hcju/isabelleps huggingface dataset:
max_id: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<succeq> y" shows "Max x y = x"
subset_iff: fixes A :: "'x set"   and B :: "'x set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
finiteRelationCharacterization: fixes Q :: "('c \<times> 'd) set" shows "(finite (Domain Q) \<and> finite (Range Q)) = finite Q"
soundness': fixes A :: "'d set"   and n :: "nat"   and branch :: "(('e, 'd) fm list \<times> 'd) list"   and M :: "('f, 'e) model"   and g :: "'d \<Rightarrow> 'f" assumes "A, n \<turnstile> branch"   and "M, g \<Turnstile>\<^sub>\<Theta> branch" shows "False"
sorted_insorta: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "FWNormalisationCore.sorted (FWNormalisationCore.insort x xs l) l"   and "all_in_list (x # xs) l"   and "all_in_list (x # xs) l"   and "distinct (x # xs)"   and "singleCombinators [x]"   and "singleCombinators xs" shows "FWNormalisationCore.sorted xs l"
ide_cod: fixes f :: "'s" assumes "S.arr f" shows "S.ide (S.cod f)"
strict_trans: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'t \<Rightarrow>\<^sub>0 'c" assumes "b \<prec>\<^sub>p a"   and "c \<prec>\<^sub>p b" shows "c \<prec>\<^sub>p a"
cases: fixes a1 :: "'d"   and a2 :: "'d" assumes "three TYPE('d)"   and "three.rel_12_21 a1 a2" obtains "a1 = three.three\<^sub>1" and "a2 = three.three\<^sub>2"   | "a1 = three.three\<^sub>2" and "a2 = three.three\<^sub>1"
subst_cls_mset_union: fixes CC :: "'a literal multiset multiset"   and DD :: "'a literal multiset multiset"   and \<sigma> :: "'s" shows "(CC + DD) \<cdot>cm \<sigma> = CC \<cdot>cm \<sigma> + DD \<cdot>cm \<sigma>"
prefix_definedD: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" assumes "prefix\<cdot>xs\<cdot>ys = TT" shows "xs \<noteq> \<bottom> \<and> ys \<noteq> \<bottom>"
simps_2: fixes n :: "String.literal"   and uu :: "int" shows "name (Var (n, uu)) = n"
finite_vimageI: fixes F :: "'d set"   and h :: "'e \<Rightarrow> 'd" assumes "finite F"   and "inj h" shows "finite (h -` F)"
is_valid_operator_sas_plus_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and op :: "('a, 'b) sas_plus_operator" shows "is_valid_operator_sas_plus \<Psi> op \<equiv> let pre = precondition_of op; eff = effect_of op; vs = \<Psi>\<^sub>\<V>\<^sub>+; D = range_of \<Psi> in list_all (\<lambda>(v, a). ListMem v vs) pre \<and> list_all (\<lambda>(v, a). D v \<noteq> None \<and> ListMem a (the (D v))) pre \<and> list_all (\<lambda>(v, a). ListMem v vs) eff \<and> list_all (\<lambda>(v, a). D v \<noteq> None \<and> ListMem a (the (D v))) eff \<and> list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') pre) pre \<and> list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') eff) eff"
surj_simplex_map: shows "f \<turnstile> X = Y"
segment_horizontal: fixes a :: "(real, 2) vec"   and b :: "(real, 2) vec"   and x :: "(real, 2) vec" assumes "a $ 2 = b $ 2" shows "(x \<in> closed_segment a b) = (x $ 2 = a $ 2 \<and> x $ 2 = b $ 2 \<and> (a $ 1 \<le> x $ 1 \<and> x $ 1 \<le> b $ 1 \<or> b $ 1 \<le> x $ 1 \<and> x $ 1 \<le> a $ 1))"
ChainsD: fixes x :: "'b"   and C :: "'b set"   and r :: "('b \<times> 'b) set"   and y :: "'b" assumes "x \<in> C"   and "C \<in> Chains r"   and "y \<in> C" shows "(x, y) \<in> r \<or> (y, x) \<in> r"
simps_1: fixes i :: "nat"   and v :: "'b" shows "[][i := v] = []"
arcs_app_iso: fixes hom :: "('g, 'h, 'f, 'e) digraph_isomorphism"   and G :: "('g, 'h) pre_digraph" shows "arcs (app_iso hom G) = iso_arcs hom ` arcs G"
