hcju/isabelleps huggingface dataset:
simps: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" shows "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = (if x \<le> n \<and> \<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) then Some (card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n)) else None)"
AxParallelCones: fixes onCone :: "'a Point \<Rightarrow> 'a Cone \<Rightarrow> bool"   and e :: "'a Point"   and econe :: "'a Cone"   and f :: "'a Point"   and fcone :: "'a Cone" shows "onCone e econe \<and> e \<noteq> vertex econe \<and> onCone f fcone \<and> f \<noteq> vertex fcone \<and> inPlane f (tangentPlane e econe) \<longrightarrow> samePlane (tangentPlane e econe) (tangentPlane f fcone) \<and> line joining vertex econe to e \<parallel>  line joining vertex fcone to f"
keys_mult_scalar_monomial_right: fixes c :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "c \<noteq> (0::'c)" shows "keys (p \<odot> monomial c v) = (\<lambda>t. t \<oplus> v) ` keys p"
comp_in_homI': fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and c :: "'a" assumes "C.arr f"   and "C.arr g"   and "C.dom f = a"   and "C.cod g = c"   and "C.dom g = C.cod f" shows "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>"
hequiv_sym_rel: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set" assumes "Hintikka A H" shows "sym (hequiv_rel A H)"
acc_lengths_merge1: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list"   and k :: "nat" assumes "merge as bs us vs" shows "list.set (acc_lengths k us) \<subseteq> list.set (acc_lengths k as)"
lam_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : local.prod a b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>\<Lambda> a b c f : a \<rightarrow> exp b c\<guillemotright>"
preserves_comp_2: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'b \<Rightarrow> 'd"   and f2' :: "'b"   and f2 :: "'b" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A2 (A2 f2' f2)" shows "(\<lambda>f1. F (f1, A2 f2' f2)) = vertical_composite.map A1 B (\<lambda>f1. F (f1, f2)) (\<lambda>f1. F (f1, f2'))"
chainTransitionAppend_4: fixes P :: "pi"   and u :: "name"   and P''' :: "pi"   and a :: "name"   and x :: "name"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>lu in P'''\<rightarrow>a<x> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sub>lu in P'''\<rightarrow>a<x> \<prec> P'"
unfoldE_2: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx> Q" shows "Q \<approx> P"
simps_1: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and e :: "char list"   and ts :: "'a MFOTL.trm list"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Pred e ts) j = j"
that: fixes j :: "nat" assumes "i < length (seq j)" shows "thesis"
convex_on_domain2: fixes f :: "'b \<Rightarrow> ereal" shows "Lower_Semicontinuous.convex_on (domain f) f = (\<forall>S. Lower_Semicontinuous.convex_on S f)"
segmap_def: fixes D :: "('c, 'd) Order_scheme" shows "segmap D = restrict (segment D) (carrier D)"
inEdges_correct': fixes g :: "'g"   and n :: "'node" assumes "invar g" shows "set (inEdges g n) = (\<lambda>(f, d). (f, d, n)) ` pred (\<alpha> g) n"
geodesic_segment_dist_le: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and a :: "'b"   and b :: "'b" assumes "geodesic_segment_between G x y"   and "a \<in> G"   and "b \<in> G" shows "dist a b \<le> dist x y"
assms_4: shows "X \<in> sets (stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y)))"
intros_1: fixes M :: "'b measure"   and N :: "'b measure" assumes "space M \<subset> space N" shows "M \<le> N"
