hcju/isabelleps huggingface dataset:
not_sum_squares_lt_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "\<not> less (plus (times x x) (times y y)) zero"
concurrent_def: fixes s1 :: "'a \<times> ('a, 'b) operation"   and s2 :: "'a \<times> ('a, 'b) operation" shows "hb.concurrent s1 s2 \<equiv> \<not> hb s1 s2 \<and> \<not> hb s2 s1"
order_lesseq_imp: fixes x :: "'a"   and y :: "'a" shows "(\<forall>z\<ge>x. y \<le> z) = (y \<le> x)"
LCOSETS_def: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set" shows "lcosets\<^bsub>G\<^esub> H = (\<Union>a\<in>carrier G. {a <#\<^bsub>G\<^esub> H})"
Compl_in_sets_lebesgue: fixes A :: "'a set" shows "(- A \<in> sets lebesgue) = (A \<in> sets lebesgue)"
assms: shows "P \<longmapsto> a<x> \<prec> P'"
mm_equiv_low_eq: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
isos_compose: fixes f :: "('a S.E.term, 'a) S.arr"   and f' :: "('a S.E.term, 'a) S.arr" assumes "S.iso f"   and "S.iso f'"   and "S.seq f' f" shows "S.iso (f' \<cdot>\<^sub>S f)"
alw_holds_pred_stream_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a stream" shows "alw (holds P) xs = pred_stream P xs"
in_oreturnOk_eq: fixes x :: "'e"   and s :: "'f"   and v :: "'d + 'e" shows "(oreturnOk x s = Some v) = (v = Inr x)"
star_sup_1: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<oplus> y)\<^sup>\<odot> = x\<^sup>\<odot> \<odot> (y \<odot> x\<^sup>\<odot>)\<^sup>\<odot>"
Gsk_def: shows "Gsk = mat k n (\<lambda>(i, j). gso i $ j)"
Quantity_eq_intro: fixes x :: "('a, 'b, 'c) Quantity_scheme"   and y :: "('a, 'b, 'c) Quantity_scheme" assumes "mag x = mag y"   and "Quantity.dim x = Quantity.dim y"   and "Quantity.more x = Quantity.more y" shows "x = y"
solutions_complete: shows "local.Minimal_Solutions \<subseteq> set (solutions a b)"
homeomorphism_compose: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and U :: "'d set"   and h :: "'c \<Rightarrow> 'd"   and k :: "'d \<Rightarrow> 'c" assumes "homeomorphism S T f g"   and "homeomorphism T U h k" shows "homeomorphism S U (h \<circ> f) (g \<circ> k)"
convolution_density: fixes f :: "real \<Rightarrow> ennreal"   and g :: "real \<Rightarrow> ennreal" assumes "f \<in> borel_measurable borel"   and "g \<in> borel_measurable borel"   and "finite_measure (density lborel f)"   and "finite_measure (density lborel g)" shows "(density lborel f \<star> density lborel g) = density lborel (\<lambda>x. \<integral>\<^sup>+ y. f (x - y) * g y \<partial>lborel)"
IH: fixes s :: "('a, 'b) fmap"   and vs :: "'a set" assumes "sat_precond_as s as" shows "exec_plan (fmrestrict_set vs s) (as_proj as vs) = fmrestrict_set vs (exec_plan s as)"
first_ind_nth: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs" shows "first_ind xs (xs ! i) j = i + j"
linorder_ow_def: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "linorder_ow U le ls \<equiv> order_ow U le ls \<and> linorder_ow_axioms U le"
stereographic_North: fixes x :: "riemann_sphere" shows "(stereographic x = \<infinity>\<^sub>h) = (x = North)"
rel_reflp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflp R" shows "reflp (list_all2 R)"
fminus_finter_distrib: fixes C :: "'b fset"   and A :: "'b fset"   and B :: "'b fset" shows "C |\<inter>| (A |-| B) = C |\<inter>| A |-| (C |\<inter>| B)"
J_def: shows "J = {i \<in> I. F \<inter> S' i \<noteq> {}}"