prob_space_distr: fixes f :: "'b \<Rightarrow> 'd"   and i :: "'i"   and M' :: "'d measure" assumes "Y.random_variable i M' f" shows "prob_space (distr (distr M borel (Y i)) M' f)"
list_induct: fixes P :: "'b SList.list \<Rightarrow> bool"   and l :: "'b SList.list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (x # xs)" shows "P l"
permutation_of_list_unique': fixes xs :: "('b \<times> 'b) list"   and x :: "'b"   and y :: "'b" assumes "distinct (map fst xs)"   and "(x, y) \<in> set xs" shows "permutation_of_list xs x = y"
bl_sbin_sign: fixes n :: "nat"   and w :: "int" shows "hd (bin_to_bl (Suc n) w) = (bin_sign (signed_take_bit n w) = - 1)"
axioms: fixes A :: "V"   and f :: "V" assumes "binop_onto A f" shows "nop_onto A (2\<^sub>\<nat>) f"
prm_unit_commutes: fixes a :: "'b"   and b :: "'b" shows "[a \<leftrightarrow> b] = [b \<leftrightarrow> a]"
ordinal_div_decreasing: fixes x :: "ordinal"   and y :: "ordinal" shows "x div y \<le> x"
func: shows "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier"
simps_3: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "quadratic_sub var a b c d (Leq p) = (let p1 = quadratic_part_1 var a b d (Leq p); p2 = quadratic_part_2 var c p1; A = isolate_variable_sparse p2 var 0; B = isolate_variable_sparse p2 var 1 in Or (And (fm.Atom (Leq A)) (fm.Atom (Leq (B\<^sup>2 * c - A\<^sup>2)))) (And (fm.Atom (Leq B)) (fm.Atom (Leq (A\<^sup>2 - B\<^sup>2 * c)))))"
rel_intros_2: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x21 :: "'a"   and y21 :: "'b"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x22 :: "'c"   and y22 :: "'d" assumes "R1 x21 y21"   and "R2 x22 y22" shows "rel_sp\<^sub>\<mu> R1 R2 (Put x21 x22) (Put y21 y22)"
inst_var: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and a :: "Core.term"   and \<tau> :: "typ"   and x :: "variable" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "term_ok \<Theta> a"   and "typ_of a = Some \<tau>"   and "(x, \<tau>) \<notin> FV \<Gamma>" shows "\<Theta>,\<Gamma> \<turnstile> subst_term [((x, \<tau>), a)] B"
on_curve_def: fixes a :: "'a"   and b :: "'a"   and p :: "'a point" shows "on_curve a b p = (case p of Infinity \<Rightarrow> True | Point x y \<Rightarrow> x \<in> carrier R \<and> y \<in> carrier R \<and> y [^] 2 = x [^] 3 \<oplus> a \<otimes> x \<oplus> b)"
simps_15: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case C of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = f3"
Cons_3: shows "list_all (\<lambda>P. index_free P i) (P # Ps)"
foldr_append: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'e list"   and a :: "'d" shows "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
mem_existence_ivl_iv_defined_2: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "x0 \<in> X"
P3: fixes j :: "nat" shows "foldl1 f1 ([One] @ replicate j Zero) = One"
map_poly_x_minus_y: fixes p :: "'a poly" shows "map_poly (map_poly hom) (poly_x_minus_y p) = poly_x_minus_y (map_poly hom p)"
simps_1: fixes u :: "nat"   and v :: "nat" shows "isPath u [] v = (u = v)"
full_imp_height: fixes n :: "nat"   and t :: "'b tree234" assumes "full n t" shows "height t = n"
simple_imp_good_ruleset: fixes rs :: "'b rule list" assumes "simple_ruleset rs" shows "good_ruleset rs"
fmdom'_notI: fixes m :: "('d, 'c) fmap"   and x :: "'d" assumes "m $$ x = None" shows "x \<notin> fmdom' m"
dominates_extend: fixes g :: "'g"   and n :: "'node"   and m :: "'node"   and m' :: "'node"   and ms :: "'node list" assumes "old.dominates g n m"   and "g \<turnstile> m'-ms\<rightarrow>m"   and "n \<notin> set (tl ms)" shows "old.dominates g n m'"
