hcju/isabelleps huggingface dataset:
dense: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.dense_order less_eq less"   and "less x y" shows "\<exists>z. less x z \<and> less z y"
round_robin_reschedule_Cons_1: fixes t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "round_robin_reschedule t0 (t0 # queue) n0 s = None"
\<gamma>_1: shows "path_image \<gamma> \<subseteq> path_image (reversepath h +++ g)"
simps_7: fixes var :: "nat"   and \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "convert_fm var (ExQ \<phi>) \<Gamma> = None"
assms: shows "fds_abs_converges (fds_norm f) s"
bound: shows "\<forall>\<^sub>F x in at_top. \<forall>y\<in>A. norm (f y x) \<le> g x"
set_preorder: fixes t :: "'b tree" shows "set (preorder t) = set_tree t"
contours_in_ve_upds_binds: fixes ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and Q :: "contour \<Rightarrow> bool"   and \<beta>' :: "nat \<Rightarrow> contour option"   and b'' :: "contour"   and ls :: "((nat \<times> char list) \<times> lambda) list" assumes "\<forall>b'\<in>contours_in_ve ve. Q b'"   and "\<forall>b'\<in>ran \<beta>'. Q b'" shows "\<forall>b'\<in>contours_in_ve (ve ++ map_of (map (\<lambda>(v, l). ((v, b''), \<A> (L l) \<beta>' ve)) ls)). Q b'"
preserves_reflects_arr: fixes f :: "'a" shows "C.arr (trg\<^sub>C f) = C.arr f"
lazy_conj: fixes A :: "bool"   and B :: "bool" shows "(if A then B else False) = (A \<and> B)"
monomials_monom_mult: fixes mon :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a"   and q :: "'a mpoly" shows "monomials (MPoly_Type.monom mon a * q) = (if a = (0::'a) then {} else (+) mon ` monomials q)"
AbGroup_inner_dirsum_el_decomp_plus: fixes Gs :: "'a set list"   and x :: "'a"   and y :: "'a" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "x \<in> (\<Oplus>G\<leftarrow>Gs. G)"   and "y \<in> (\<Oplus>G\<leftarrow>Gs. G)" shows "\<Oplus>Gs\<leftarrow> (x + y) = map2 (+) (\<Oplus>Gs\<leftarrow> x) (\<Oplus>Gs\<leftarrow> y)"
eq_LConsD: fixes xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" assumes "xs = y ## ys" shows "xs \<noteq> LNil \<and> lhd xs = y \<and> ltl xs = ys"
hypreal_of_real_isLub_iff: fixes Q :: "real set"   and Y :: "real" shows "isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y) = isLub UNIV Q Y"
hyps_2: assumes "consistent t\<^sub>2" shows "cost t\<^sub>2 = (\<Sum>a\<in>alphabet t\<^sub>2. freq t\<^sub>2 a * depth t\<^sub>2 a)"
eq_iff: fixes x :: "'c"   and y :: "'c" shows "(x = y) = (x \<le> y \<and> y \<le> x)"
list_all2_append: fixes xs :: "'c list"   and ys :: "'d list"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and us :: "'c list"   and vs :: "'d list" assumes "length xs = length ys" shows "list_all2 P (xs @ us) (ys @ vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
\<alpha>: shows "4 / 3 \<le> \<alpha>"
zero_neq_one_ow_axioms: shows "zero_neq_one_ow U 1\<^sub>o\<^sub>w 0\<^sub>o\<^sub>w"
linear_add: fixes k :: "enat"   and f :: "('c \<Rightarrow> real) \<Rightarrow> real"   and b1 :: "'c \<Rightarrow> real"   and b2 :: "'c \<Rightarrow> real" assumes "\<forall>x. x \<in> manifold_eucl.diff_fun_space k \<longrightarrow> f x \<in> UNIV"   and "b1 \<in> manifold_eucl.diff_fun_space k"   and "b2 \<in> manifold_eucl.diff_fun_space k"   and "linear_on (manifold_eucl.diff_fun_space k) UNIV (*\<^sub>R) (*\<^sub>R) f" shows "f (b1 + b2) = f b1 + f b2"
wfVE: shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
lexord_append_rightI: fixes y :: "'a list"   and x :: "'a list"   and r :: "('a \<times> 'a) set" assumes "\<exists>b z. y = b # z" shows "(x, x @ y) \<in> lexord r"