lappend_assoc: fixes xs :: "'b llist"   and ys :: "'b llist"   and zs :: "'b llist" shows "(xs $ ys) $ zs = xs $ ys $ zs"
chain2: fixes y' :: "'b" assumes "y' \<in> Y" shows "Complete_Partial_Order.chain (\<le>) (f y' ` Y)"
sqrt_code: fixes n :: "nat" shows "local.sqrt n = Max (Set.filter (\<lambda>m. m\<^sup>2 \<le> n) {0..n})"
vimage2pI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and g :: "'d \<Rightarrow> 'b"   and y :: "'d" assumes "R (f x) (g y)" shows "BNF_Def.vimage2p f g R x y"
zero_smallest: fixes a :: "'a" shows "(0::'a) \<le> a"
Hermite_reduce_above_works: fixes n :: "nat"   and i :: "'a"   and a :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and j :: "'c"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'c" assumes "n \<le> mod_type_class.to_nat i"   and "mod_type_class.to_nat a < n" shows "Hermite_reduce_above A n i j res $ a $ b = row_add A a i ((res (A $ i $ j) (A $ a $ j) - A $ a $ j) div A $ i $ j) $ a $ b"
object_ptr_kinds_M_def: shows "object_ptr_kinds_M \<equiv> l_ptr_kinds_M.a_ptr_kinds_M object_ptr_kinds"
comm_add: fixes p :: "'a"   and x :: "'a"   and y :: "'a" assumes "test p"   and "p \<cdot> x = x \<cdot> p"   and "p \<cdot> y = y \<cdot> p" shows "p \<cdot> (x + y) = (x + y) \<cdot> p"
delta_gt_def: fixes \<delta> :: "'a" shows "delta_gt \<delta> \<equiv> \<lambda>x y. \<delta> \<le> x - y"
homeomorphic_def: fixes s :: "'b set"   and t :: "'c set" shows "s homeomorphic t \<equiv> \<exists>f g. homeomorphism s t f g"
add_less_le_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less a b"   and "less_eq c d" shows "less (plus a c) (plus b d)"
path_exists_if_no_separation: fixes S :: "'a set" assumes "S \<subseteq> V"   and "v0 \<notin> S"   and "v1 \<notin> S"   and "\<not> Separation G v0 v1 S" shows "\<exists>xs. v0 \<leadsto>xs\<leadsto> v1 \<and> set xs \<inter> S = {}"
uncurry_rec_nat: fixes fn :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and fs :: "nat \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'c"   and n :: "nat"   and a :: "'d"   and b :: "'e" shows "rec_nat fn fs n a b = rec_nat (\<lambda>(a, b). fn a b) (\<lambda>n rr (a, b). fs n (\<lambda>a b. rr (a, b)) a b) n (a, b)"
ex1_eqvt2: fixes p :: "perm"   and P :: "'a \<Rightarrow> bool" shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))"
nn_integral_fst_count_space: fixes f :: "'a \<times> 'b \<Rightarrow> ennreal" shows "(\<Sum>\<^sup>+ x. \<Sum>\<^sup>+ y. f (x, y)) = integral\<^sup>N (count_space UNIV) f"
subgraph_finite: fixes G :: "nat set \<times> nat set set"   and G' :: "nat set \<times> nat set set" assumes "finite_graph G"   and "subgraph G' G" shows "finite_graph G'"
abs_eq: shows "id_moebius_mmat = Abs_moebius_mat id_moebius_cmat"
oreturn_synth: fixes x :: "'a" shows "ospec (Some x) (\<lambda>r. r = x)"
Rep_lazy_inject: fixes x :: "'a lazy"   and y :: "'a lazy" shows "(Rep_lazy x = Rep_lazy y) = (x = y)"
fully_faithful_functor_axioms: shows "fully_faithful_functor (local.Right a') (local.Right a') (H\<^sub>R a')"
pathsConverge': fixes g :: "'g"   and x :: "'node"   and xs :: "'node list"   and y :: "'node"   and ys :: "'node list"   and z :: "'node" shows "old.pathsConverge g x xs y ys z = old.pathsConverge' g x xs y ys z"
succsr_def: shows "KBPAlg.succsr \<equiv> {(x, y). y \<in> set (k_succs x)}"
