hcju/isabelleps huggingface dataset:
forest_subsets: fixes G :: "('c, 'd) graph"   and H :: "('c, 'd) graph" assumes "forest G"   and "valid_graph H"   and "edges H \<subseteq> edges G"   and "nodes H \<subseteq> nodes G" shows "forest H"
1: shows "type_update tn' (to_ct_list [t]) (Type [arity_1_result e]) = tm'"
in_aG: fixes x :: "'b" assumes "x \<in> segment G.face_cycle_succ a' a" shows "x \<in> arcs G - {a, a'}"
dotprod_append: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" assumes "length a = length b" shows "(a @ x) \<bullet> (b @ y) = a \<bullet> b + x \<bullet> y"
Di_def: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and y :: "nat list" shows "hlde_ops.Di a b i y = {Linear_Diophantine_Equations.dij a b i j - 1 |j. j < length y \<and> Linear_Diophantine_Equations.eij a b i j \<le> y ! j}"
assms_2: shows "big_d' a b y i \<noteq> []"
square_free_factor: fixes a :: "'a poly"   and p :: "'a poly" assumes "a dvd p"   and "square_free p" shows "square_free a"
index_vec: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "i < n" shows "vec n f $ i = f i"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> CC.arr f" shows "local.Prod f = null"
Gauss_Jordan_wrapper_def: fixes i :: "'c"   and j :: "'d"   and A :: "(('a, 'b) vec, 'c) vec"   and B :: "(('a, 'd) vec, 'c) vec" shows "Gauss_Jordan_wrapper i j A B = (\<chi>s. if s = i then A $ s else row_add A s i (- B $ s $ j) $ s)"
invf_closed1: fixes x :: "'a" assumes "x \<in> carrier K - {\<zero>}" shows "x\<^bsup>\<hyphen> K\<^esup> \<in> carrier K - {\<zero>}"
disc_martingale_charact: fixes M :: "'b measure"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and F :: "nat \<Rightarrow> 'b measure" assumes "\<forall>n. integrable M (X n)"   and "Filtration.filtration M F"   and "\<forall>m. sigma_finite_subalgebra M (F m)"   and "\<forall>m. X m \<in> borel_measurable (F m)"   and "\<forall>n. AEeq M (real_cond_exp M (F n) (X (Suc n))) (X n)" shows "martingale M F X"
fin_cut_sameE: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "fin_cut_same x xs = ys" shows "\<exists>m. xs = ys @ replicate m x"
dim_ty_sem_DimInv_def: fixes x :: "('e\<^sup>-\<^sup>1) itself" shows "dim_ty_sem x = inverse QD('e)"
prev: fixes f :: "complex \<Rightarrow> complex" assumes "continuous_on S f"   and "\<forall>z\<in>S. f z \<noteq> 0" shows "\<exists>g. continuous_on S g \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)"
simps_1: fixes S :: "nat \<Rightarrow> bool option" shows "share [] S = S"
LSC_BigAndL_inv: fixes F :: "'b formula list"   and \<Gamma> :: "'b formula multiset" assumes "\<^bold>\<And>F, \<Gamma> \<Rightarrow>\<^sub>n"   and "\<forall>f\<in>set F. is_nnf f"   and "is_nnf_mset \<Gamma>" shows "mset F + \<Gamma> \<Rightarrow>\<^sub>n"
map_disjD: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 \<bottom> h\<^sub>1" shows "dom h\<^sub>0 \<inter> dom h\<^sub>1 = {}"
irreducible_dict: shows "irreducible \<equiv> comm_semiring_1.irreducible (*) (1::'a) (0::'a)"
proj2_eq_sqn: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>2 (the (rt dip)) = sqn rt dip"
approx_trans3: fixes x :: "'b star"   and r :: "'b star"   and s :: "'b star" assumes "x \<approx> r"   and "x \<approx> s" shows "r \<approx> s"
left_commute: fixes b :: "'g"   and a :: "'g"   and c :: "'g" shows "min b (min a c) = min a (min b c)"
hyp_sub: fixes I :: "qtname"   and i :: "iface" assumes "iface G I = Some i"   and "\<forall>J\<in>set (isuperIfs i). (I, J) \<in> subint1 G \<and> P J \<and> is_iface G J" shows "P I"
