hcju/isabelleps huggingface dataset:
rel_sym: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a" assumes "x \<sim> y"   and "x \<in> R \<times> S"   and "y \<in> R \<times> S" shows "y \<sim> x"
scalar_eq_scaleR: fixes c :: "real"   and x :: "(real, 'a) vec" shows "c *s x = c *\<^sub>R x"
lin_indpt_list_length_le_n: fixes xs :: "'a vec list" assumes "lin_indpt_list xs" shows "length xs \<le> n"
bij_betw_iff_has_inverse: shows "bij_betw \<alpha> S S = (\<exists>\<beta>\<in>S \<rightarrow>\<^sub>E S. compose S \<beta> \<alpha> = identity S \<and> compose S \<alpha> \<beta> = identity S)"
d\<^sub>\<kappa>_inject: fixes o\<^sub>1 :: "\<nu>"   and x :: "\<kappa>"   and y :: "\<kappa>" assumes "Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>1 = d\<^sub>\<kappa> y" shows "x = y"
that: fixes \<D> :: "'a set set" assumes "countable \<D>"   and "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"   and "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"   and "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2 ^ n"   and "\<And>K. \<lbrakk>K \<in> \<D>; box a b \<noteq> {}\<rbrakk> \<Longrightarrow> interior K \<noteq> {}"   and "S \<subseteq> \<Union> \<D>"   and "\<Union> \<D> \<in> lmeasurable"   and "Sigma_Algebra.measure lebesgue (\<Union> \<D>) \<le> Sigma_Algebra.measure lebesgue S + e" shows "thesis"
has_derivative_subset: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set"   and t :: "'b set" assumes "(f has_derivative f') (at x within s)"   and "t \<subseteq> s" shows "(f has_derivative f') (at x within t)"
numeral_One: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
collapse_1: fixes step :: "('e, 'f) step" assumes "is_Skip step" shows "Skip (un_Skip step) = step"
2422: fixes x :: "'a" shows "- x \<sqinter> - - - x = - - - x"
ind: assumes "n \<le> length lm" shows "{\<lambda>nl. nl = lm} empty_boxes n {\<lambda>nl. nl = 0 \<up> n @ drop n lm}"
rtrancl_on_converseI: fixes y :: "'a"   and x :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set" assumes "(y, x) \<in> rtrancl_on F r" shows "(x, y) \<in> rtrancl_on F (r\<inverse>)"
retraction_map_eq: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "retraction_map X Y f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "retraction_map X Y g"
flow_initial_time_if: fixes x0 :: "'a" shows "flow0 x0 0 = (if 0 \<in> UNIV \<and> x0 \<in> X then x0 else (0::'a))"
unilateral_substitutes_on_def: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" shows "unilateral_substitutes_on A f = (\<not> (\<exists>B\<subseteq>A. \<exists>a b. {a, b} \<subseteq> A \<and> Xd b \<notin> Xd ` B \<and> b \<notin> f (B \<union> {b}) \<and> b \<in> f (B \<union> {a, b})))"
classic_Local_indep: fixes Y :: "loc"   and Y' :: "loc"   and G :: "'b triple set"   and P :: "'b assn"   and c :: "com"   and d :: "nat"   and a :: "state \<Rightarrow> nat" assumes "Y \<noteq> Y'"   and "G|-{P}. c .{\<lambda>Z s. s<Y'> = d}" shows "G|-{\<lambda>Z s. P Z (s[Loc Y::=a s])}. LOCAL Y:=a IN c .{\<lambda>Z s. s<Y'> = d}"
assms_2: shows "u \<turnstile>\<^bsub>v,n\<^esub> M2"
nyinitcls_gext: fixes s :: "state"   and s' :: "state"   and G :: "prog" assumes "snd s\<le>|snd s'" shows "nyinitcls G s' \<subseteq> nyinitcls G s"
assms: shows "density_context V V' \<Gamma> (\<lambda>x. ennreal (extract_real (cexpr_sem x \<delta>)))"
inf_None_1: fixes y :: "'b option" shows "None \<sqinter> y = None"