SFAccNonStatic: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> C has F,NonStatic:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},s\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,s\<rangle>"
goal3_5: shows "Rmi_g (Suc (Suc n)) (IF v t e) bdd"
interval_union_mono2: fixes A :: "'a interval"   and a :: "'a interval" shows "set_of A \<subseteq> set_of (sup a A)"
divide_real_def: fixes x :: "real"   and y :: "real" shows "x div y = x * inverse y"
errMOD_igSwapIPresIGWlsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igSwapIPresIGWls MOD"   and "igWlsDisj MOD" shows "igSwapIPresIGWlsSTR (errMOD MOD)"
mask_AND_less_0: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" assumes "x && mask n = 0"   and "m \<le> n" shows "x && mask m = 0"
add_cancel_left_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus a b = a) = (b = zero)"
rel_chain_tl: fixes l :: "'b list"   and R :: "('b \<times> 'b) set" assumes "rel_chain l R" shows "rel_chain (tl l) R"
irreducible_mono: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.irreducible times one zero b"   and "dvd.dvd times a b"   and "\<not> dvd.dvd times a one" shows "comm_semiring_1.irreducible times one zero a"
del_list_sorted4: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and b :: "'b"   and zs :: "'b list"   and c :: "'b"   and us :: "'b list"   and x :: "'b" assumes "sorted (xs @ a # ys @ b # zs @ c # us)"   and "x < c" shows "del_list x (xs @ a # ys @ b # zs @ c # us) = del_list x (xs @ a # ys @ b # zs) @ c # us"
subgroup_axioms: shows "subgroup (kernel G (H\<lparr>carrier := h ` carrier G\<rparr>) h) G"
simps_3: fixes p1 :: "'a"   and v :: "bool" shows "isReceiverOf p1 <\<bottom>, outM v> = False"
f2f2'': shows "1 \<le> (f2 - f2'') * real l0"
continuous_def: fixes f :: "'b set \<Rightarrow> 'c set" shows "continuous f = (\<forall>C. chain C \<longrightarrow> chain (f \<circ> C) \<and> f (natUnion C) = natUnion (f \<circ> C))"
projection_split_first_with_suffix: fixes \<tau> :: "'a list"   and X :: "'a set"   and x :: "'a"   and xs :: "'a list" assumes "\<tau> \<upharpoonleft> X = x # xs" shows "\<exists>\<alpha> \<beta>. \<tau> = \<alpha> @ [x] @ \<beta> \<and> \<alpha> \<upharpoonleft> X = [] \<and> \<beta> \<upharpoonleft> X = xs"
int_le_real_less: fixes n :: "int"   and m :: "int" shows "(n \<le> m) = (real_of_int n < real_of_int m + 1)"
indep_sets_finite_X: fixes i :: "'i \<Rightarrow> 'i" assumes "\<And>ia. ia \<in> I \<Longrightarrow> i ia \<in> I"   and "\<And>ia. ia \<in> I \<Longrightarrow> i ia \<in> I" shows "indep_sets (\<lambda>ia. {X (i ia) -` A \<inter> space M |A. A \<in> E (i ia)}) I = (\<forall>A\<in>\<Pi> ia\<in>I. {X (i ia) -` A \<inter> space M |A. A \<in> E (i ia)}. prob (\<Inter> (A ` I)) = (\<Prod>j\<in>I. prob (A j)))"
divergentST_STCal_divergent: fixes P :: "('procS, 'procT) Proc" shows "P \<longmapsto>STCal Source Target\<omega> = P \<longmapsto>ST\<omega>"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "B.dom (g \<cdot>\<^sub>B f) = B.dom f"
dgrad_p_set_le_uminus: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "dgrad_p_set_le d {- p} {p}"
HPow_mono_strict: fixes x :: "hf"   and y :: "hf" assumes "x < y" shows "HPow x < HPow y"
change_respecting_dom: fixes cms :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val"   and X :: "'Var set"   and g :: "('Var \<Rightarrow> 'Val option) \<Rightarrow> 'Var \<Rightarrow> 'Val option"   and \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "change_respecting (cms, mem) (cms', mem') X g"   and "dom \<sigma> = X" shows "dom (g \<sigma>) \<subseteq> X"