divmod_abs_def: fixes k :: "integer"   and l :: "integer" shows "Code_Numeral.divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
merge_guards_res_While: fixes c :: "('d, 'e, 'f) com"   and b :: "'d set"   and c' :: "('d, 'e, 'f) com" assumes "merge_guards c = While b c'" shows "\<exists>c''. c = While b c'' \<and> merge_guards c'' = c'"
sel_1: fixes x1 :: "((('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) list"   and x2 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x3 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x4 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x5 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x6 :: "('h strand_label \<times> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step) list" shows "transaction_fresh (Transaction x1 x2 x3 x4 x5 x6) = x1"
that: fixes z :: "(real, 2) vec" assumes "z \<in> path_image f"   and "z \<in> path_image g" shows "thesis"
contD: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lubb :: "'b set \<Rightarrow> 'b"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and Y :: "'a set" assumes "cont luba orda lubb ordb f"   and "Complete_Partial_Order.chain orda Y"   and "Y \<noteq> {}" shows "f (luba Y) = lubb (f ` Y)"
1: shows "a\<^sub>1 = a\<^sub>2 \<and> w\<^sub>1 = w\<^sub>2"
sup_right_divisibility: fixes y :: "'a"   and x :: "'a" shows "(y \<le> x) = (\<exists>z. z \<sqinter> x = y)"
rcos_self: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier G"   and "subgroup H G" shows "x \<in> H #> x"
SubstTermP_vquot_dbtm: fixes w :: "name"   and V :: "name set"   and V' :: "name set"   and dbtm :: "dbtm" assumes "w \<in> Vs - V"   and "V \<subseteq> Vs"   and "V' = p \<bullet> V"   and "supp dbtm \<subseteq> atom ` Vs" shows "insert (ConstP (F w)) {ConstP (F i) |i. i \<in> V} \<turnstile> SubstTermP \<guillemotleft>Var w\<guillemotright> (F w) (ssubst (vquot_dbtm V dbtm) V F) (subst w (F w) (ssubst (vquot_dbtm (insert w V) dbtm) V F))"
less_bf: fixes j' :: "nat"   and i' :: "nat"   and j :: "nat"   and i :: "nat"   and k :: "nat"   and M :: "nat set"   and r :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "((j', i'), j, i) \<in> pair_less"   and "(j', i') \<in> IJ k"   and "infinite M" shows "fst (bf M r (j', i')) \<lless> fst (bf M r (j, i))"
monomial_pmdl_field: fixes B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and v :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'b" assumes "is_monomial_set B"   and "p \<in> local.punit.pmdl B"   and "v \<in> keys p" shows "monomial c v \<in> local.punit.pmdl B"
ccSUP_le_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and u :: "'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A" shows "less_eq (Sup (f ` A)) u = (\<forall>i\<in>A. less_eq (f i) u)"
Enabled_disj: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> Enabled (F \<or> G) = (Enabled F \<or> Enabled G)"
max_step_def: fixes t :: "nat \<Rightarrow> real"   and t1 :: "real"   and p :: "nat"   and L :: "real"   and B :: "real"   and r :: "real" shows "max_step t t1 p L B r \<equiv> grid t \<and> max_step_axioms t t1 p L B r"
index_of_r_to_l_def: shows "index_of_r_to_l = pair_by_index (pair_by_index index_of_c_fst (comp_by_index index_of_c_fst index_of_c_snd)) (comp_by_index index_of_c_snd index_of_c_snd)"