kernel_padl: fixes d :: "'a vec"   and D :: "'a mat"   and A :: "'a mat"   and nr1 :: "nat"   and nc1 :: "nat"   and C :: "'a mat"   and nr2 :: "nat"   and nc2 :: "nat" assumes "d \<in> mat_kernel D"   and "A \<in> carrier_mat nr1 nc1"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "0\<^sub>v nc1 @\<^sub>v d \<in> mat_kernel (four_block_mat A (0\<^sub>m nr1 nc2) C D)"
makeAF_def: shows "makeAF \<equiv> (0, \<lambda>index. fillByte)"
inner_prod_with_unitary_mat: fixes U :: "complex mat"   and u :: "complex vec"   and v :: "complex vec" assumes "unitary U"   and "dim_vec u = dim_col U"   and "dim_vec v = dim_col U" shows "\<langle>col_fst (U * |u\<rangle>)|col_fst (U * |v\<rangle>)\<rangle> = \<langle>u|v\<rangle>"
Arg_of_real: fixes r :: "real" shows "Arg (complex_of_real r) = (if r < 0 then pi else 0)"
simps_1: fixes ms :: "nat list" shows "Cantor_sum [] ms = 0"
iprev_nth_Suc_mono2: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "\<exists>x\<in>I. x < I \<leftarrow> n" shows "I \<leftarrow> Suc n < I \<leftarrow> n"
ints_le_add_frac2: fixes x :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> x"   and "x < 1"   and "a \<in> \<int>"   and "b \<in> \<int>"   and "b \<le> a + x" shows "b \<le> a"
hom_boundary_trivial: fixes p :: "int" assumes "p \<le> 0" shows "hom_boundary p = (\<lambda>q r t. undefined)"
lea_in_angle: 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 P :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B P A B C"   and "Tarski_neutral_dimensionless.OS Bet A B C P" shows "Tarski_neutral_dimensionless.InAngle Bet P A B C"
span_sum: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> module.span scale S" shows "sum f A \<in> module.span scale S"
I_R_mult_atom: fixes c :: "real"   and a :: "atom"   and xs :: "real list" assumes "0 \<le> c"   and "I\<^sub>R a xs" shows "I\<^sub>R (c *\<^sub>a a) xs"
map_simp_ide: fixes a :: "'a \<times> 'a \<times> 'a" assumes "CCC.ide a" shows "\<alpha>.map a = assoc (fst a) (fst (snd a)) (snd (snd a))"
get_root_node_si_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_root_node_si ptr"
preserves_comp: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "F (D g f) = C (F g) (F f)"
onorm_scaleR: fixes f :: "'a \<Rightarrow> 'b"   and r :: "real" assumes "bounded_linear f" shows "onorm (\<lambda>x. r *\<^sub>R f x) = \<bar>r\<bar> * onorm f"
assms_3: shows "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> n"
list_all_append: shows "list_all is_const (es @ [e]) = (list_all is_const es \<and> list_all is_const [e])"
R_1: fixes nb :: "nat" shows "emeasure (P (J nb)) (B nb) = Sup (emeasure (P' nb) ` {K. K \<subseteq> fm nb ` B nb \<and> compact K})"
r2: shows "\<forall>w. w \<noteq> z \<and> dist w z < r2 \<longrightarrow> f w \<noteq> 0"
single_subset_iff: fixes a :: "'b"   and M :: "'b multiset" shows "(\<LM> a  \<RM> \<subseteq># M) = (a \<in># M)"
remove1_list_rel2: fixes xs :: "'a list"   and ys :: "'b list"   and R :: "('a \<times> 'b) set"   and a :: "'a"   and b :: "'b" assumes "(xs, ys) \<in> \<langle>R\<rangle>list_rel"   and "(a, b) \<in> R"   and "\<And>c. (a, c) \<in> R \<Longrightarrow> c = b"   and "\<And>c. (c, b) \<in> R \<Longrightarrow> c = a" shows "(remove1 a xs, remove1 b ys) \<in> \<langle>R\<rangle>list_rel"
components_are_iso: fixes a :: "'a" assumes "ide a" shows "S.iso (S.UP a)"
mem_val_alt_8_unchanged3: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 ((pa AND 68719476732) + 3) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 8 ((pa AND 68719476732) + 3) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