partn_lst_triv0: fixes \<alpha> :: "V list"   and i :: "nat"   and n :: "nat"   and r :: "('a \<times> 'a) set"   and B :: "'a set" assumes "\<alpha> ! i = 0"   and "i < length \<alpha>"   and "n \<noteq> 0" shows "Partitions.partn_lst r B \<alpha> n"
mix_pmf_in_lotteries: fixes p :: "'b pmf"   and A :: "'b set"   and q :: "'b pmf"   and a :: "real" assumes "p \<in> lotteries_on A"   and "q \<in> lotteries_on A"   and "a \<in> {0<..<1}" shows "mix_pmf a p q \<in> lotteries_on A"
intro: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "\<And>x y. \<lbrakk>x \<in> D; y \<in> D; cmp x y = EQUAL\<rbrakk> \<Longrightarrow> x = y" shows "eq_linorder_on_axioms D cmp"
permutations_of_setI: fixes xs :: "'a list"   and A :: "'a set" assumes "set xs = A"   and "distinct xs" shows "xs \<in> permutations_of_set A"
ideD_1: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.arr a"
unset_return_status_def: shows "unset_return_status = (\<lambda>\<sigma>. Some ((), \<sigma>\<lparr>return_status := False\<rparr>))"
simps_1: fixes ys :: "'b llist" shows "lnil :++ ys = ys"
finite_surj: fixes A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "B \<subseteq> f ` A" shows "finite B"
Pmod_0_principal: fixes t :: "'a"   and g :: "'a" assumes "Idomain S"   and "t \<in> carrier S"   and "g \<in> carrier R"   and "P_mod R S X (S \<diamondsuit>\<^sub>p t) g" shows "\<exists>h\<in>carrier R. g = t \<cdot>\<^sub>r h"
square_free_iff_separable: fixes f :: "'a poly" shows "square_free f = separable f"
pdo12: shows "partial_density_operator (\<rho>\<^sub>1 + \<rho>\<^sub>2)"
sup_assoc: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
is_weak_rankingI: fixes xs :: "'b set list" assumes "{} \<notin> set xs"   and "\<And>i j. \<lbrakk>i < length xs; j < length xs; i \<noteq> j\<rbrakk> \<Longrightarrow> xs ! i \<inter> xs ! j = {}" shows "is_weak_ranking xs"
termMOD_igSubstCls: shows "igSubstCls termMOD"
IH: shows "\<forall>m<n. \<forall>\<Gamma> E. ( \<Gamma> \<Rightarrow>* E, m) \<in> derivable R* \<longrightarrow> ( \<Gamma> + \<Gamma>' \<Rightarrow>* E, m) \<in> derivable R*"
forest_alt1: fixes a :: "'b"   and b :: "'b"   and F :: "'b uprod set" assumes "Upair a b \<in> F"   and "forest F"   and "\<And>e. e \<in> F \<Longrightarrow> proper_uprod e" shows "(a, b) \<notin> uconnected (F - {Upair a b})"
reachable0: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "system.reachable s\<^sub>0 lstep s\<^sub>0"
one_le_inverse: fixes a :: "'a" assumes "(0::'a) < a"   and "a \<le> (1::'a)" shows "(1::'a) \<le> inverse a"
unity_def: shows "D'.unity \<equiv> D.cod \<iota>\<^sub>D"
list_all2_trans: fixes P1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and P3 :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and as :: "'d list"   and bs :: "'e list"   and cs :: "'f list" assumes "\<And>a b c. \<lbrakk>P1 a b; P2 b c\<rbrakk> \<Longrightarrow> P3 a c"   and "list_all2 P1 as bs"   and "list_all2 P2 bs cs" shows "list_all2 P3 as cs"
mat_adjoint_def': fixes M :: "'b mat" shows "mat_adjoint M = (map_mat conjugate M)\<^sup>T"
fields_mono: fixes G :: "prog"   and C :: "qtname"   and fn :: "fspec"   and f :: "Decl.field"   and D :: "qtname" assumes "table_of (DeclConcepts.fields G C) fn = Some f"   and "G\<turnstile>D\<preceq>\<^sub>C C"   and "is_class G D"   and "wf_prog G" shows "table_of (DeclConcepts.fields G D) fn = Some f"
wivs_index_integer_code: shows "wivs_index_integer = dflt_size_integer - 1"
strongSimWeakSim: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto>[Rel] Q" shows "P \<leadsto>\<^sup>^<Rel> Q"