wf_ps_triple_nodeI: fixes v :: "node"   and \<sigma>1 :: "node \<rightharpoonup>\<^sub>f topbot"   and \<sigma>2 :: "node \<rightharpoonup>\<^sub>f topbot" assumes "\<exists>\<sigma>. v |\<notin>| fmdom \<sigma> \<and> \<sigma>1 = [ {|v|} |=> Top ] ++\<^sub>f \<sigma> \<and> \<sigma>2 = [ {|v|} |=> Bot ] ++\<^sub>f \<sigma>" shows "wf_ps_triple (\<sigma>1, Inl v, \<sigma>2)"
closedin_limpt: fixes T :: "'b set"   and S :: "'b set" shows "closedin (top_of_set T) S = (S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S))"
insert_noteq_member: fixes b :: "'a"   and B :: "'a multiset"   and c :: "'a"   and C :: "'a multiset" assumes "b, B = c, C"   and "b \<noteq> c" shows "c \<in># B"
GodExImpliesNecEx_v2: shows "\<lfloor>(\<lambda>w. (\<exists>x. \<forall>xa. \<P> xa w \<longrightarrow> xa x) \<longrightarrow> (\<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa w \<longrightarrow> xa x))) w)\<rfloor>"
allocationCharacterization: fixes a :: "('c \<times> 'd set) set"   and \<Omega> :: "'d set"   and N :: "'c set" shows "(a \<in> allAllocations N \<Omega>) = (a \<in> injectionsUniverse \<and> Domain a \<subseteq> N \<and> Range a \<in> all_partitions \<Omega>)"
simps_15: fixes a :: "floatarith"   and n :: "nat"   and vs :: "real list" shows "interpret_floatarith (Power a n) vs = interpret_floatarith a vs ^ n"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "i < length xs" shows "xs[i := x] ! i = x"
DirProd_def: fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "G \<times>\<times> H = \<lparr>carrier = carrier G \<times> carrier H, mult = \<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h'), one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
after_forall: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "|~ (\<forall>x. P x)$ = (\<forall>x. P x$)"
IH: fixes V :: "'a set" assumes "wf\<^sub>s\<^sub>t V (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S')"   and "t \<in> ik\<^sub>s\<^sub>t S"   and "Unifier \<delta> (Fun f X) t"   and "bvars\<^sub>s\<^sub>t (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S') \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "wf\<^sub>s\<^sub>t (fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)) (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>)"
af_G_free: fixes \<phi> :: "'a ltl"   and w :: "'a set list" assumes "\<^bold>G \<phi> = {}" shows "af \<phi> w = af\<^sub>G \<phi> w"
return_transfer: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "state_dp_consistency.rel_fun_lifted R R Wrap State_Monad.return"
nslim_def: fixes X :: "nat \<Rightarrow> 'a" shows "nslim X = (THE L. X \<longlonglongrightarrow>\<^sub>N\<^sub>S L)"
Tree\<^sub>\<alpha>_abs_rep: fixes t\<^sub>\<alpha> :: "('b, 'c, 'd, 'e) Tree\<^sub>\<alpha>" shows "abs_Tree\<^sub>\<alpha> (rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
row_add_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and a :: "'d"   and b :: "'d"   and q :: "'b" shows "row_add A a b q = (\<chi>i j. if i = a then A $ a $ j + q * A $ b $ j else A $ i $ j)"
polyhedral_cone_def: fixes A :: "'a mat" shows "polyhedral_cone A = {x \<in> carrier_vec n. A *\<^sub>v x \<le> 0\<^sub>v (dim_row A)}"
\<epsilon>o_def: fixes a :: "'c" shows "\<epsilon>o a = (SOME e. terminal_arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (Go a) a e)"
mmake_affine_def: fixes q :: "int"   and p :: "int \<times> int \<times> int" shows "mmake_affine q p = (let (x, y, z) = p in if z = 0 then Infinity else let (a, b) = bezout_coefficients z q in Point (a **\<^bsub>q\<^esub> x) (a **\<^bsub>q\<^esub> y))"