degree_eq_length_coeffs: fixes p :: "'c poly" shows "degree p = length (coeffs p) - 1"
flip_other: fixes y :: "nat"   and X :: "bool list"   and z :: "nat" assumes "y < length X"   and "z < length X"   and "z \<noteq> y" shows "flip z X ! y = X ! y"
balance_right_rbt_greater: fixes k :: "'a"   and a :: "('a, 'c) rbt"   and b :: "('a, 'c) rbt"   and x :: "'a"   and t :: "'c" assumes "k \<guillemotleft>| a"   and "k \<guillemotleft>| b"   and "k < x" shows "k \<guillemotleft>| balance_right a x t b"
F: shows "F \<in> Pi I (S.set \<circ> D) \<inter> S.extensional' I"
top_higher: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<sqsubseteq> \<top>"
rangeI: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "f x \<in> range f"
finfun_default_aux_infinite: fixes f :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "infinite UNIV"   and "finite {a. f a \<noteq> b}" shows "finfun_default_aux f = b"
psubset_eq: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subset> B) = (A \<subseteq> B \<and> A \<noteq> B)"
prv_imp_eqvEL: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "prv (eqv \<phi>1 \<phi>2)" shows "prv (imp \<phi>1 \<phi>2)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>B) F' F' F'"
add_idem': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.near_dioid plus times less_eq less" shows "plus x x = x"
dg_Obj_in_Vset: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<CC>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
isTrue_imp: fixes \<phi> :: "fm"   and \<psi> :: "fm" assumes "\<phi> \<in> UNIV"   and "\<psi> \<in> UNIV"   and "Fvars \<phi> = {}"   and "Fvars \<psi> = {}"   and "eval_fm e0 \<phi>"   and "eval_fm e0 (\<phi> IMP \<psi>)" shows "eval_fm e0 \<psi>"
val_Zp_of_minus: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "val_Zp a = val_Zp (\<ominus> a)"
invert_opposite_sides_of_square: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and h :: "('a, 'a set) Maps.arr"   and k :: "('a, 'a set) Maps.arr" assumes "Maps.seq f g"   and "f \<odot> g = h \<odot> k"   and "Maps.iso f"   and "Maps.iso k" shows "Maps.seq g (Maps.inv k) \<and> Maps.seq (Maps.inv f) h \<and> g \<odot> Maps.inv k = Maps.inv f \<odot> h"
jkbpCn_jkbpC_inc: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and n :: "nat" assumes "t \<in> SPR.jkbpCn n" shows "t \<in> SPR.jkbpC"
If_cases: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite A" shows "(\<Prod>x\<in>A. if P x then h x else g x) = prod h (A \<inter> {x. P x}) * prod g (A \<inter> - {x. P x})"
Cons_shuffles_subset1: fixes x :: "'g"   and xs :: "'g list"   and ys :: "'g list" shows "(#) x ` shuffles xs ys \<subseteq> shuffles (x # xs) ys"
simps: fixes x :: "'a ext"   and y :: "'a ext" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
image_mono: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
simps_1: fixes edge_rev :: "'c \<Rightarrow> 'c"   and edge_succ :: "'c \<Rightarrow> 'c"   and more :: "'d" shows "edge_rev \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_rev"
the_int_def: shows "the_int \<equiv> map_fun Word.rep id (take_bit LENGTH('k))"
measurable_Pair2': fixes y :: "'c"   and M2 :: "'c measure"   and M1 :: "'d measure" assumes "y \<in> space M2" shows "(\<lambda>x. (x, y)) \<in> M1 \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
y_2: shows "\<Gamma> (Var y) = \<Gamma> (Var x)"
elims: fixes x :: "ereal"   and y :: "ereal" assumes "\<bar>x\<bar> = y" obtains r :: "real" where "x = ereal r" and "y = ereal \<bar>r\<bar>"   | "x = - \<infinity>" and "y = \<infinity>"   | "x = \<infinity>" and "y = \<infinity>"
