hcju/isabelleps huggingface dataset:
\<tau>J_def: fixes \<psi> :: "'fmla" shows "\<tau>J \<psi> \<equiv> instInp (encF (ssap \<psi>)) \<langle>encF (ssap \<psi>)\<rangle>"
subformulas\<^sub>\<nu>_subset: fixes \<psi> :: "'a ltln"   and \<phi> :: "'a ltln" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "subformulas\<^sub>\<nu> \<psi> \<subseteq> subformulas\<^sub>\<nu> \<phi>"
psp_ensures: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set" assumes "F \<in> A ensures A'"   and "F \<in> B co B'" shows "F \<in> A \<inter> B' ensures A' \<inter> B \<union> (B' - B)"
t2: shows "flow0 y t2 \<in> {a<--<b}"
poly_mapping_eqI: fixes f :: "'q \<Rightarrow>\<^sub>0 'r"   and g :: "'q \<Rightarrow>\<^sub>0 'r" assumes "\<And>k. lookup f k = lookup g k" shows "f = g"
sum_deriv: fixes u :: "'a" assumes "u \<in> closed_segment w z" shows "((\<lambda>v. \<Sum>i\<le>n. f i v * (z - v) ^ i / fact i) has_field_derivative f (Suc n) u * (z - u) ^ n / fact n) (at u within S)"
is_volatile_Write\<^sub>s\<^sub>b_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_volatile_Write\<^sub>s\<^sub>b (Ghost\<^sub>s\<^sub>b A L R W) = False"
extremum_strict: fixes a :: "'k" shows "\<not> \<top> < a"
continuous_within_cong: fixes y :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous (at y within S) f"   and "\<forall>\<^sub>F x in at y within S. f x = g x"   and "f y = g y" shows "continuous (at y within S) g"
epred_min: fixes a :: "enat"   and b :: "enat" shows "epred (min a b) = min (epred a) (epred b)"
Rep_bit0_def: fixes x :: "'b bit0" shows "Rep x = Rep_bit0 x"
nat_power_of_real: fixes n :: "nat"   and x :: "real" shows "nat_power n (of_real x) = of_real (real n powr x)"
Un_insert_right: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" shows "A \<union> ({a} \<union> B) = {a} \<union> (A \<union> B)"
toList_def: shows "toList \<equiv> rec_FingerTreeStruc [] nodeToList (\<lambda>_ pr m sf ma. digitToList pr @ ma @ digitToList sf)"
lfp_F_stable_pair_on: fixes ds :: "'b set" shows "stable_pair_on ds (lfp_F ds)"
intros_36: fixes c :: "i32"   and "is" :: "nat list"   and i :: "nat" assumes "nat_of_int c < length is" shows "\<lparr>[$C ConstInt32 c, $Br_table is i]\<rparr> \<leadsto> \<lparr>[$Br (is ! nat_of_int c)]\<rparr>"
prog_instrs_append_Ghost\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "prog_instrs (sb @ [Ghost\<^sub>s\<^sub>b A L R W]) = prog_instrs sb"
arr_cod: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.arr (B.VV.cod f)"
Complex_eq: fixes a :: "real"   and b :: "real" shows "Complex a b = complex_of_real a + \<i> * complex_of_real b"
simps_2: fixes f :: "'y \<Rightarrow> 'z \<Rightarrow> 'x"   and x1 :: "'y"   and x2 :: "'z" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
quadratic_sub_eq: fixes var :: "nat"   and L :: "real list"   and Dv :: "real"   and Cv :: "real"   and c :: "real mpoly"   and a :: "real mpoly"   and Av :: "real"   and b :: "real mpoly"   and Bv :: "real"   and d :: "real mpoly"   and p :: "real mpoly" assumes "var < length L"   and "Dv \<noteq> 0"   and "0 \<le> Cv"   and "var \<notin> vars c"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = Bv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) c = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = Dv" shows "aEval (Eq p) (L[var := (Av + Bv * sqrt Cv) / Dv]) = eval (quadratic_sub var a b c d (Eq p)) (L[var := sqrt Cv])"