append_queue_def: shows "append_queue \<equiv> map_fun id (map_fun Rep_queue Abs_queue) (\<lambda>a q. case q of (fs, ls) \<Rightarrow> (fs, a # ls))"
measurable_on_diff: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f measurable_on S"   and "g measurable_on S" shows "(\<lambda>x. f x - g x) measurable_on S"
gbg_to_idx_ext_correct: fixes ecnv :: "('Q, 'L, 'more) gba_rec_scheme \<Rightarrow> 'b" assumes "\<And>A. A \<in> F \<Longrightarrow> finite A" shows "gbg_to_idx_ext ecnv G \<le> SPEC (\<lambda>G'. igb_graph.is_acc_run G' = is_acc_run \<and> g_V G' = V \<and> g_E G' = E \<and> g_V0 G' = V0 \<and> igb_graph_rec.more G' = ecnv G \<and> igb_graph G')"
val_size_mem_l: fixes a :: "val"   and b :: "val"   and t :: "(val \<times> val) list" assumes "(a, b) \<in> set t" shows "val_size a < fun_size t"
le_supI1: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x a" shows "less_eq x (sup a b)"
invKey_image_eq: fixes x :: "key"   and A :: "key set" shows "(invKey x \<in> invKey ` A) = (x \<in> A)"
EntryPath_butlast_less_last: fixes g :: "'g"   and ns :: "'node list"   and z :: "'node" assumes "old.EntryPath g ns"   and "z \<in> set (butlast ns)" shows "old.shortestPath g z < old.shortestPath g (last ns)"
simps: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('b \<times> 'c) list"   and ox :: "'a"   and ys :: "('b \<times> 'd) list \<times> 'a" shows "oalist_raw.prod_ord_raw rep_key_order f (xs, ox) ys = comparator.prod_ord_pair (key_compare (rep_key_order ox)) f xs (oalist_raw.sort_oalist_aux rep_key_order ox ys)"
bit_not_exp_iff: fixes m :: "nat"   and n :: "nat" shows "bit (NOT ((2::'a) ^ m)) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> n \<noteq> m)"
assms_2: shows "prob0 n v i \<noteq> 0"
mat_rel_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c mat"   and B :: "'d mat" shows "mat_rel R A B \<equiv> dim_row A = dim_row B \<and> dim_col A = dim_col B \<and> (\<forall>i j. i < dim_row B \<longrightarrow> j < dim_col B \<longrightarrow> R (A $$ (i, j)) (B $$ (i, j)))"
sympartp_equivpartpI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "sympartp r x y" shows "equivpartp r x y"
\<sigma>2's: shows "set (labels \<sigma>2') \<subseteq> {fst \<alpha>_step}"
hyps_1: shows "g \<turnstile> n-ns\<rightarrow>m"
get_even_double: fixes n :: "nat" shows "\<exists>i. get_even n = 2 * i"
start_2_start: fixes m :: "nat"   and lm :: "nat list"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_loop_start m lm (b, Oc # list)" shows "wprepare_loop_start m lm (Oc # b, list)"
perm_fun_def: fixes pi :: "('c \<times> 'c) list"   and f :: "'a \<Rightarrow> 'b" shows "pi \<bullet> f = (\<lambda>x. pi \<bullet> f (rev pi \<bullet> x))"
tendsto_zero: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "bounded_linear f"   and "(g \<longlongrightarrow> (0::'a)) F" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> (0::'b)) F"
measurable_e2ennreal: shows "e2ennreal \<in> borel_measurable borel"
inf_greatest: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "x \<le> z" shows "x \<le> y \<sqinter> z"
mat1_vert_equiv: shows "mat1 2 = brickmat vert"
row_submatrix_UNIV: fixes i :: "nat"   and A :: "'b mat"   and I :: "nat set" assumes "i < card {i. i < dim_row A \<and> i \<in> I}" shows "row (submatrix A I UNIV) i = row A (pick I i)"
continuous_on_compose: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "continuous_on s f"   and "continuous_on (f ` s) g" shows "continuous_on s (g \<circ> f)"