dependent_explicit: fixes s :: "'b set" shows "local.dependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *s v) = (0::'b) \<and> (\<exists>v\<in>t. u v \<noteq> (0::'a)))"
limit: shows "((\<lambda>b. (\<Sum>k = a..b. f (real_of_int k)) - F (real_of_int b) - (\<Sum>i<2 * N + 1. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real_of_int b))) \<longlongrightarrow> C) at_top"
insort_insert_key_def: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" shows "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
tr_init: fixes c :: "('a, 'b, 'c) configuration" shows "trace c [] c"
affine_hyperplane: fixes a :: "'c"   and b :: "real" shows "affine {x. a \<bullet> x = b}"
mult_right_inj: fixes n :: "nat" assumes "0 < n" shows "inj (\<lambda>x. x * n)"
berlekamp_basis_irreducible: fixes f :: "'a mod_ring poly"   and n :: "nat"   and us :: "'a mod_ring poly list"   and u :: "'a mod_ring poly" assumes "square_free f"   and "n = length (berlekamp_basis f)"   and "monic f"   and "f = prod_list us"   and "length us = n"   and "\<And>u. u \<in> set us \<Longrightarrow> 0 < degree u"   and "u \<in> set us" shows "irreducible u"
minus_mat_def: fixes A :: "'b mat"   and B :: "'b mat" shows "A - B \<equiv> mat (dim_row B) (dim_col B) (\<lambda>ij. A $$ ij - B $$ ij)"
convergentE: fixes T :: "'b set set"   and F :: "'b set set" assumes "F convergent\<^bsub>T\<^esub>" obtains x :: "'b" where "F \<longlongrightarrow>\<^bsub>T\<^esub> x" and "x \<in> carrier\<^bsub>T\<^esub>"
bdd_below_image_antimono: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "antimono f"   and "bdd_above A" shows "bdd_below (f ` A)"
hcomp_simps_1: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "C.hseq \<nu> \<mu>" shows "src\<^sub>C (\<nu> \<star>\<^sub>C \<mu>) = src\<^sub>C \<mu>"
\<rho>'_def: shows "\<rho>' = hd (renamings_apart [mset D, mset C])"
nn_integral_measure_pmf: fixes p :: "'a pmf"   and f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf p) f = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
rel_ANNOT_eq: fixes t :: "'a"   and R :: "('b \<times> 'a) set" shows "t \<equiv> t ::: R"
arrI: fixes f :: "'c \<times> 'c" assumes "C.arr (fst f) \<and> C.arr (snd f) \<and> src\<^sub>C (fst f) = trg\<^sub>C (snd f)" shows "C.VV.arr f"
sup_absorb2: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type) = y"
l11_44_2_b: fixes B :: "'p"   and A :: "'p"   and C :: "'p" assumes "B A C LtA B C A" shows "B C Lt B A"
assms_2: shows "length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<le> l"
a: shows "(X, Branch A B) \<in> set G \<and> subword w i k \<in> Lang G A \<and> subword w (i + k) (j - k) \<in> Lang G B \<and> Suc 0 \<le> k \<and> k < j"
vsubstAll_swapAll: fixes P :: "('d, 'e, 'varSort, 'var, 'f) param"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> (\<forall>y1 y2. {y1, y2} \<subseteq> varsOf P \<and> fresh ys y1 X \<longrightarrow> X #[y1 // y2]_ys = X #[y1 \<and> y2]_ys)) \<and> (goodAbs A \<longrightarrow> (\<forall>y1 y2. {y1, y2} \<subseteq> varsOf P \<and> freshAbs ys y1 A \<longrightarrow> A $[y1 // y2]_ys = A $[y1 \<and> y2]_ys))"
tendsto_mult_Pair: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.topological_semigroup_mult times open" shows "LIM x topological_space.nhds open a \<times>\<^sub>F topological_space.nhds open b. times (fst x) (snd x) :> topological_space.nhds open (times a b)"