lhd_llist_of: fixes xs :: "'b list" shows "lhd (llist_of xs) = hd xs"
RETURN_RES_refine_iff: fixes x :: "'d"   and R :: "('d \<times> 'e) set"   and Y :: "'e set" shows "(RETURN x \<le> \<Down> R (RES Y)) = (\<exists>y\<in>Y. (x, y) \<in> R)"
finprod_def: fixes G :: "('a, 'b) monoid_scheme"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "finprod G f A = (if finite A then foldD (carrier G) ((\<otimes>\<^bsub>G\<^esub>) \<circ> f) \<one>\<^bsub>G\<^esub> A else \<one>\<^bsub>G\<^esub>)"
uniformity_fls_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
non_zero_vectors_def: shows "non_zero_vectors \<equiv> {x. x \<noteq> (0::'a)}"
assms_2: fixes P :: "pi"   and R :: "pi" assumes "(P, Q) \<in> Rel"   and "(R, !Q) \<in> bangRel Rel" shows "F (P \<parallel> R)"
op_intersects_def: fixes X :: "'c set"   and sctn :: "'c sctn" shows "op_intersects X sctn = Inf_inner X (normal sctn) \<bind> (\<lambda>ii. Sup_inner X (normal sctn) \<bind> (\<lambda>si. RETURN (ii \<le> pstn sctn \<and> pstn sctn \<le> si)))"
simps_1: fixes x21 :: "'c"   and x22 :: "'c"   and x23 :: "'d"   and y21a :: "'c"   and y22a :: "'c"   and y23a :: "'d" shows "(MkArr x21 x22 x23 = MkArr y21a y22a y23a) = (x21 = y21a \<and> x22 = y22a \<and> x23 = y23a)"
alphaYXyx: shows "Y #[[y \<and> x]]_xs #= X #[[y \<and> x]]_xs"
arr_dom: fixes f :: "'d" assumes "D.arr f" shows "D.arr (D.dom f)"
Mapping_lookup_map: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'a"   and m :: "('c, 'd) mapping"   and a :: "'b" shows "Mapping.lookup (Mapping.map f g m) a = map_option g (Mapping.lookup m (f a))"
\<alpha>_simp: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.arr f"   and "D.arr g"   and "D.arr h" shows "\<a>\<^sub>D[f, g, h] = (f \<otimes>\<^sub>D g \<otimes>\<^sub>D h) \<cdot>\<^sub>D \<a>\<^sub>D[D.dom f, D.dom g, D.dom h]"
TOTAL: fixes r :: "('d \<times> 'd) set" assumes "wo_rel r" shows "Total r"
aform_err_to_aform_def: fixes X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" shows "aform_err_to_aform X n = (fst (fst X), pdev_upd (snd (fst X)) n (snd X))"
ide_map\<^sub>0_obj: fixes a :: "'c" assumes "C.obj a" shows "C.ide (\<epsilon>'.map\<^sub>0 a)"
map_simp: shows "map (\<lambda>i. R Quot I i) [0..<Suc (Suc n)] = (R Quot I 0) # map (\<lambda>i. R Quot I (Suc i)) [0..<Suc n]"
lm060: fixes P :: "('a \<times> 'b) set" shows "inj_on fst P = inj_on (snd \<circ> flip) P"
cccFix_nonrec_def: fixes x :: "Vars.var"   and e :: "exp" shows "cccFix_nonrec x e = (\<Lambda> i. (Afix_nonrec x e\<cdot>i, CCfix_nonrec x e\<cdot>i))"
g: shows "\<guillemotleft>g : a' \<rightarrow>\<^sub>C a\<guillemotright>"
assms: fixes a :: "'a" assumes "a \<in> as" shows "monic (f a)"
overlap_intro: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "PM"   and "P z x"   and "P z y" shows "O x y"
n: shows "\<^bold>g [^] n = \<^bold>g [^] a \<otimes> \<^bold>g [^] b"
zero_pnat_def: shows "0 \<equiv> pnat.PInfty"
collapse_empty_iff_empty: fixes s :: "nat \<Rightarrow> 'b" shows "emptyseq (\<natural> s) = emptyseq s"
a: shows "\<forall>x\<in>S. i < IArray.length (f x)"
at_phl_while: fixes p :: "'a"   and s :: "'a"   and x :: "'a"   and w :: "'a" assumes "\<lbrace>\<alpha> p \<cdot> \<tau> s\<rbrace>x\<lbrace>\<alpha> p\<rbrace>" shows "\<lbrace>\<alpha> p\<rbrace>it (\<tau> s \<cdot> x) \<cdot> \<tau> w\<lbrace>\<alpha> p \<cdot> \<tau> w\<rbrace>"
not_isFault_iff: fixes t :: "('e, 'f) xstate" shows "(\<not> isFault t) = (\<forall>f. t \<noteq> Fault f)"
le_defined_1: fixes x :: "Integer"   and y :: "Integer" assumes "le\<cdot>x\<cdot>y = TT" shows "x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>"