diff_diff_eq2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus a (minus b c) = minus (plus a c) b"
inorder_combine: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree" shows "Tree2.inorder (combine l r) = Tree2.inorder l @ Tree2.inorder r"
simps: fixes k :: "nat"   and x :: "'a"   and t :: "'a btree" shows "local.insert k x t = tree\<^sub>i (ins k x t)"
minWait_ex: fixes s :: "'state"   and rs :: "'rule stream" assumes "s \<in> S"   and "fair rs" shows "\<exists>n. enabled (shd (sdrop n rs)) s"
induct: fixes P :: "bool list \<Rightarrow> bool list \<Rightarrow> bool"   and a0 :: "bool list"   and a1 :: "bool list" assumes "P [] []"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
inf_Sec_def: fixes d1 :: "Sec"   and d2 :: "Sec" shows "d1 \<sqinter> d2 = (if d1 = Low \<or> d2 = Low then Low else High)"
Q_def: shows "Q \<equiv> reciprocal_poly p P \<circ>\<^sub>p [:1, 1:]"
simps_26: fixes x1 :: "'b" shows "set_flat_complete_lattice (flat_complete_lattice.Value x1) = {x1}"
box_eucl_less: fixes a :: "'b"   and b :: "'b" shows "box a b = {x. a <e x \<and> x <e b}"
assms: fixes c :: "complex"   and x :: "'a" shows "f *\<^sub>v c *\<^sub>C x = c *\<^sub>C (f *\<^sub>v x)"
reachablep_reachable_eq: fixes A :: "('d, 'c) nba"   and p :: "'c" shows "transition_system.reachablep (\<lambda>a p. snd a) (\<lambda>a p. fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) p = (\<lambda>x. x \<in> reachable A p)"
zero_smallest: fixes r :: "('a \<times> 'a) set"   and j :: "'a" assumes "wo_rel r"   and "j \<in> Field r" shows "(zero r, j) \<in> r"
Fin_lemma: fixes G :: "prog"   and s1 :: "st"   and c2 :: "stmt"   and x2 :: "abopt"   and s2 :: "st"   and a :: "abrupt"   and L :: "lenv" assumes "G\<turnstile>Norm s1 \<midarrow>c2\<rightarrow> (x2, s2)"   and "wf_prog G"   and "(Some a, s1)\<Colon>\<preceq>(G, L)"   and "(x2, s2)\<Colon>\<preceq>(G, L)"   and "dom (locals s1) \<subseteq> dom (locals s2)" shows "(abrupt_if True (Some a) x2, s2)\<Colon>\<preceq>(G, L)"
eq2: shows "z \<cdot> w = [a] \<^sup>@ m \<cdot> [b] \<cdot> z' \<cdot> [a] \<^sup>@ k"
approx_trans: fixes x :: "'a star"   and y :: "'a star"   and z :: "'a star" assumes "x \<approx> y"   and "y \<approx> z" shows "x \<approx> z"
c1_def: fixes y :: "'a" shows "c1 y = f (x - y) * g y"
PiE: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and x :: "'d" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "\<not> x \<in> A"
prems_in: shows "{DA0} \<union> set CAs0 \<subseteq> M"
Inf_fun_def: fixes A :: "('c \<Rightarrow> 'd) set" shows "Inf A = (\<lambda>x. INF f\<in>A. f x)"
divide_self_if: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a a = (if a = zero then zero else one)"
load_sub2_result: fixes address :: "word32"   and asi :: "word8"   and rd :: "word5"   and curr_win :: "'b word"   and word0 :: "word32"   and s :: "'b sparc_state" shows "snd (load_sub2 address asi rd curr_win word0 s) = False"
is_subseq_of_def: fixes s :: "nat \<Rightarrow> 'b"   and s' :: "nat \<Rightarrow> 'b" shows "is_subseq_of s s' = (\<exists>f. is_increasing f \<and> s' = take_subseq s f)"
chine_assoc_def: shows "gfg.chine_assoc \<equiv> \<langle>gfg.Prj\<^sub>1\<^sub>1 \<lbrakk>g.leg0, f.leg1 \<cdot> fg.prj\<^sub>1\<rbrakk> \<langle>gfg.Prj\<^sub>0\<^sub>1 \<lbrakk>f.leg0, g.leg1\<rbrakk> gfg.Prj\<^sub>0\<rangle>\<rangle>"