separating_hyperplane_closed_point_inset: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "closed S"   and "S \<noteq> {}"   and "z \<notin> S" obtains a :: "'a" and b :: "real" where "a \<in> S" and "(a - z) \<bullet> z < b" and "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
eSuc_enat: fixes n :: "nat" shows "eSuc (enat n) = enat (Suc n)"
MP': fixes ps :: "('a, 'b) fm list"   and p :: "('a, 'b) fm"   and q :: "('a, 'b) fm" assumes "ps \<turnstile> p \<^bold>\<longrightarrow> q"   and "ps \<turnstile> p" shows "ps \<turnstile> q"
prime_implicates_def: fixes S :: "'at Literal set set" shows "prime_implicates S = simplify (implicates S)"
constT_def: fixes D :: "ast_domain" shows "ast_domain.constT D \<equiv> map_of (consts D)"
memberdecl_fdecl_acc_modi_simp: fixes m :: "fdecl" shows "accmodi (fdecl m) = accmodi m"
return_edges_only: fixes a :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<forall>a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<longrightarrow> (\<exists>Qx fx. kind a' = Qx\<hookleftarrow>\<^bsub>p\<^esub>fx)"
absolutely_integrable_spike_set: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'a set" assumes "f absolutely_integrable_on S"   and "negligible {x \<in> S - T. f x \<noteq> (0::'b)}"   and "negligible {x \<in> T - S. f x \<noteq> (0::'b)}" shows "f absolutely_integrable_on T"
simps_2: fixes f :: "'e \<Rightarrow> ('c, 'd) state dual"   and l :: "'e Tree_Relabelling.tree"   and r :: "'e Tree_Relabelling.tree" shows "traverse_rev f (Tree_Relabelling.tree.Node l r) = pure Tree_Relabelling.tree.Node \<diamondop> traverse_rev f l \<diamondop> traverse_rev f r"
distinct_11: fixes x3 :: "'f"   and x41 :: "'g"   and x42 :: "('e, 'f, 'g) Tree" shows "tPred x3 \<noteq> tAct x41 x42"
inject: fixes x1 :: "('a \<times> real) msllist"   and x2 :: "real \<Rightarrow> real"   and y1 :: "('a \<times> real) msllist"   and y2 :: "real \<Rightarrow> real" shows "(MS x1 x2 = MS y1 y2) = (x1 = y1 \<and> x2 = y2)"
simps_234: fixes f :: "'c \<Rightarrow> 'b"   and x121 :: "'c ltlc"   and x122 :: "'c ltlc" shows "map_ltlc f (x121 R\<^sub>c x122) = map_ltlc f x121 R\<^sub>c map_ltlc f x122"
in_carrier2: shows "inv \<^bold>g [^] u1 \<in> carrier \<G>"
perm: shows "\<forall>xs ys. xs <~~> ys \<longrightarrow> P xs = P ys"
ffold_set_union: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and A :: "'d fset"   and B :: "'d fset" assumes "comp_fun_idem f" shows "ffold f x (A |\<union>| B) = ffold f (ffold f x A) B"
rep_eq: fixes x :: "'d fset" shows "fBall x = Ball (fset x)"
from_hma\<^sub>v_add: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec"   and w :: "('c, 'd) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>v v + from_hma\<^sub>v w = from_hma\<^sub>v (v + w)"
a_repr_independence': fixes y :: "'a"   and H :: "'a set"   and x :: "'a" assumes "y \<in> H #>\<^bsub>add_monoid G\<^esub> x"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "H #>\<^bsub>add_monoid G\<^esub> x = H #>\<^bsub>add_monoid G\<^esub> y"
trivial_RSubmodule: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "RModule R smult M" shows "RModule R smult 0 \<and> 0 \<subseteq> M"
inv_notame: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g"   and "notame7 g" shows "notame7 g'"
v'_t: shows "v' \<Turnstile>\<^sub>t Tableau (preprocess' t start)"
path_singleton: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c" assumes "Graph G"   and "v \<in> V\<^bsub>G\<^esub>" shows "Graph.walk G [v] \<and> distinct [v]"
sample_cong: fixes p :: "int pmf"   and q :: "int pmf"   and f :: "int \<Rightarrow> 'm"   and g :: "int \<Rightarrow> 'm" assumes "p = q"   and "\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x" shows "sample p f = sample q g"