Agent_Exp: fixes X :: "agent"   and X' :: "msg"   and Y' :: "msg" shows "Messages.Agent X \<noteq> Exp X' Y'"
NewElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and d :: "Class" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (new d)" shows "\<exists>a. a = nextLoc h \<and> n = Suc 0 \<and> t = (RVal (Loc a) # os, S, h[a\<mapsto>(d, emp)]) \<and> ll = l + 1"
karatsuba_cost_pos: fixes n :: "nat" assumes "1 \<le> n" shows "0 < karatsuba_cost n"
sub1_preserves_triangle: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "sub1 mf q k l (r, A) = (r', A')"   and "triangular_to l A"   and "l < dim_row A"   and "l < dim_col A"   and "l + k < dim_row A" shows "triangular_to l A'"
diff_eq_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.group_add minus plus zero uminus" shows "(minus a b = c) = (a = plus c b)"
pairwise_orthogonal_proj_set: fixes C :: "'a set"   and a :: "'a" assumes "finite C"   and "pairwise orthogonal C" shows "pairwise orthogonal (insert (a - proj_onto a C) C)"
Disj_cong: fixes H :: "fm set"   and A :: "fm"   and A' :: "fm"   and B :: "fm"   and B' :: "fm" assumes "H \<turnstile> A IFF A'"   and "H \<turnstile> B IFF B'" shows "H \<turnstile> A OR B IFF A' OR B'"
eq: fixes U :: "'c set" shows "{x \<in> topspace X. f x \<in> topspace X' \<and> g (f x) \<in> U} = {x \<in> topspace X. g (f x) \<in> U}"
arrE: fixes f :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f" shows "\<F>C.arr f \<and> (\<F>C.ide (\<F>C.dom f) \<and> Diag (\<F>C.DOM (\<F>C.dom f))) \<and> \<F>C.ide (\<F>C.cod f) \<and> Diag (\<F>C.DOM (\<F>C.cod f))"
a_Roll_inv_V: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV' :: "Syntax.term"   and \<alpha> :: "Syntax.tvar"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> e, eP, Roll eV' : Mu \<alpha> \<tau>" obtains e' :: "Syntax.term" and eP' :: "Syntax.term" where "e = Roll e'" and "eP = Roll eP'" and "\<Gamma> \<turnstile> e', eP', eV' : subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>"
update_by_pair_not_0: fixes xs :: "('b \<times> 'a) list"   and ko :: "'b key_order"   and kv :: "'b \<times> 'a" assumes "(0::'a) \<notin> snd ` set xs" shows "(0::'a) \<notin> snd ` set (update_by_pair_ko ko kv xs)"
comm_semiring_1_def: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" shows "class.comm_semiring_1 times one plus zero \<equiv> class.comm_monoid_mult times one \<and> class.comm_semiring_0 plus zero times \<and> class.zero_neq_one one zero"
that: fixes c6 :: "real" assumes "\<And>x. x\<^sub>0 \<le> x \<Longrightarrow> f x \<le> c6 * f_approx x"   and "0 < c6" shows "thesis"
SeqRedsThrow: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>throw e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e;; e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>throw e',s',b'\<rangle>"
deleteMin_correct_aux_1: fixes q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q)"   and "q \<noteq> []" shows "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt (BsSkewBinomialHeapStruc.deleteMin q))"
IVT_two_functions: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on {a..b} f"   and "continuous_on {a..b} g"   and "f a < g a"   and "g b < f b"   and "a \<le> b" shows "\<exists>x\<in>{a..b}. f x = g x"
bin_snd_mismatch: shows "symcode.\<alpha> \<cdot> [c\<^sub>1] \<le>p u\<^sub>1 \<cdot> symcode.\<alpha>"
nn_integral_spmf: fixes A :: "'c set"   and p :: "'c spmf" shows "\<integral>\<^sup>+ x. ennreal (spmf p x) \<partial>count_space A = emeasure (measure_spmf p) A"