induct: fixes x1 :: "('exp, 'id) MWLsCom"   and x2 :: "'id \<Rightarrow> 'val"   and x3 :: "('exp, 'id) MWLsCom list"   and x4 :: "('exp, 'id) MWLsCom option"   and x5 :: "'id \<Rightarrow> 'val"   and P :: "('exp, 'id) MWLsCom \<Rightarrow> ('id \<Rightarrow> 'val) \<Rightarrow> ('exp, 'id) MWLsCom list \<Rightarrow> ('exp, 'id) MWLsCom option \<Rightarrow> ('id \<Rightarrow> 'val) \<Rightarrow> bool" assumes "\<langle>x1,x2\<rangle> \<rightarrow>\<lhd>x3\<rhd> \<langle>x4,x5\<rangle>"   and "\<And>\<iota> m. P (skip\<^bsub>\<iota>\<^esub>) m [] None m"   and "\<And>e m v x \<iota>. E e m = v \<Longrightarrow> P (x:=\<^bsub>\<iota>\<^esub> e) m [] None (m(x := v))"   and "\<And>c1 m \<alpha> m' c2. \<lbrakk>\<langle>c1,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>None,m'\<rangle>; P c1 m \<alpha> None m'\<rbrakk> \<Longrightarrow> P (c1;c2) m \<alpha> (Some c2) m'"   and "\<And>c1 m \<alpha> c1' m' c2. \<lbrakk>\<langle>c1,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>Some c1',m'\<rangle>; P c1 m \<alpha> (Some c1') m'\<rbrakk> \<Longrightarrow> P (c1;c2) m \<alpha> (Some (c1';c2)) m'"   and "\<And>b m \<iota> c1 c2. BMap (E b m) = True \<Longrightarrow> P (if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi) m [] (Some c1) m"   and "\<And>b m \<iota> c1 c2. BMap (E b m) = False \<Longrightarrow> P (if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi) m [] (Some c2) m"   and "\<And>b m \<iota> c. BMap (E b m) = True \<Longrightarrow> P (while\<^bsub>\<iota>\<^esub> b do c od) m [] (Some (c;while\<^bsub>\<iota>\<^esub> b do c od)) m"   and "\<And>b m \<iota> c. BMap (E b m) = False \<Longrightarrow> P (while\<^bsub>\<iota>\<^esub> b do c od) m [] None m"   and "\<And>\<iota> V m. P (spawn\<^bsub>\<iota>\<^esub> V) m V None m" shows "P x1 x2 x3 x4 x5"
DOWN_def: shows "replete_setcat.DOWN \<equiv> setcat.DOWN"
arc_set_balance_def: fixes G :: "('c, 'd) pre_digraph"   and w :: "'c"   and A :: "'d set" shows "pre_digraph.arc_set_balance G w A = int (card (in_arcs G w \<inter> A)) - int (card (out_arcs G w \<inter> A))"
RTL: shows "X \<noteq> {} \<and> (\<forall>x\<in>X. rank_of (X - {x}) = card (X - {x}) \<and> card (X - {x}) = rank_of X)"
sel_3: fixes x1 :: "'b"   and x2 :: "'b tree"   and x3 :: "'b tree" shows "right (Node x1 x2 x3) = x3"
snd_if_conv: fixes P :: "bool"   and A :: "'e"   and B :: "'d"   and C :: "'e"   and D :: "'d" shows "snd (if P then (A, B) else (C, D)) = (if P then B else D)"
measurable_snd: fixes M1 :: "'c measure"   and M2 :: "'d measure" shows "snd \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M2"
find_adds_SomeD3: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and u :: "'t"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "find_adds fs u = Some f" shows "lt f adds\<^sub>t u"
vec_list: fixes v :: "'c vec" shows "vec_of_list (list_of_vec v) = v"
basis_card: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool"   and B\<^sub>1 :: "'c set"   and B\<^sub>2 :: "'c set" assumes "matroid carrier indep"   and "indep_system.basis carrier indep B\<^sub>1"   and "indep_system.basis carrier indep B\<^sub>2" shows "card B\<^sub>1 = card B\<^sub>2"
conv_radius_conv_Sup: fixes f :: "nat \<Rightarrow> 'b" shows "conv_radius f = Sup {r. \<forall>z. ereal (norm z) < r \<longrightarrow> summable (\<lambda>n. f n * z ^ n)}"
simps_1: fixes a :: "nat \<Rightarrow> complex" shows "FFT 0 a = a"
sup_bool_def: fixes P :: "bool"   and Q :: "bool" shows "P \<sqinter> Q = (P \<or> Q)"
not_ephemeralD: fixes C :: "'s set"   and x :: "'s" assumes "not_ephemeral C"   and "x \<in> C" shows "\<exists>n>0. 0 < p x x n"
naturally_isomorphic_symmetric: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd" assumes "naturally_isomorphic A B F G" shows "naturally_isomorphic A B G F"