PiM_uniform_measure_permute: fixes g :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and a :: "real"   and b :: "real" assumes "g permutes A"   and "a < b" shows "distr (Pi\<^sub>M A (\<lambda>_. uniform_measure lborel {a..b})) (Pi\<^sub>M A (\<lambda>_. lborel)) (\<lambda>f. f \<circ> g) = Pi\<^sub>M A (\<lambda>_. uniform_measure lborel {a..b})"
union_single_eq_diff: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "add_mset x M = N" shows "M = remove1_mset x N"
simps_7: fixes Vs :: "char list list"   and e :: "char list exp"   and F :: "char list"   and D :: "char list" shows "compE\<^sub>1 Vs (e\<bullet>F{D}) = compE\<^sub>1 Vs e\<bullet>F{D}"
integral_nonneg: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a \<le> b"   and "integrable f a b"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x" shows "0 \<le> integral f a b"
\<b>_le: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and d :: "nat"   and i :: "nat" assumes "\<a> ps \<le> d"   and "\<And>h' U'. \<lbrakk>(h', U') \<in> set ps; i \<le> card U'\<rbrakk> \<Longrightarrow> poly_deg h' < d" shows "\<b> ps i \<le> d"
assms_13: shows "a \<le> - forest_components h \<sqinter> - - g"
value_n0_n_val_n0: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x \<noteq> 0" shows "n_val K v x \<noteq> 0"
simps_2: fixes \<sigma> :: "node \<rightharpoonup>\<^sub>f topbot"   and e :: "node fset \<times> command_gadget \<times> node fset" shows "next_ps \<sigma> (Inr e) = \<sigma> \<ominus> fst3 e ++\<^sub>f [ thd3 e |=> Top ]"
abs_eq: fixes x :: "'c set"   and xa :: "'c" assumes "eq_onp countable x x" shows "(xa \<in>\<^sub>c acset x) = (xa \<in> x)"
vimage_Un: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" shows "f -` (A \<union> B) = f -` A \<union> f -` B"
added_ids_Deliver_Store_collapse: fixes i :: "'c"   and e :: "'b"   and mo :: "'c"   and j :: "'c"   and e' :: "'b" shows "added_ids [Deliver (i, Store e mo j)] e' = []"
trace_core_eq_def: fixes core1 :: "('a, 'b, 'c, 'd, 'e, 'f) core"   and core2 :: "('g, 'b, 'c, 'd, 'e, 'f) core"   and E :: "'b set"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'g spmf" shows "trace_core_eq core1 core2 E IA IU p q = (\<forall>tr. set tr \<subseteq> E <+> IA \<times> UNIV <+> IU \<times> UNIV \<longrightarrow> rel_prod (rel_fun (eq_onp (\<lambda>e. e \<in> E)) (=)) (rel_prod (rel_fun (eq_onp (\<lambda>ia. ia \<in> IA)) (=)) (rel_fun (eq_onp (\<lambda>iu. iu \<in> IU)) (=))) (trace_core' core1 p tr) (trace_core' core2 q tr))"
Type_System_axioms: shows "Type_System E BMap DA lH AssignSideCondition WhileSideCondition IfSideCondition"
Fvars_ldsj: fixes \<phi>s :: "'fmla list" assumes "set \<phi>s \<subseteq> fmla" shows "Fvars (ldsj \<phi>s) = \<Union> (set (map Fvars \<phi>s))"
card_of_Times_ordLeq_infinite: fixes C :: "'b set"   and A :: "'c set"   and B :: "'d set" assumes "infinite C"   and "|A| \<le>o |C|"   and "|B| \<le>o |C|" shows "|A \<times> B| \<le>o |C|"
rule0: fixes a :: "'a" shows "a \<equiv> a + (0::'a)"
eq: shows "quot_of_ratfps (ratfps_shift n x) = (let (a, b) = quot_of_ratfps (x - ratfps_of_poly (ratfps_cutoff n x)) in (poly_shift n a, b))"
elims: fixes x :: "('c \<times> 'd) tree"   and y :: "'c set" assumes "Tree2.set_tree x = y" obtains "x = \<langle>\<rangle>" and "y = {}"   | l :: "('c \<times> 'd) tree" and a :: "'c" and uu_ :: "'d" and r :: "('c \<times> 'd) tree" where "x = \<langle>l, (a, uu_), r\<rangle>" and "y = {a} \<union> Tree2.set_tree l \<union> Tree2.set_tree r"