parts_mpair_2: fixes X :: "msg"   and Y :: "msg" shows "insert \<lbrace>X, Y\<rbrace> (parts {X} \<union> parts {Y}) \<subseteq> parts {\<lbrace>X, Y\<rbrace>}"
val2cont_def: shows "val2cont \<equiv> \<Lambda> a c. c\<cdot>a"
wls_Var_simp: fixes s :: "'sort"   and xs :: "'varSort"   and x :: "'var" shows "wls s (Var xs x) = (s = asSort xs)"
indRelSTEQ_symm: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "sym SRel"   and "sym TRel" shows "sym (indRelSTEQ SRel TRel)"
simps_16: fixes vb :: "'b"   and vc :: "value list"   and v :: "'b gexp"   and va :: "'b gexp" shows "GExp_Lexorder.less_gexp_aux (In vb vc) (Nor v va) = True"
isDiagonal_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "isDiagonal A = (\<forall>a b. to_nat a \<noteq> to_nat b \<longrightarrow> A $ a $ b = (0::'a))"
td_nat_int: shows "type_definition int nat (Collect ((\<le>) 0))"
reals_Archimedean2: fixes x :: "'a" shows "\<exists>n. x < of_nat n"
trans_not_greater: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b \<noteq> Greater"   and "cmp b c \<noteq> Greater" shows "cmp a c \<noteq> Greater"
residuated_multr: fixes x :: "'a" shows "residuated ((\<cdot>) x)"
gSwapBinp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'b, 'j) Recursion.model_scheme"   and zs :: "'e"   and z1 :: "'h"   and z2 :: "'h"   and binp' :: "'a \<Rightarrow> ('c, 'd, 'e, 'h, 'g) qAbs set option"   and binp :: "'a \<Rightarrow> 'b option" shows "gSwapBinp MOD zs z1 z2 binp' binp \<equiv> lift2 (gSwapAbs MOD zs z1 z2) binp' binp"
U_3: fixes n :: "nat" shows "emeasure lborel (U n - B) < ennreal (e * (1 / 2) ^ Suc n)"
assms_1: shows "\<tau> \<star> \<mu> \<noteq> null"
bind_map_spmf: fixes f :: "'n \<Rightarrow> 'm"   and p :: "'n spmf"   and g :: "'m \<Rightarrow> 'l spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
simps_1: fixes 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"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "P \<turnstile> typeof\<^bsub>h\<^esub> Unit = \<lfloor>Void\<rfloor>"
simps: fixes i :: "nat" shows "local.b i = (if i \<le> 2 then 1 else if i = 3 then (- 1) ^ (degree (local.F 1) - degree (local.F 2) + 1) else if local.F (i - 2) = 0 then 1 else (- 1) ^ (degree (local.F (i - 2)) - degree (local.F (i - 1)) + 1) * lead_coeff (local.F (i - 2)) * local.h (i - 2) ^ (degree (local.F (i - 2)) - degree (local.F (i - 1))))"
wtC_wax_unprot: fixes \<sigma> :: "'tp" assumes "\<not> unprot \<sigma>"   and "\<not> isRes \<sigma> \<or> protCl \<sigma>" shows "GE.wtC (wax \<sigma>)"
expectation_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('i, 'j) \<I>"   and f :: "'h \<Rightarrow> ennreal" shows "expectation_gpv fail \<I> f \<equiv> lfp.fixp_fun (\<lambda>expectation_gpv gpv. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv (c r)) \<partial>measure_spmf (the_gpv gpv) + fail * ennreal (pmf (the_gpv gpv) None))"
simps_1: fixes p :: "predicate"   and vs :: "'ent list" shows "wf_atom (predAtm p vs) = wf_pred_atom (p, vs)"
Kdelta_in_Zinf: fixes j :: "nat"   and n :: "nat"   and k :: "nat"   and z :: "int" assumes "j \<le> Suc n"   and "k \<le> Suc n" shows "z *\<^sub>a (\<delta>\<^bsub>j k\<^esub>) \<in> Z\<^sub>\<infinity>"
hyps: fixes l :: "'a list" assumes "n + 1 < length l" shows "l[n := l ! (n + 1), n + 1 := l ! n] = take n l @ [l ! (n + 1), l ! n] @ drop (n + 2) l"
assms_1: shows "dom_trm (t \<lhd> \<theta>) (subst_cl C \<theta>)"