is_inj_def: fixes x :: "'a" shows "is_inj x \<equiv> x ; x\<^sup>T \<le> 1'"
collision_restrict: shows "collision {0..} = collision {0..max ego.t_stop other.t_stop}"
2: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
matrix_mult_to_times_mat: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" assumes "0 < dim_col A"   and "0 < dim_col B"   and "dim_col A = dim_row B" shows "A * B = Tensor.mat_of_cols_list (dim_row A) (mat_mult (mult.row_length (mat_to_cols_list A)) (mat_to_cols_list A) (mat_to_cols_list B))"
Bell_altdef: fixes A :: "'a set" assumes "finite A" shows "Bell (card A) = card {P. partition_on A P}"
in_set_list_format: fixes e :: "'a"   and l :: "'a list" assumes "e \<in> set l" obtains l1 :: "'a list" and l2 :: "'a list" where "l = l1 @ e # l2"
singleton_nonsplit: fixes p :: "'a"   and x :: "'a" assumes "singleton p"   and "x \<le> p" shows "x = 0 \<or> x = p"
chamberI: fixes X :: "'b set set"   and x :: "'b set" assumes "ChamberComplex X"   and "x \<in> X"   and "\<And>z. \<lbrakk>z \<in> X; order.greater_eq z x\<rbrakk> \<Longrightarrow> z = x" shows "SimplicialComplex.maxsimp X x"
is_weak_composition: shows "weak_composition (\<cdot>\<^sub>D) (\<star>\<^sub>D)"
pickFT_pred: fixes cl :: "('d, 'e, 'f) cmd list" assumes "WtFT cl = 1" shows "\<exists>n. pickFT_pred cl n"
in_spanI: fixes v :: "'a vec"   and a :: "'a vec \<Rightarrow> 'a"   and A :: "'a vec set"   and W :: "'a vec set" assumes "v = MN.lincomb a A"   and "finite A"   and "A \<subseteq> W" shows "v \<in> MN.span W"
cproper_interval_word_def: shows "cproper_interval = proper_interval"
diagram_axioms: shows "diagram J.comp (\<cdot>) (\<lambda>i. if i = m then a else null)"
SPAN_def: fixes x :: "complex vec list" shows "SPAN x = (let n = length canonical_basis in ccspan (basis_enum_of_vec ` Set.filter (\<lambda>v. dim_vec v = n) (set x)))"
trivial_implies_finite: fixes X :: "'b set" assumes "trivial X" shows "finite X"
simps_5: fixes v :: "'b poly"   and va :: "'b poly" shows "isnpolyh (Sub v va) = (\<lambda>k. False)"
trancl_into_trancl: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r\<^sup>+\<^sup>+ a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
K_DisE: fixes A :: "'b fm"   and G :: "'b fm list"   and C :: "'b fm"   and B :: "'b fm" assumes "\<turnstile> imply (A # G) C"   and "\<turnstile> imply (B # G) C"   and "\<turnstile> imply G (A \<^bold>\<or> B)" shows "\<turnstile> imply G C"
asymmetric_axioms: shows "asymmetric A (\<bowtie>)"
oalist_inv_raw_sort_oalist: fixes xs :: "('a \<times> 'b) list" shows "tc.oalist_inv_raw (sort_oalist_tc xs)"
lp: assumes "w1 \<noteq> NEG_INF \<and> w1 <s 0 \<and> w2 = POS_INF" shows "P w1 w2"
intros_2: fixes xs :: "'c option list"   and P :: "'c option process"   and X :: "'c option set"   and Y :: "'c option set"   and Q :: "'c option process" assumes "xs \<in> sentences P"   and "(xs, X) \<in> failures P"   and "([], Y) \<in> failures Q" shows "(xs, insert None X \<inter> Y) \<in> seq_comp_failures P Q"
revis_vacuity: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and ".\<not> \<phi> \<notin> K" shows "K \<oplus> \<phi> \<subseteq> K \<^bold>* \<phi>"
that: fixes q :: "nat" assumes "w = push_bit n (word_of_nat q)"   and "q < 2 ^ (LENGTH('a) - n)" shows "thesis"
sup_not_bot: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linear_bounded_lattice inf less_eq less sup bot top"   and "x \<noteq> bot" shows "sup x y \<noteq> bot"