valid_ConsI: fixes rest :: "(recf \<times> nat list \<times> nat list) list"   and f :: "recf"   and xs :: "nat list"   and locs :: "nat list" assumes "valid rest"   and "recfn (length xs) f" shows "valid ((f, xs, locs) # rest)"
edge_start: fixes p :: "'a"   and q :: "'a" assumes "is_point p"   and "is_point q"   and "p \<noteq> q" shows "start_points (p ; q\<^sup>T) = p"
le_n: fixes k :: "nat" assumes "0 < k"   and "ln (real k + sqrt ((real k)\<^sup>2 - 1)) / pi \<le> x" shows "k \<le> n"
dist_snd_le: fixes x :: "'e \<times> 'd"   and y :: "'e \<times> 'd" shows "dist (snd x) (snd y) \<le> dist x y"
numeral_1_eq_Suc_0: shows "Numeral1 = Suc 0"
phi_equiv_mappingE: fixes g :: "'g"   and m\<^sub>1 :: "('val, ('node \<times> 'val) set) mapping"   and b :: "'node \<times> 'val"   and x :: "'node \<times> 'val"   and vs :: "'val list" assumes "g \<turnstile> m\<^sub>1 \<approx>\<^sub>\<phi> uninst_code.ssa.phiNodes_of phis g"   and "b \<in> Mapping.keys (phis g)"   and "Mapping.lookup (phis g) x = Some vs"   and "snd b \<in> set vs" obtains ns :: "'a" where "Mapping.lookup m\<^sub>1 (snd b) = Some {n \<in> Mapping.keys (phis g). snd b \<in> set (the (Mapping.lookup (phis g) n))}"
int_xor_extra_simps_2: fixes w :: "int" shows "w XOR - 1 = NOT w"
triangular_to_triangular: fixes A :: "'a mat" shows "upper_triangular A = local.triangular_to (dim_row A) A"
subset_imp: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and B :: "('x \<Rightarrow>\<^sub>0 nat) set" assumes "A \<subseteq> B"   and "A \<noteq> {}"   and "finite B" shows "ordered_powerprod_lin.Max A \<preceq> ordered_powerprod_lin.Max B"
order_bot_ow_axioms: shows "order_bot_ow U\<^sub>2 bot\<^sub>2 (\<le>\<^sub>o\<^sub>w\<^sub>.\<^sub>2) (<\<^sub>o\<^sub>w\<^sub>.\<^sub>2)"
bind_left_identity: fixes a :: "'d"   and f :: "'d \<Rightarrow> ('b, 'c) state" shows "return a \<bind> f = f a"
invertible_image_lemma_2: fixes a :: "'a" assumes "invertible a"   and "a \<in> G" shows "\<eta> (inverse a) \<cdot>' \<eta> a = \<one>'"
msg_Nonce_supply: fixes msg :: "msg" shows "\<exists>N. \<forall>n\<ge>N. Nonce n \<notin> parts {msg}"
harm_Suc: fixes n :: "nat" shows "harm (Suc n) = harm n + inverse (of_nat (Suc n))"
l_new_shadow_root_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_shadow_root_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_new_shadow_root_get_child_nodes_axioms get_child_nodes get_child_nodes_locs"
aux: fixes b :: "'b option" shows "(\<bottom> \<noteq> b) = (\<exists>y. b = \<lfloor>y\<rfloor>)"
fmdom_notD: fixes x :: "'d"   and m :: "('d, 'e) fmap" assumes "x |\<notin>| fmdom m" shows "fmlookup m x = None"
msetext_huet_compat_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "msetext_huet gt ys xs" shows "msetext_huet gt (ys @ [x]) (xs @ [x])"
hIm_minus: fixes z :: "complex star" shows "hIm (- z) = - hIm z"
ExecCondEvalError1: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s\<^sub>1 :: "stmt"   and s\<^sub>2 :: "stmt" assumes "eval \<sigma> e\<^sub>1 = None" shows "IfEq e\<^sub>1 e\<^sub>2 s\<^sub>1 s\<^sub>2 \<rhd> \<sigma> \<leadsto> None"
simps_1: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "typeof\<^bsub>h\<^esub> Unit = \<lfloor>Void\<rfloor>"
EF_induct: fixes a :: "'a"   and f :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "a \<in> EF f"   and "mono (\<lambda>Z. f \<union> EX' Z)"   and "\<And>x. x \<in> f \<union> EX' (EF f \<inter> {x. P x}) \<Longrightarrow> P x" shows "P a"