infinite_AE: shows "AE x in M. \<exists>delta. (\<forall>l. 0 < delta l) \<and> delta \<longlonglongrightarrow> 0 \<and> infinite {n. \<forall>l\<in>{1..n}. - delta l * real l < w n x - w (n - l) ((T ^^ l) x)}"
smallD: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and c :: "real" assumes "f \<in> l F g"   and "0 < c" shows "\<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))"
real_cond_exp_int_1: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "integrable M (real_cond_exp M Invariants f)"
neg_distr_cond_bset_def: fixes C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and C' :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and uu :: "'h itself" shows "neg_distr_cond_bset C C' uu = (rel_bset (C OO C') \<le> rel_bset C OO rel_bset C')"
termdiffs_strong_converges_everywhere: fixes c :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)" shows "((\<lambda>x. \<Sum>n. c n * x ^ n) has_field_derivative (\<Sum>n. diffs c n * x ^ n)) (at x)"
group_by_complete: fixes a :: "'a"   and as :: "'a set"   and f :: "'a \<Rightarrow> 'c \<times> 'b" assumes "a \<in> as" obtains cs :: "'b set" where "(fst (f a), cs) \<in> group_by f as" and "snd (f a) \<in> cs"
fr_rename_ext_aimpl_def: fixes ecnv :: "('a \<Rightarrow> 'c option) \<Rightarrow> ('c, 'd) graph_rec_scheme \<Rightarrow> 'b nres"   and f :: "'c \<Rightarrow> 'a"   and G :: "('c, 'd) graph_rec_scheme" shows "fr_rename_ext_aimpl ecnv f G \<equiv> ASSERT (inj_on f (g_V G)) \<bind> (\<lambda>_. ASSERT (inj_on f (g_V0 G)) \<bind> (\<lambda>_. let fi_map = the_inv_into_map (g_V G) f in ecnv fi_map G \<bind> (\<lambda>e. RETURN \<lparr>g_V = f ` g_V G, g_E = E_of_succ (\<lambda>v. case fi_map v of None \<Rightarrow> {} | Some u \<Rightarrow> f ` succ_of_E (g_E G) u), g_V0 = f ` g_V0 G, \<dots> = e\<rparr>)))"
lookup_def: fixes x :: "'b" shows "SFI_lookup x = get_state (\<lambda>s. case s x of None \<Rightarrow> fail_state (fail_option return_id) | Some y \<Rightarrow> return_state (return_option return_id) y)"
Perp_cases: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "A B Perp C D \<or> B A Perp C D \<or> A B Perp D C \<or> B A Perp D C \<or> C D Perp A B \<or> C D Perp B A \<or> D C Perp A B \<or> D C Perp B A" shows "A B Perp C D"
summable: shows "(\<lambda>f. if \<forall>x. x \<notin> A \<longrightarrow> f x = dflt then \<Prod>x\<in>A. pmf (p x) (f x) else 0) abs_summable_on UNIV"
induct: fixes x :: "nat set"   and P :: "nat set \<Rightarrow> bool" assumes "x \<in> i_set0_ind"   and "P {}"   and "\<And>I. I \<in> i_set_ind \<Longrightarrow> P I" shows "P x"
respects_hcomp: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C g = trg\<^sub>C f" shows "(\<tau>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D (\<tau>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv \<a>\<^sub>D[G g, \<tau>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<tau>\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[G g, G f, \<tau>\<^sub>0 (src\<^sub>C f)] = \<tau>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f))"
well_base\<^sub>h_encode: fixes e :: "nat"   and n :: "nat" shows "local.well_base\<^sub>h (local.encode e n)"
less_irrefl: fixes x :: "'t" shows "\<not> lt_of_nat_term_order cmp_term x x"
zero_le_square: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a a)"
mono_set_mult: fixes H :: "'a set"   and H' :: "'a set"   and K :: "'a set"   and K' :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "H \<subseteq> H'"   and "K \<subseteq> K'" shows "H <#>\<^bsub>G\<^esub> K \<subseteq> H' <#>\<^bsub>G\<^esub> K'"