t: fixes xs :: "V list"   and x :: "V" assumes "distinct xs"   and "x \<notin> list.set xs" shows "distinct (x # xs)"
ChamberComplexFolding_axioms: shows "ChamberComplexFolding X f"
assms_2: fixes b :: "'a" assumes "b \<in> B" shows "b .\<in> A"
hom_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "act x \<in> carrier (BijGroup X)"
\<sigma>'_2: fixes v :: "'a" assumes "v \<in> winning_region p" shows "winning_strategy p \<sigma>' v"
unit_simps_4: shows "src\<^sub>C \<eta> = src\<^sub>C f"
main_part_def: fixes M :: "'b measure"   and A :: "'b set" shows "main_part M A = fst (Eps (split_completion M A))"
\<phi>2s: shows "set \<phi>2s \<subseteq> fmla"
mdl_carrier: fixes H :: "'a set" assumes "submodule R M H" shows "carrier (mdl M H) = H"
fls_integral_of_nat': fixes n :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "fls_integral (of_nat n) = of_nat n * fls_X"
l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes to_tree_order \<equiv> to_tree_order = l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_to_tree_order get_child_nodes"
fds_truncate_divide: fixes m :: "nat"   and f :: "'b fds"   and g :: "'b fds" shows "fds_truncate m (fds_truncate m f / fds_truncate m g) = fds_truncate m (f / g)"
AEf1: shows "AE x in M. real_cond_exp M Invariants f1 x = 0"
arr_simps_2: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "trg \<mu> = a"
Abinds_reorder1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and v :: "Vars.var"   and e :: "exp" assumes "map_of \<Gamma> v = Some e" shows "ABinds \<Gamma> = ABind v e \<squnion> ABinds (delete v \<Gamma>)"
IH_2: assumes "avl l" shows "height (AVL.insert x l) = height l \<or> height (AVL.insert x l) = height l + 1"
prod_mono: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "\<forall>i\<in>A. (0::'a) \<le> f i \<and> f i \<le> g i" shows "prod f A \<le> prod g A"
axioms_3: fixes step1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'c \<Rightarrow> bool"   and order :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and match :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "backward_simulation step1 step2 final1 final2 order match" shows "well_founded order"
inverse_less_imp_less: fixes a :: "'a"   and b :: "'a" assumes "inverse a < inverse b"   and "(0::'a) < a" shows "b < a"
simps_2: fixes n :: "nat" shows "Dijkstra_Abstract.enat_of_option (Some n) = enat n"
pr_unique_passwd_parts: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and K :: "rat"   and m' :: "nat"   and m :: "nat" assumes "(evs, S, A, U) \<in> protocol"   and "Crypt (sesK K) (Passwd m') \<in> parts (A \<union> spies evs)"   and "Crypt (sesK K) (Passwd m) \<in> parts (A \<union> spies evs)" shows "m' = m"
intro: fixes set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')" shows "l_set_child_nodes_get_shadow_root set_child_nodes_locs get_shadow_root_locs"
inorder_lookup_empty_iff: fixes m :: "'t" assumes "invar m" shows "(lookup m = Map.empty) = (inorder m = [])"
rat_roots2: fixes p :: "rat poly" assumes "degree p = 2" shows "{x. poly p x = 0} = set (rat_roots2 p)"
fmmap_fmupd: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'a"   and y :: "'c"   and m :: "('a, 'c) fmap" shows "fmmap f m(x $$:= y) = (fmmap f m)(x $$:= f y)"
min_elems5: fixes x :: "'a list"   and A :: "'a list set" assumes "x \<in> A" shows "\<exists>s\<le>x. s \<in> min_elems A"
indeps_7: shows "mylens.base\<^sub>L \<bowtie> mylens.more\<^sub>L"
simps_1: fixes x1 :: "form multiset"   and x2 :: "form multiset"   and y1 :: "form multiset"   and y2 :: "form multiset" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