mm_equiv_step: fixes cms\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "(cms\<^sub>1, mem\<^sub>1) \<approx> (cms\<^sub>2, mem\<^sub>2)"   and "snd cms\<^sub>1 = snd cms\<^sub>2"   and "(cms\<^sub>1, mem\<^sub>1) \<leadsto> (cms\<^sub>1', mem\<^sub>1')" shows "\<exists>c\<^sub>2' mem\<^sub>2'. (cms\<^sub>2, mem\<^sub>2) \<leadsto> \<langle>c\<^sub>2', snd cms\<^sub>1', mem\<^sub>2'\<rangle> \<and> (cms\<^sub>1', mem\<^sub>1') \<approx> \<langle>c\<^sub>2', snd cms\<^sub>1', mem\<^sub>2'\<rangle>"
sel_3: fixes x11 :: "char list"   and x12 :: "'a list" shows "get_terms (Pos x11 x12) = x12"
setPair5: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "{a, b} = {c, d}"   and "a \<noteq> c" shows "a = d"
has_integral_reflect_lemma: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and a :: "'b"   and b :: "'b" assumes "(f has_integral i) (cbox a b)" shows "((\<lambda>x. f (- x)) has_integral i) (cbox (- b) (- a))"
Nil_is_map_conv: fixes f :: "'g \<Rightarrow> 'f"   and xs :: "'g list" shows "([] = map f xs) = (xs = [])"
filter_mset_False: fixes M :: "'a multiset" shows "{#y \<in># M. False#} = {#}"
eventually_at_left_field: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "eventually P (at_left x) = (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
opt_bst_wpl_eq_pair: fixes i :: "int"   and j :: "int" shows "opt_bst_wpl i j = (local.opt_bst i j, local.wpl i j (local.opt_bst i j))"
domintros_3: fixes n :: "nat"   and m :: "nat"   and l :: "nat list" assumes "ackloop_dom (n # Suc m # m # l)" shows "ackloop_dom (Suc n # Suc m # l)"
wf_eq_minimal: fixes r :: "('e \<times> 'e) set" shows "wf r = (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q))"
usubstappf_geqr: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and \<eta> :: "trm" assumes "usubstappf \<sigma> U (Geq \<theta> \<eta>) \<noteq> undeff" shows "usubstappf \<sigma> U (Geq \<theta> \<eta>) = Afml (Geq (the (usubstappt \<sigma> U \<theta>)) (the (usubstappt \<sigma> U \<eta>)))"
real_of_float_eq: fixes f1 :: "float"   and f2 :: "float" shows "(f1 = f2) = (real_of_float f1 = real_of_float f2)"
assms_5: shows "X #[y \<and> x]_xs = X' #[y \<and> x']_xs"
nat_cut_less_Max_less: fixes I :: "nat set"   and t :: "nat" assumes "I \<down>< t \<noteq> {}" shows "Max (I \<down>< t) < t"
even_mask_div_iff: fixes m :: "nat"   and n :: "nat" shows "even (((2::'a) ^ m - (1::'a)) div (2::'a) ^ n) = ((2::'a) ^ n = (0::'a) \<or> m \<le> n)"
insert_DiffM: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "add_mset x (M - {#x#}) = M"
a_l_coset_carrier: fixes G :: "('c, 'd) ring_scheme"   and y :: "'c"   and x :: "'c"   and H :: "'c set" assumes "abelian_group G"   and "y \<in> x <+\<^bsub>G\<^esub> H"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "y \<in> carrier G"
assms_2: shows "table (args_n args) (args_L args) rel1"
FrameImp_def: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and F :: "'d frame"   and \<Phi> :: "'e" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.FrameImp SImp F \<Phi> = (\<exists>A\<^sub>F \<Psi>\<^sub>F. F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle> \<and> A\<^sub>F \<sharp>* \<Phi> \<and> SImp \<Psi>\<^sub>F \<Phi>)"
edges_of_comp1: fixes xs :: "'a list"   and ys :: "'a list" shows "edges_of_walk xs \<subseteq> edges_of_walk (xs @ ys)"
assms_1: shows "w \<in> path_image (part_circlepath z r st tt)"
assm2: shows "mn \<in> set (normalize_primitive_extract (disc1, sel1) C f m)"