remap_lang: fixes f :: "'Q \<Rightarrow> 'a" assumes "inj_on f Q" shows "ta_lang (ta_remap f TA) = ta_lang TA"
args_def: fixes X :: "exp" shows "args X = the_elem (\<Union>U\<in>Rep_Exp X. {Abs_ExpList (freeargs U)})"
fmfilter_true: fixes m :: "('d, 'e) fmap"   and P :: "'d \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x" shows "fmfilter P m = m"
wellformed_items_Complete: fixes I :: "item set"   and k :: "nat" assumes "wellformed_items I" shows "wellformed_items (Complete k I)"
compose: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h' :: "'c \<Rightarrow> 'a"   and G :: "'c filter" assumes "f \<in> O[F](g)"   and "filterlim h' F G" shows "(\<lambda>x. f (h' x)) \<in> O[G](\<lambda>x. g (h' x))"
inv\<^sub>1E_1: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V" shows "bp (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V})"
ratfps_of_poly_eq_0_iff: fixes x :: "'b poly" shows "(ratfps_of_poly x = 0) = (x = 0)"
has_field_derivative_powr_right: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0" shows "((powr) w has_field_derivative Ln w * w powr z) (at z)"
find_nonzeros_from_input_vec_def: fixes lhs_vec :: "rat vec" shows "find_nonzeros_from_input_vec lhs_vec = filter (\<lambda>i. lhs_vec $ i \<noteq> 0) [0..<dim_vec lhs_vec]"
ideal_pOp_closed: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "y \<in> I" shows "x \<plusminus>\<^bsub>R\<^esub> y \<in> I"
trivial_singleton_group: fixes a :: "'a" shows "trivial_group (singleton_group a)"
eval_tm_HPair: fixes e :: "(name, hf) finfun"   and a :: "tm"   and b :: "tm" shows "\<lbrakk>HPair a b\<rbrakk>e = \<langle>\<lbrakk>a\<rbrakk>e, \<lbrakk>b\<rbrakk>e\<rangle>"
nameTermFresh: fixes x :: "name"   and y :: "name" shows "x \<sharp> nameTerm y = (x \<noteq> y)"
exhaustive_fusion_def: fixes proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and g :: "('a, 'b) generator"   and s :: "'b" shows "proper_intrvl.exhaustive_fusion proper_interval g s = proper_intrvl.exhaustive proper_interval (list.unfoldr g s)"
cauchy_ine\<^sub>N_def: fixes N :: "'c quasinorm"   and u :: "nat \<Rightarrow> 'c" shows "cauchy_ine\<^sub>N N u = (\<forall>e>0. \<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. eNorm N (u n - u m) < e)"
grow_mining: fixes n :: "nat" assumes "length (bc n) < length (bc (Suc n))" shows "mining (Suc n)"
fps_Gcd_def: shows "Gcd = normalization_euclidean_semiring_class.Gcd"
intros_3: fixes a :: "'a"   and x :: "'b" shows "BPS (bsingleton (a, x)) \<sim> PMF (return_pmf (a, x))"
attach_shadow_root_is_weakly_component_safe_step: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and shadow_root_mode :: "shadow_root_mode"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> attach_shadow_root element_ptr shadow_root_mode \<rightarrow>\<^sub>h h'"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> attach_shadow_root element_ptr shadow_root_mode|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
istopology_base_eq: fixes P :: "'c set \<Rightarrow> bool" shows "istopology (arbitrary union_of P) = (\<forall>S T. P S \<and> P T \<longrightarrow> (arbitrary union_of P) (S \<inter> T))"
h6: shows "4 * k + 3 dvd a'\<^sup>2 + b'\<^sup>2"
CCexp_subst: fixes x :: "Vars.var"   and S :: "Vars.var set"   and y :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "x \<notin> S"   and "y \<notin> S" shows "ccExp_syn a e[y::=x] G|` S = ccExp_syn a e G|` S"