proj_in_setD: fixes a :: "'d strand_label \<times> 'e"   and l :: "'d"   and A :: "('d strand_label \<times> 'e) list" assumes "a \<in> set (proj l A)" obtains k :: "'d strand_label" and b :: "'e" where "a = (k, b)" and "k = ln l \<or> k = \<star>"
igWlsAbsDisj_fromMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) Recursion.model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.gWlsAbsDisj Delta MOD" shows "FixSyn.igWlsAbsDisj Delta (FixSyn.fromMOD Delta MOD)"
member_Id_on: fixes dxs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "DList_Set.member (DList_Set.Id_on dxs) = (\<lambda>(x, y). x = y \<and> DList_Set.member dxs x)"
card_le: fixes X :: "'a set" shows "card (linear_cf X) \<le> 1"
sentences_def: fixes P :: "'a option process" shows "sentences P \<equiv> {xs. xs @ [None] \<in> traces P}"
is_none_simps_2: fixes x :: "'m" shows "\<not> Option.is_none (Some x)"
select_cong: fixes k :: "nat"   and k' :: "nat"   and xs :: "'b list"   and xs' :: "'b list" assumes "k = k'"   and "mset xs = mset xs'" shows "select k xs = select k' xs'"
suminf_eq_zero_iff: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f"   and "\<And>n. (0::'b) \<le> f n" shows "(suminf f = (0::'b)) = (\<forall>n. f n = (0::'b))"
list_conj_nb: fixes ps :: "fm list" assumes "\<forall>p\<in>set ps. bound0 p" shows "bound0 (list_conj ps)"
real_v: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "enn2real (v cfg) = T.prob cfg {\<omega> \<in> space (T cfg). (HLD S1 suntil HLD S2) (state cfg ## \<omega>)}"
simple_path2_unsnoc: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n \<noteq> m" obtains ns' :: "'node list" where "g \<turnstile> n-ns'\<rightarrow>last ns'" and "last ns' \<in> set (predecessors g m)" and "distinct ns'" and "set ns' \<subseteq> set ns" and "m \<notin> set ns'"
ps_1: shows "set ps = (SIGMA i:{..<length ms}. if ms ! i = 0 then {0} else (\<lambda>p. p ^ multiplicity p (ms ! i)) ` prime_factors (ms ! i))"
inverse_le_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse a) (inverse b) = ((less zero (times a b) \<longrightarrow> less_eq b a) \<and> (less_eq (times a b) zero \<longrightarrow> less_eq a b))"
hyps_1: fixes s :: "'a set" assumes "n = card s" shows "finite s \<longrightarrow> (\<exists>l. set l = s \<and> sorted_wrt (to_ord r) l \<and> distinct l)"
prod_is_node_def: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list" shows "prod_is_node A B = (\<lambda>(i, j). dfa_is_node A i \<and> dfa_is_node B j)"
higher_differentiable_on: fixes S :: "'e set"   and f :: "'e \<Rightarrow> 'a"   and n :: "nat"   and g :: "'e \<Rightarrow> 'b" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on S g n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. f x ** g x) n"
2: shows "(\<box>a) \<^bold>\<cdot> a \<cong> a"
SN_iff_wf: fixes A :: "('a \<times> 'a) set" shows "SN A = wf (A\<inverse>)"
restrict_def: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
SeqThrow: fixes E :: "char list \<Rightarrow> ty option"   and r :: "nat \<times> char list list"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P,E \<turnstile> \<langle>Throw r;; e\<^sub>2,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
balanced_net_Pool: fixes m1 :: "(nat \<times> nat) convnet"   and m2 :: "(nat \<times> nat) convnet" assumes "balanced_net m1"   and "balanced_net m2"   and "count_weights True m1 = count_weights True m2" shows "balanced_net (Pool m1 m2)"
IH: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "match test r i j = (\<Squnion> (match test ` lpd test i r)) (i + 1) j"