assms_1: shows "S \<equiv> \<lparr>g_V = UNIV, g_E = rel_of_enex (en, ex), g_V0 = {init}, sa_L = int\<rparr>"
distinct_1: fixes x1 :: "nat"   and x2 :: "nat" shows "I x1 \<noteq> R x2"
agree_on_trans: fixes X :: "('e, 'f) labeled_graph"   and f :: "('f \<times> 'g) set"   and g :: "('f \<times> 'g) set"   and h :: "('f \<times> 'g) set" assumes "agree_on X f g"   and "agree_on X g h" shows "agree_on X f h"
wf_typeE1_1: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"
borel_measurable_minus_ennreal: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
hom_dvd: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q" shows "poly.coeff p 0 dvd poly.coeff q 0"
finite_induct: fixes P :: "'e\<cdot>('f, 'g) writerT \<Rightarrow> bool"   and n :: "nat"   and x :: "'e\<cdot>('f, 'g) writerT" assumes "P \<bottom>"   and "\<And>app. app \<noteq> \<bottom> \<Longrightarrow> P (WriterT\<cdot>app)" shows "P (writerT_take n\<cdot>x)"
rsided_bres_eq: fixes x :: "'a" shows "rsd x = (x \<rightarrow> x = \<top>)"
encode_positive_transition_frame_axiom_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and v :: "'b" shows "encode_positive_transition_frame_axiom \<Pi> t v \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O>; adding_operators = filter (\<lambda>op. ListMem v (add_effects_of op)) ops in Atom (State t (index vs v)) \<^bold>\<or> (\<^bold>\<not> (Atom (State (Suc t) (index vs v))) \<^bold>\<or> \<^bold>\<Or>map (\<lambda>op. Atom (Operator t (index ops op))) adding_operators)"
lifted_bracket_axioms: shows "lifted_bracket term_bracket extra_rule itrm_bracket"
sum_of_prideals1: fixes n :: "nat" shows "\<forall>f. (\<forall>l\<le>n. f l \<in> carrier R) \<longrightarrow> f ` {i. i \<le> n} \<subseteq> sum_pr_ideals R f n"
foldseq_significant_positions: fixes N :: "nat"   and S :: "nat \<Rightarrow> 'b"   and T :: "nat \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<forall>i\<le>N. S i = T i" shows "foldseq f S N = foldseq f T N"
hyps: assumes "xs \<noteq> []" shows "\<exists>a\<in>set xs. Missing_List.min_list xs = a"
that: fixes pq :: "('a \<times> 'b) spmf" assumes "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> P x y"   and "p = map_spmf fst pq"   and "q = map_spmf snd pq" shows "thesis"
pre_oldest_configuration_Min_in: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir.pre_oldest_tokens \<delta> q\<^sub>0 w q n \<noteq> {}" shows "Min (semi_mojmir.pre_oldest_tokens \<delta> q\<^sub>0 w q n) \<in> semi_mojmir.pre_oldest_tokens \<delta> q\<^sub>0 w q n"
h: shows "h \<in> hom (free_Abelian_group S) (free_Abelian_group T)"
islist_def: fixes h :: "'c \<Rightarrow> 'c"   and p :: "'c" shows "islist h p = (\<exists>as. List h p as)"
permute_list_nth: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'b list"   and i :: "nat" assumes "f permutes {..<length xs}"   and "i < length xs" shows "permute_list f xs ! i = xs ! f i"
ans_d_def: fixes x :: "('a \<times> 'a) set" shows "rel_antidomain_kleene_algebra.ans_d x = rel_ad (rel_ad x)"
3: shows "sets (tree_sigma M) = sigma_sets (space (tree_sigma M)) (trees_cyl ` trees (sets M))"
simps_5: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c"   and x2 :: "real"   and y1 :: "'d"   and y2 :: "real" shows "rel_sctn R (Sctn x1 x2) (Sctn y1 y2) = (R x1 y1 \<and> x2 = y2)"
big_mono: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. R (norm (f x)) (norm (g x))" shows "f \<in> L F g"
assms_1: shows "(S ===> C ===> rel_spmf (rel_prod R S)) callee1 callee2"
