), b, T). (F, D, b, T)) t) F = \<lfloor>(C, y)\<rfloor>"
hyps: fixes y :: "nat"   and e :: "expr" assumes "y < n"   and "card (radicals e) = y"   and "\<lbrace>e\<rbrace> ^ 3 + a * \<lbrace>e\<rbrace>\<^sup>2 + b * \<lbrace>e\<rbrace> + c = 0" shows "\<exists>x\<in>\<rat>. x ^ 3 + a * x\<^sup>2 + b * x + c = 0"
ipurge_fail_aux_t_eq_tr: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "fst (ipurge_fail_aux_t I D U xs X) = ipurge_tr_aux I D U xs"
is_valid_problem_strips_def: fixes \<Pi> :: "'a strips_problem" shows "is_valid_problem_strips \<Pi> \<equiv> let ops = \<Pi>\<^sub>\<O>; vs = \<Pi>\<^sub>\<V>; I = \<Pi>\<^sub>I; G = \<Pi>\<^sub>G in list_all (is_valid_operator_strips \<Pi>) ops \<and> (\<forall>v. (I v \<noteq> None) = ListMem v vs) \<and> (\<forall>v. G v \<noteq> None \<longrightarrow> ListMem v vs)"