fps_subdegree_mult_fps_X_2: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "subdegree (f * fps_X) = subdegree f + 1"
case_rews_2: fixes f1 :: "'r"   and f2 :: "'s \<rightarrow> 's tree \<rightarrow> 's tree \<rightarrow> 'r" shows "tree_case\<cdot>f1\<cdot>f2\<cdot>Null = f1"
add_0_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.monoid_add plus zero" shows "plus zero a = a"
sup_left_idem: fixes x :: "'a"   and y :: "'a" shows "x + (x + y) = x + y"
tendsto_at_right_realI_sequentially: fixes c :: "real"   and f :: "real \<Rightarrow> 'a"   and y :: "'a" assumes "\<And>X. filterlim X (at_right c) sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y" shows "(f \<longlongrightarrow> y) (at_right c)"
finite_measure_restr_to_subalg: fixes M :: "'c measure"   and F :: "'c measure" assumes "subalgebra M F"   and "finite_measure M" shows "finite_measure (restr_to_subalg M F)"
rel_witness_gpv2: fixes A :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and C :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and R :: "'n \<Rightarrow> 'p \<Rightarrow> bool"   and R' :: "'p \<Rightarrow> 'o \<Rightarrow> bool"   and gpv :: "('j, 'l, 'n) gpv"   and gpv' :: "('k, 'm, 'o) gpv" assumes "rel_gpv'' A C (R OO R') gpv gpv'"   and "left_unique R"   and "right_total R"   and "right_unique R'"   and "left_total R'" shows "rel_gpv'' (\<lambda>(a, b') b. b = b' \<and> A a b') (\<lambda>(c, d') d. d = d' \<and> C c d') R' (rel_witness_gpv A C R R' (gpv, gpv')) gpv'"
assms: shows "hs \<noteq> \<langle>\<rangle>"
basic: fixes x :: "'b set"   and B :: "'b set set" assumes "x \<in> B" shows "x \<in> topo B"
or_lta2_conga: 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 "A \<noteq> B"   and "C \<noteq> B"   and "D \<noteq> E"   and "F \<noteq> E" shows "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F \<or> Tarski_neutral_dimensionless.LtA Bet Cong D E F A B C \<or> Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F"
infinite_UNIV_vec: assumes "infinite UNIV" shows "infinite UNIV"
mopup_false1: fixes s :: "nat"   and n :: "nat"   and RR :: "bool" assumes "0 < s"   and "s \<le> 2 * n"   and "s mod 2 = Suc 0"   and "\<not> Suc s \<le> 2 * n" shows "RR"
de_morgan4: fixes x :: "'a"   and y :: "'a" shows "!(\<tt>\<tt> x \<oplus> \<tt>\<tt> y) = !x \<cdot> !y"
intros_1: fixes u :: "'b list"   and C :: "'b list set"   and b :: "'b list" assumes "u \<in> C"   and "u \<noteq> b" shows "u \<in> no_head_gen C b"
ind_mult_str_def: fixes V :: "'b set" shows "ind_mult_str V \<equiv> mult_str (X \<inter> V)"
p: shows "p = hd_prog p\<^sub>s\<^sub>b ((r # sb) @ sb')"
fg_start_round_smallest: fixes rho :: "nat \<Rightarrow> ('f, 'g, 'h, 'i) config_scheme"   and k :: "nat"   and p :: "'g"   and r :: "nat" assumes "round (rho k) p = r" shows "fg_start_round rho p r \<le> k"
iter_widen_def: fixes f :: "'a acom \<Rightarrow> 'a acom" shows "iter_widen f = while_option (\<lambda>c. \<not> f c \<sqsubseteq> c) (\<lambda>c. c \<nabla>\<^sub>c f c)"
intro: assumes "\<And>\<sigma> L. \<lbrakk>finite L; unifier\<^sub>l\<^sub>s \<sigma> L\<rbrakk> \<Longrightarrow> \<exists>\<theta>. mgu\<^sub>l\<^sub>s \<theta> L" shows "unification"
codomain_unique: fixes b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and b' :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "b \<in> codomains f"   and "b' \<in> codomains f" shows "b = b'"
fcard_finsert_if: fixes x :: "'b"   and A :: "'b fset" shows "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