intro: fixes A :: "V"   and f :: "V" assumes "nop A (2\<^sub>\<nat>) f" shows "binop A f"
priority_def: fixes q :: "('c, 'd) pq" shows "priority q = map_of (alist_of q)"
h_def: shows "h \<equiv> \<lambda>(s, b). if b then f s else - f s"
iallI: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "\<And>t. t \<in> I \<Longrightarrow> P t" shows "\<box> t I. P t"
fresh_iff_non_subst_dbfm: fixes i :: "name"   and t :: "dbfm" shows "(subst_dbfm DBZero i t = t) = atom i \<sharp> t"
"6_2": shows "infs (\<lambda>x. x \<in> V \<and> gaccepting A x) r"
pred_subset_eq: fixes R :: "'d set"   and S :: "'d set" shows "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) = (R \<subseteq> S)"
lappend_ltake_enat_ldropn: fixes n :: "nat"   and xs :: "'b llist" shows "lappend (ltake (enat n) xs) (ldropn n xs) = xs"
mrbst_union_Leaf_left: shows "mrbst_union \<langle>\<rangle> = return_pmf"
gfinprod_mono_neutral_cong_right: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and h :: "'b \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "A \<subseteq> B"   and "\<And>i. i \<in> B - A \<Longrightarrow> g i = \<one>\<^bsub>homology_group p X\<^esub>"   and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"   and "g \<in> B \<rightarrow> carrier (homology_group p X)" shows "gfinprod (homology_group p X) g B = gfinprod (homology_group p X) h A"
wary_arg: fixes t :: "('s, 'v) tm" assumes "wary t" shows "wary (arg t)"
equivalence_map_preserved_by_iso: fixes f :: "'b"   and f' :: "'b" assumes "D.equivalence_map f"   and "D.isomorphic f f'" shows "D.equivalence_map f'"
delete_id: fixes k :: "'c"   and al :: "('c \<times> 'd) list" assumes "k \<notin> fst ` set al" shows "delete k al = al"
add_less_mono1: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j" shows "i + k < j + k"
eqI: fixes k :: "int"   and l :: "int" assumes "\<And>n. bit k (numeral n) = bit l (numeral n)"   and "even k = even l" shows "k = l"
cos_squared_eq: fixes x :: "'a" shows "(cos x)\<^sup>2 = (1::'a) - (sin x)\<^sup>2"
tendsto_at_left_sequentially: fixes b :: "'b"   and a :: "'b"   and X :: "'b \<Rightarrow> 'c"   and L :: "'c" assumes "b < a"   and "\<And>S. \<lbrakk>\<And>n. S n < a; \<And>n. b < S n; incseq S; S \<longlonglongrightarrow> a\<rbrakk> \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" shows "(X \<longlongrightarrow> L) (at_left a)"
closed_segment_eq_sing: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(closed_segment a b = {c}) = (a = c \<and> b = c)"
loop_orientation_split_inf_1: fixes y :: "'a"   and x :: "'a" shows "loop_orientation x y = split (y \<sqinter> (1::'a)) y x"
lowEquivalence_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and Exit :: "'b"   and H :: "'d set"   and L :: "'d set"   and High :: "'b"   and Low :: "'b"   and s :: "'c"   and s' :: "'c" assumes "NonInterferenceIntraGraph sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice Exit H L High Low" shows "NonInterferenceIntraGraph.lowEquivalence state_val L s s' \<equiv> \<forall>V\<in>L. state_val s V = state_val s' V"
error_msg_ne_SUCCES_4: fixes i :: "'a"   and m :: "char list" shows "\<not> is_cfound (error_msg i m)"
Collect_mem_eq: fixes A :: "'c set" shows "{x. s2p A x} = A"
pow_one: fixes x :: "nat" shows "b ^ x \<le> (1::'a)"
fw_shortest_path_up_to: fixes n :: "nat"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (0::'a) \<le> len m i i xs"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "D m i j k = fw m n k i j"
uint_range': fixes x :: "'b word" shows "0 \<le> uint x \<and> uint x < 2 ^ LENGTH('b)"