gmnl_correct: fixes xs :: "('a, 'b) Node list" assumes "\<forall>x\<in>set xs. FingerTreeStruc.is_measured_node x" shows "FingerTreeStruc.gmnl xs = sum_list (map snd (FingerTreeStruc.nlistToList xs))"
"case": fixes f :: "fol fm \<Rightarrow> 'a"   and x :: "fol fm" shows "(case Thm x of Thm x \<Rightarrow> f x) = f x"
assms: shows "demand f = C\<^bsup>n\<^esup>"
RootState_HARootState: fixes ST :: "('d, 'e, 'f) status" shows "RootState ST \<in> States (HARoot (HA ST))"
wlp_while_n_close: fixes WS :: "complex mat \<Rightarrow> complex mat"   and M :: "nat \<Rightarrow> complex mat"   and P :: "complex mat"   and k :: "nat" assumes "\<And>P. is_quantum_predicate P \<Longrightarrow> is_quantum_predicate (WS P)"   and "measurement d 2 M"   and "is_quantum_predicate P" shows "is_quantum_predicate (wlp_while_n (M 0) (M 1) WS k P)"
zero_def: shows "0\<^sub>h \<equiv> abs_complex_homo 0\<^sub>h\<^sub>c"
a1: shows "Collect Q \<subseteq> (S\<inverse> O R)\<inverse> `` Collect P"
l: shows "hi' + 1 < hi \<or> hi \<le> hi' + 1"
asEnv_def: fixes qrho :: "'f \<Rightarrow> 'g \<Rightarrow> ('h, 'i, 'varSort, 'var, 'j) qTerm option" shows "asEnv qrho \<equiv> \<lambda>xs. lift asTerm (qrho xs)"
onorm_sphere: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "onorm f = Sup {norm (f x) |x. norm x = 1}"
mertens_first_theorem: fixes x :: "real" assumes "1 \<le> x" shows "\<bar>\<MM> x - ln x\<bar> < 2"
ind_False: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<not> P n"   and "0 < n" shows "ind P n = (0::'b)"
hypnat_gt_zero_iff: fixes n :: "nat star" shows "(0 < n) = (1 \<le> n)"
simps_1: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "name \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = name"
fmdrop_fset_fmdom: fixes A :: "('a, 'b) fmap" shows "fmdrop_fset (fmdom A) A = fmempty"
SUP_subset_mono: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and A :: "'f set"   and B :: "'f set"   and f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
reachable_def: fixes g :: "(node \<times> node) list"   and xs :: "node list" shows "reachable g xs = (set g)\<^sup>* `` set xs"
WT_gpv_coinduct: fixes X :: "('d, 'e, 'f) gpv \<Rightarrow> bool"   and gpv :: "('d, 'e, 'f) gpv"   and \<Gamma> :: "('e, 'f) \<I>" assumes "X gpv"   and "\<And>gpv out c. \<lbrakk>X gpv; IO out c \<in> set_spmf (the_gpv gpv)\<rbrakk> \<Longrightarrow> out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. X (c input) \<or> \<Gamma> \<turnstile>g c input \<surd>)" shows "\<Gamma> \<turnstile>g gpv \<surd>"
simps_10: fixes a :: "'a expr1"   and i :: "'a expr1"   and e :: "'a expr1" shows "compE2 (a\<lfloor>i\<rceil> := e) = compE2 a @ compE2 i @ compE2 e @ [AStore, Push Unit]"
list_splits_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "list_splits (map f xs) = map (\<lambda>(xs, x, ys). (map f xs, f x, map f ys)) (list_splits xs)"
Restr_Field: fixes r :: "('c \<times> 'c) set" shows "Restr r (Field r) = r"
summable_mono_reindex: fixes g :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b" assumes "strict_mono g"   and "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0::'b)" shows "summable (\<lambda>n. f (g n)) = summable f"
update_sqn: fixes dsn :: "nat"   and dsk :: "k"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and rip :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "(dsn = 0) = (dsk = unk)" shows "sqn rt dip \<le> sqn (update rt rip (dsn, dsk, val, hops, nhip)) dip"
complex_roots_unity: fixes n :: "nat" assumes "1 \<le> n" shows "{z. z ^ n = 1} = {exp (2 * complex_of_real pi * \<i> * of_nat j / of_nat n) |j. j < n}"