wset_find_path_correct: fixes E :: "('b \<times> 'b) set"   and U0 :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "wset_find_path E U0 P \<le> find_path E U0 P"
simps_1: fixes uu :: "'b"   and x :: "'b" shows "Lib.the uu (Some x) = x"
g_nonneg: fixes x :: "nat" assumes "x\<^sub>1 \<le> x" shows "0 \<le> g x"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<Phi>\<Psi>'.map f) = Cop.arr f"
t_find_closest_pair_eq_time_find_closest_pair_tm: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ps :: "(int \<times> int) list" shows "t_find_closest_pair (c\<^sub>0, c\<^sub>1) ps = time (find_closest_pair_tm (c\<^sub>0, c\<^sub>1) ps)"
ok': shows "\<turnstile> (TVarB X T\<^sub>2 # \<Gamma>) ok"
invariantsHoldInValidRuns: fixes stateA :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set"   and stateB :: "State" assumes "invariantsHoldInState stateA F0 decisionVars"   and "(stateA, stateB) \<in> transitionRelation decisionVars" shows "invariantsHoldInState stateB F0 decisionVars"
neval_Suc_simp: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and x :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and y :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.neval eval x (Suc 0) y = ((x, y) \<in> eval)"
simps_2: fixes M :: "('fun, 'var) Term.term set"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) stateful_strand_step list" shows "\<lbrakk>M; D; send\<langle>t\<rangle> # S\<rbrakk>\<^sub>s = (\<lambda>\<I>. M \<turnstile> t \<cdot> \<I> \<and> \<lbrakk>M; D; S\<rbrakk>\<^sub>s \<I>)"
v2: shows "(\<lambda>x. x - P *\<^sub>V x) -` {0::'a} \<subseteq> range ((*\<^sub>V) P)"
gc_fair: fixes Ns :: "('a literal multiset \<times> label) set llist" assumes "chain (\<leadsto>GC) Ns"   and "FL.active_subset (lhd Ns) = {}"   and "FL.passive_subset (Liminf_llist Ns) = {}" shows "FL.empty_ord.fair Ns"
span_explicit': fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d set" assumes "module scale" shows "module.span scale b = {\<Sum>v | f v \<noteq> (0::'c). scale (f v) v |f. finite {v. f v \<noteq> (0::'c)} \<and> (\<forall>v. f v \<noteq> (0::'c) \<longrightarrow> v \<in> b)}"
strong_complement_character: fixes x :: "i" assumes "CEMC"   and "x \<noteq> u" shows "\<forall>v. P v (\<midarrow> x) = (\<not> O v x)"
f_g: fixes a :: "nat"   and b :: "nat" shows "f a / nat_power a s * (g b / nat_power b s) = f a * g b / nat_power (a * b) s"
simps_7: fixes x3 :: "'b" shows "Inner x3 \<noteq> vertex.SINK"
g_def: shows "g = foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<Suc k]"
gen_Image_def: fixes tol1 :: "'i \<Rightarrow> ('j \<times> 'k) list"   and mem2 :: "'j \<Rightarrow> 'l \<Rightarrow> bool"   and emp3 :: "'h"   and ins3 :: "'k \<Rightarrow> 'h \<Rightarrow> 'h"   and X :: "'i"   and Y :: "'l" shows "gen_Image tol1 mem2 emp3 ins3 X Y \<equiv> fold (\<lambda>(a, b). if mem2 a Y then ins3 b else id) (tol1 X) emp3"
bdd: fixes x :: "'e" assumes "x \<in> space M" shows "bdd_below ((\<lambda>i. F i x) ` I)"
length_subsets_smash: fixes n :: "nat"   and subs1 :: "nat list list"   and subs2 :: "nat list list" shows "length (subsets_smash n subs1 subs2) = length subs1 * length subs2"
mult_1_left: fixes a :: "'c" shows "comp_inf.covector a"
sum_SucD: fixes f :: "'k \<Rightarrow> nat"   and A :: "'k set"   and n :: "nat" assumes "sum f A = Suc n" shows "\<exists>a\<in>A. 0 < f a"
N_3: shows "invar \<langle>l, (xy, n), r\<rangle>"
d: shows "sing_reg (punit.lt (rep_list q) \<oplus> lt f2) (punit.lt (rep_list f2) \<oplus> lt q)"