preserves_local_guarantee_compliance_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" shows "preserves_local_guarantee_compliance \<R> \<equiv> \<forall>cm\<^sub>A mem\<^sub>A cm\<^sub>C mem\<^sub>C. abs.respects_own_guarantees cm\<^sub>A \<and> ((cm\<^sub>A, mem\<^sub>A), cm\<^sub>C, mem\<^sub>C) \<in> \<R> \<longrightarrow> conc.respects_own_guarantees cm\<^sub>C"
floor_zero: shows "\<lfloor>0::'c\<rfloor> = 0"
factorize_int_poly_generic_def: fixes alg :: "int_poly_factorization_algorithm"   and f :: "int poly" shows "factorize_int_poly_generic alg f = (case x_split f of (n, g) \<Rightarrow> if g = 0 then (0, []) else case factorize_int_last_nz_poly alg g of (a, fs) \<Rightarrow> if n = 0 then (a, fs) else (a, (Polynomial.monom 1 1, n - 1) # fs))"
init_comp_ty_def: fixes T :: "ty" shows "init_comp_ty T = (if \<exists>C. T = Class C then Init (the_Class T) else Skip)"
subt_irrefl: fixes s :: "'b trm" shows "\<not> s \<prec> s"
group_axioms: shows "group (\<cdot>) (1::'a) inverse"
assms_5: fixes s :: "'b" assumes "s \<in> set ss" shows "(0::'a) \<in> f s"
span_redundant: fixes x :: "('a, 'b) vec"   and S :: "('a, 'b) vec set" assumes "x \<in> vec.span S" shows "vec.span (insert x S) = vec.span S"
abs_summable_on_reindex: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and A :: "'b set" assumes "(\<lambda>x. f (g x)) abs_summable_on A" shows "f abs_summable_on g ` A"
l7_9_bis: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and X :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A P X"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A X Q" shows "P = Q"
C: fixes a :: "('a, 'b) alpha_rp_scheme"   and b :: "('a, 'b) alpha_rp_scheme" assumes "P (a, b)" shows "P' (a, b)"
rot_diff_commute: fixes b :: "'a"   and a :: "'a" shows "rot (b - a) = - rot (a - b)"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
Cd_mono: fixes x :: "'a"   and d :: "'b"   and X :: "'a set"   and Y :: "'a set" assumes "x \<in> Cd d X" shows "\<exists>y\<in>Cd d (X \<union> Y). (x, y) \<in> Pd d"
HEndPhase2_valueChosen: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and v :: "InputsOrNi" assumes "HEndPhase2 s s' q"   and "valueChosen s v"   and "v \<in> Inputs" shows "valueChosen s' v"
inter_guards_Spec: fixes r :: "('d \<times> 'd) set"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Spec r \<inter>\<^sub>g c2) = Some c) = (c2 = Spec r \<and> c = Spec r)"
stateinterpol_FVT: fixes z :: "variable"   and t :: "trm"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real"   and \<omega>' :: "variable \<Rightarrow> real"   and S :: "variable set" assumes "z \<notin> FVT t" shows "term_sem I t (stateinterpol \<omega> \<omega>' S) = term_sem I t (stateinterpol \<omega> \<omega>' (insert z S))"
map_of_clearjunk: fixes al :: "('a \<times> 'b) list" shows "map_of (clearjunk al) = map_of al"
small_big_asymmetric: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<omega>[F](g)"   and "g \<in> \<Omega>[F](f)" shows "\<forall>\<^sub>F x in F. f x = (0::'c)"
intgbl: shows "set_integrable M (\<Union> (range A)) f"
w1case1a: shows "sint (SCAST(32 \<rightarrow> 64) w1 + - 0x7FFFFFFF) = sint (SCAST(32 \<rightarrow> 64) w1) + sint (- 0x7FFFFFFF)"
vector_smult_lzero: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(0::'a) *s x = 0"
simps_2: fixes s :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool"   and n :: "nat" shows "seq_filter s P (Suc n) = (LEAST k. P (s k) \<and> seq_filter s P n < k)"
eint_minus_distr: fixes b :: "eint"   and d :: "eint"   and c :: "eint" assumes "b \<noteq> d" shows "c * (b - d) = c * b - c * d"