generated_ring_def: shows "G.generated_ring = {\<Union> C |C. C \<subseteq> generated_ring \<and> finite C \<and> disjoint C}"
subst_support_iff_mgt_if_subst_idem: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "subst_idem \<theta>" shows "(\<theta> \<preceq>\<^sub>\<circ> \<delta>) = (\<theta> supports \<delta>)"
wellformed_transaction_wf\<^sub>s\<^sub>s\<^sub>t_3: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" assumes "wellformed_transaction T" shows "set (transaction_fresh T) \<inter> bvars_transaction T = {}"
intros_2: fixes R :: "pi"   and T :: "pi"   and Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi" assumes "(R, T) \<in> Rel"   and "(P, Q) \<in> bangRel Rel" shows "(R \<parallel> P, T \<parallel> Q) \<in> bangRel Rel"
2: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" assumes "b = receive\<langle>t\<rangle>" shows "unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # A) \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>) = send\<langle>t \<cdot> \<theta>\<rangle> # unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)"
angle_action_def: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<diamond>\<langle>P\<rangle>_v \<equiv> \<not> \<box>[\<not> P]_v"
IH_2: fixes Rs :: "('a \<times> 'a) set set" assumes "finite Rs" shows "card ((\<lambda>R. eval_sorter R (f False)) ` Rs) \<le> 2 ^ count_wc_queries Rs (f False)"
distinct_1: fixes x1 :: "fresh_t"   and x2 :: "ltkey" shows "sesK x1 \<noteq> ltK x2"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (gcd x xa) = gcd (int_of_integer x) (int_of_integer xa)"
adjacent_refl: fixes x :: "'b set" assumes "x \<noteq> {}" shows "x \<sim> x"
suminf: fixes X :: "nat \<Rightarrow> complex" assumes "summable X" shows "cnj (\<Sum>n. X n) = (\<Sum>n. cnj (X n))"
assms_1: shows "indexed_const \<in> ring_hom R L"
HPhase1or2ReadElse_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Inv2a s"   and "HPhase1or2ReadElse s s' p d q" shows "Inv2a s'"
closed_1: shows "has_all_wt_instances_of \<Gamma> (subterms\<^sub>s\<^sub>e\<^sub>t (set M)) (set M)"
true_cls_def: fixes I :: "'b set"   and C :: "'b literal multiset" shows "(I \<TTurnstile> C) = (\<exists>L\<in>#C. I \<TTurnstile>l L)"
respects_natural_isomorphism: fixes F' :: "'b \<Rightarrow> 'a"   and \<tau> :: "'b \<Rightarrow> 'a"   and G' :: "'a \<Rightarrow> 'b"   and \<mu> :: "'a \<Rightarrow> 'b" assumes "natural_isomorphism D C F' F \<tau>"   and "natural_isomorphism C D G G' \<mu>" shows "meta_adjunction C D F' G' (\<lambda>y f. D (\<mu> (\<phi>\<psi>.C.cod f)) (\<phi> y (C f (inverse_transformation.map D C F \<tau> y)))) (\<lambda>x g. C (\<psi> x (D (inverse_transformation.map C D G' \<mu> x) g)) (\<tau> (\<phi>\<psi>.D.dom g)))"
linear_fun_on_add: fixes S :: "'c set"   and x :: "'c \<Rightarrow> real"   and y :: "'c \<Rightarrow> real" assumes "linear_fun_on S x"   and "linear_fun_on S y"   and "subspace S" shows "linear_fun_on S (x + y)"
assms: fixes h' :: "'a"   and h'' :: "'a" assumes "h \<oplus> h' = Some h''"   and "P h'" shows "Q h''"
elims: fixes x :: "('c, 'd) lit"   and y :: "Preliminaries.var set" assumes "nvL x = y" obtains at :: "('c, 'd) atm" where "x = Pos at__" and "y = nvA at__"   | at :: "('c, 'd) atm" where "x = Neg at" and "y = {}"
minus_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
map_option_eq_Some: fixes f :: "'e \<Rightarrow> 'd"   and xo :: "'e option"   and y :: "'d" shows "(map_option f xo = \<lfloor>y\<rfloor>) = (\<exists>z. xo = \<lfloor>z\<rfloor> \<and> f z = y)"
iso_set_refl: fixes G :: "('a, 'b) monoid_scheme" shows "(\<lambda>x. x) \<in> Group.iso G G"