that: fixes x :: "'b" assumes "x \<in> S"   and "f x = Some y" shows "thesis"
assms: shows "\<forall>a. (\<exists>A. (a, A) \<in> R) \<longrightarrow> (\<forall>A. (a, A) \<in> R)"
dim_le_card': fixes s :: "'b set" assumes "finite s" shows "local.dim s \<le> card s"
NORMAL_Var: fixes a :: "name" shows "NORMAL (Var a)"
axiom_exists_def: fixes x :: "String.literal"   and p :: "fol fm" shows "axiom_exists x p \<equiv> Thm (Iff (Exists x p) (fm.Not (Forall x (fm.Not p))))"
simps_4: fixes \<alpha> :: "'b"   and t :: "('a, 'c, 'b) Tree" shows "S_transform_Tree (tAct \<alpha> t) = Act\<^sub>\<alpha> (S_action.Act \<alpha>) (S_transform_Tree t)"
split_face_empty_ram1_ram2_in_f21: fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face oldF ram1 ram2 []"   and "(f12, f21) = split_face oldF ram1 ram2 []" shows "(ram1, ram2) \<in> \<E> f21"
is_num_normalize_1: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.neg_numeral minus plus zero uminus" shows "plus (plus a b) c = plus a (plus b c)"
OrL: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and G :: "'b formula" assumes "F, \<Gamma> \<Rightarrow> \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<or> G, \<Gamma> \<Rightarrow> \<Delta>"
drop_Suc_Cons: fixes n :: "nat"   and x :: "'d"   and xs :: "'d list" shows "drop (Suc n) (x # xs) = drop n xs"
r': fixes msrFn :: "'a \<Rightarrow> nat"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<And>x. \<forall>y. (case (y, x) of (x, y) \<Rightarrow> msrFn x < msrFn y) \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
imp_recovery0: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "A \<turnstile> p .\<or>. (p .\<longrightarrow>. q)"
sum_list_map_cong: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.monoid_add plus zero"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "monoid_add.sum_list plus zero (map f xs) = monoid_add.sum_list plus zero (map g xs)"
snd_echelon_form_of_column_k_det: fixes det_P :: "'d"   and A :: "'d iarray iarray"   and i :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and k :: "nat" shows "snd (echelon_form_of_column_k_det_iarrays (det_P, A, i, bezout) k) = echelon_form_of_column_k_iarrays (A, i, bezout) k"
Lyndon_concat: fixes u :: "'a list"   and v :: "'a list" assumes "Lyndon u"   and "Lyndon v"   and "u <lex v" shows "Lyndon (u \<cdot> v)"
hnorm_minus_commute: fixes a :: "'a star"   and b :: "'a star" shows "hnorm (a - b) = hnorm (b - a)"
comp_null_1: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" shows "S.VV.comp S.VV.null f = S.VV.null"
in_keys_plusI2: fixes t :: "'b"   and q :: "'b \<Rightarrow>\<^sub>0 'c"   and p :: "'b \<Rightarrow>\<^sub>0 'c" assumes "t \<in> keys q"   and "t \<notin> keys p" shows "t \<in> keys (p + q)"
is_bicategory_of_spans: shows "bicategory_of_spans (\<bullet>) (\<star>) assoc unit src trg"
is_dghm_axioms: shows "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
image_mset_strict_mono: fixes M :: "'a multiset"   and N :: "'a multiset"   and f :: "'a \<Rightarrow> 'b" assumes "\<forall>x\<in>#M. \<forall>y\<in>#N. x < y \<longrightarrow> f x < f y"   and "M < N" shows "image_mset f M < image_mset f N"
a: shows "\<forall>b\<in>basis. \<forall>(k, \<gamma>)\<in>one_chain2. line_integral_exists F {b} \<gamma>"
SN_imp_SN_pow: fixes R :: "('b \<times> 'b) set"   and n :: "nat" assumes "SN R" shows "SN (R ^^ Suc n)"