2: fixes x :: "state \<Rightarrow> 'b" assumes "basevars (x, vs)" shows "F x sigma"
healthy_def: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "healthy t = (feasible t \<and> mono_trans t \<and> scaling t)"
Quotient_to_transfer: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'d"   and c' :: "'e" assumes "Quotient R Abs Rep T"   and "R c c"   and "c' \<equiv> Abs c" shows "T c c'"
run_one_step_basic_unreachable_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Unreachable) = (s', vs', res)" shows "\<exists>r. res = RSNormal r"
dd0_ent: shows "DD0 \<union> CC \<Turnstile> {E}"
singleDSourceEmpty4isNotDSourceLevel: fixes i :: "AbstrLevelsID"   and C :: "CSet"   and S :: "CSet" assumes "DAcc i C = {S}" shows "\<forall>Z\<in>AbstrLevel i. Z \<noteq> S \<longrightarrow> C \<notin> DSources i Z"
divide_inverse: fixes a :: "'a"   and b :: "'a" shows "a / b = a * inverse b"
is_trail_rev: fixes v' :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v :: "'v" shows "is_trail v' (rev_path ps) v = is_trail v ps v'"
isomorphism_expand: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f" shows "(f \<circ> g = id \<and> g \<circ> f = id) = ((\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x))"
strict_coboundedI2: fixes c :: "'b multiset"   and b :: "'b multiset"   and a :: "'b multiset" assumes "c \<subset># b" shows "c \<subset># a \<union># b"
neutral: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k"   and A :: "'k multiset" assumes "comm_monoid_mset f z"   and "\<forall>x\<in>#A. x = z" shows "comm_monoid_mset.F f z A = z"
simps: fixes xs :: "'a list"   and n :: "nat" shows "abs_array (xs, n) = take n xs"
simps_4: fixes t :: "('c, 'd) bintree"   and r :: "('c, 'd) bintree"   and xs :: "('c, 'd) bintree option list" shows "add (Some t) (Some r # xs) = None # add (Some (merge t r)) xs"
rel_cong: fixes x :: "'c llist"   and ya :: "'c llist"   and y :: "'d llist"   and xa :: "'d llist"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> lset ya; yb \<in> lset xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "llist_all2 R x y = llist_all2 Ra ya xa"
aux1: fixes a' :: "bool list" shows "nlists UNIV \<eta> \<times> nlists UNIV \<eta> \<inter> {x. snd x \<noteq> a'} \<noteq> {}"
unwind_trace: fixes \<Delta> :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'state"   and s1 :: "'state"   and vl :: "'value list"   and vl1 :: "'value list"   and tr :: "('state, 'act, 'out) trans list" assumes "unwind \<Delta>"   and "reachNT s"   and "reach s1"   and "\<Delta> s vl s1 vl1"   and "validFrom s tr"   and "never T tr"   and "V tr = vl" shows "\<exists>tr1. validFrom s1 tr1 \<and> O tr1 = O tr \<and> V tr1 = vl1"
lift_p_P: fixes P :: "'a \<Rightarrow> bool"   and f :: "('a, 'b) state"   and heap :: "'a"   and v :: "'b"   and heap' :: "'a" assumes "lift_p P f"   and "P heap"   and "run_state f heap = (v, heap')" shows "P heap'"
edges_irrefl': fixes u :: "'a"   and g :: "'a ugraph" shows "(u, u) \<notin> edges g"
final_step_of_path: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> \<P> k u"   and "p \<noteq> []" shows "\<exists>q ts k' u'. p = q @ ts \<and> indexlt k' u' k u \<and> is_continuation k' u' q ts"
assms_2: shows "Counterclockwise_2D_Arbitrary.lex t s"
ifte: fixes i :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a formula"   and a :: "'b"   and y :: "'a formula"   and b :: "'b" assumes "i \<notin> S"   and "hom_graph f S x a"   and "hom_graph f S y b" shows "hom_graph f (insert i S) (ifte (var i) x y) (ifte (f i) a b)"