trans: fixes P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and SRel :: "('a \<times> 'a) set"   and TRel :: "('b \<times> 'b) set"   and R :: "('a, 'b) Proc" assumes "(P, Q) \<in> encoding.indRelLSTPO Enc SRel TRel"   and "(Q, R) \<in> encoding.indRelLSTPO Enc SRel TRel" shows "(P, R) \<in> encoding.indRelLSTPO Enc SRel TRel"
sup0: fixes a :: "'b::bounded_semilattice_sup_bot" shows "a::'b::bounded_semilattice_sup_bot \<equiv> a \<squnion> \<bottom>"
prj_simps_3: fixes h :: "'a"   and k :: "'a" assumes "ide h"   and "is_left_adjoint k"   and "trg h = trg k" shows "trg (prj\<^sub>1 h k) = src k"
OPT2_B: fixes x :: "'a"   and y :: "'a"   and qs :: "'a list"   and u :: "'a list"   and v :: "'a list" assumes "x \<noteq> y"   and "qs = u @ v"   and "u = [] \<or> u = [x]"   and "v \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "T\<^sub>p [x, y] qs (OPT2 qs [x, y]) = length v div 2"
simps_1: fixes uu :: "'b set" shows "difference_list uu [] = []"
labels_not_empty: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>" shows "labels \<Gamma> p \<noteq> {}"
bigtheta_trans2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)"   and "g \<in> O[F](h)" shows "f \<in> O[F](h)"
keys_eq_empty_iff: fixes c :: "'c \<Rightarrow>\<^sub>0 'd" shows "(keys c = {}) = (c = 0)"
iwlsFSbSwTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "iwlsFSbSwTR MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWlsSTR MOD \<and> igSubstAllIPresIGWlsAllSTR MOD \<and> igFreshClsSTR MOD \<and> igSubstClsSTR MOD \<and> igAbsRenSTR MOD"
fp_def: fixes p :: "fm"   and n :: "int"   and s :: "MIR.num"   and j :: "int" shows "fp p n s j \<equiv> if 0 < n then And p (And (Ge (CN 0 n (Sub s (Add (Floor s) (C j))))) (Lt (CN 0 n (Sub s (Add (Floor s) (C (j + 1))))))) else And p (And (Le (CN 0 (- n) (Add (Neg s) (Add (Floor s) (C j))))) (Gt (CN 0 (- n) (Add (Neg s) (Add (Floor s) (C (j + 1)))))))"
open_interval_def: fixes X :: "'b interval set" shows "open X = (\<forall>x\<in>X. \<exists>A B. open A \<and> open B \<and> lower x \<in> A \<and> upper x \<in> B \<and> Interval ` (A \<times> B) \<subseteq> X)"
inorder_del: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (tree (del x t)) = del_list x (Tree2.inorder t)"
SuccD: fixes k :: "'d"   and Kl :: "'d list set"   and kl :: "'d list" assumes "k \<in> BNF_Greatest_Fixpoint.Succ Kl kl" shows "kl @ [k] \<in> Kl"
prefix_slength_eq: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" shows "eq\<cdot>xs\<cdot>ys = (eq\<cdot>(slength\<cdot>xs)\<cdot>(slength\<cdot>ys) andalso prefix\<cdot>xs\<cdot>ys)"
r_result_empty_stack_Some: fixes i :: "nat"   and f :: "recf"   and x :: "nat"   and t :: "nat"   and v :: "nat" assumes "i = encode f"   and "recfn (e_length x) f"   and "iterate t step ([(f, Partial_Recursive.list_decode x, [])], None) = ([], Some v)" shows "eval r_result [t, i, x] \<down>= Suc v"
invpst_baliR: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invpst l"   and "invpst r" shows "invpst (baliR l a r)"
ide_cod: fixes f :: "'a" assumes "\<phi>.A.arr f" shows "\<phi>.A.ide (\<phi>.A.cod f)"
coboundedI2: fixes c :: "'a word"   and b :: "'a word"   and a :: "'a word" assumes "c \<le>s b" shows "c \<le>s signed.max a b"
indecomposable_succ: fixes \<alpha> :: "V" shows "indecomposable (ZFC_in_HOL.succ \<alpha>) = (\<alpha> = 0)"
set_rel_br: fixes a :: "'a \<Rightarrow> 'c"   and I :: "'a \<Rightarrow> bool" shows "\<langle>br a I\<rangle>set_rel = br ((`) a) (\<lambda>X. Ball X I)"