czeroE: fixes r :: "('c \<times> 'c) set" assumes "r =o czero" shows "Field r = {}"
set_error_consistent: shows "\<forall>s t u a. vpeq u s t \<longrightarrow> vpeq u (set_error_code s a) (set_error_code t a)"
exhaust: fixes y :: "lit" obtains x1 :: "bool" and x2 :: "nat" where "y = Lit x1 x2"
hyps: fixes s' :: "nat"   and r :: "nat \<Rightarrow>f value option" assumes "obtains s c e s' r t" shows "recognises_execution e s' r t"
append_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(list_all2 A ===> list_all2 A ===> list_all2 A) (@) (@)"
Cong_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "Cong A B C D \<and> Cong A B D C \<and> Cong B A C D \<and> Cong B A D C \<and> Cong C D A B \<and> Cong C D B A \<and> Cong D C A B \<and> Cong D C B A"
rel_fset_alt_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fset R = (\<lambda>A B. (\<forall>x. \<exists>y. x |\<in>| A \<longrightarrow> y |\<in>| B \<and> R x y) \<and> (\<forall>y. \<exists>x. y |\<in>| B \<longrightarrow> x |\<in>| A \<and> R x y))"
bdd_all_is_node_subsetbdd: fixes A :: "bool list bdd list \<times> bool list"   and q :: "bool list" assumes "list_all (bdd_all (nfa_is_node A)) (fst A)"   and "nfa_is_node A q" shows "bdd_all (nfa_is_node A) (subsetbdd (fst A) q (nfa_emptybdd (length q)))"
f: fixes p :: "nat" assumes "p < n" shows "f p \<le> K"
assms_3: shows "\<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>"
butlast_eq_cons_conv: fixes l :: "'a list"   and x :: "'a"   and xs :: "'a list" shows "(butlast l = x # xs) = (\<exists>xl. l = x # xs @ [xl])"
has_derivative_inverse_strong: fixes S :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a" assumes "open S"   and "x \<in> S"   and "continuous_on S f"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"   and "(f has_derivative f') (at x)"   and "f' \<circ> g' = id" shows "(g has_derivative g') (at (f x))"
is_Delete_def: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" shows "is_Delete stateful_strand_step = (\<exists>x51 x52. stateful_strand_step = delete\<langle>x51,x52\<rangle>)"
in_set_zipE: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "(x, y) \<in> set (zip xs ys)" obtains "x \<in> set xs" and "y \<in> set ys"
cexpr_sem_marg_dens: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and x :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "x \<in> set vs"   and "\<rho> \<in> space (state_measure (set vs') \<Gamma>)" shows "AE v in stock_measure (\<Gamma> x). ennreal (extract_real (cexpr_sem (case_nat v \<rho>) (marg_dens_cexpr \<Gamma> vs x \<delta>))) = marg_dens (dens_ctxt_\<alpha> (vs, vs', \<Gamma>, \<delta>)) x \<rho> v"
Int_insert_left_if0: fixes a :: "'c"   and C :: "'c set"   and B :: "'c set" assumes "a \<notin> C" shows "insert a B \<inter> C = B \<inter> C"
keys_RBT_Impl_diag: fixes t :: "('c, 'd) rbt" shows "RBT_Impl.keys (RBT_Impl_diag t) = map (\<lambda>k. (k, k)) (RBT_Impl.keys t)"
square_fact_le_2_fact: fixes n :: "nat" shows "fact n * fact n \<le> fact (2 * n)"
take_suc: fixes ln :: "nat" shows "\<exists>lna. Bk # Bk \<up> ln = Bk \<up> lna"
backward_finite_path_root_path_root: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x" shows "path_root r x"
fps_divide_1': fixes a :: "'a fps" assumes "inverse (1::'a) = (1::'a)" shows "a / 1 = a"
f_bdd: fixes x :: "real" shows "norm (f x) \<le> B"