hcju/isabelleps huggingface dataset:
subdistributivity: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Ea :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Eb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Ec :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Eab :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Ebc :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and \<pi>_Faba :: "'g \<Rightarrow> 'e"   and \<pi>_Fabb :: "'g \<Rightarrow> 'd"   and \<pi>_Fbcb :: "'h \<Rightarrow> 'd"   and \<pi>_Fbcc :: "'h \<Rightarrow> 'f"   and rel_Fab :: "('i \<Rightarrow> 'j \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> bool"   and rel_Fbc :: "('j \<Rightarrow> 'k \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and rel_Fac :: "('i \<Rightarrow> 'k \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and set_Fab :: "'g \<Rightarrow> ('i \<times> 'j) set"   and set_Fbc :: "'h \<Rightarrow> ('j \<times> 'k) set"   and A :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and B :: "'j \<Rightarrow> 'k \<Rightarrow> bool" assumes "confluent_quotient R Ea Eb Ec Eab Ebc \<pi>_Faba \<pi>_Fabb \<pi>_Fbcb \<pi>_Fbcc rel_Fab rel_Fbc rel_Fac set_Fab set_Fbc" shows "rel_Fab A OO Eb OO rel_Fbc B \<le> Ea OO rel_Fac (A OO B) OO Ec"
urel64_def: fixes pp :: "uint64"   and x :: "uint64"   and y :: "int" assumes "p = int_of_uint64 pp"   and "p \<le> 4294967295" shows "urel64 x y = (y = int_of_uint64 x \<and> y < p)"
assms_2: shows "b \<in>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Obj\<rparr>"
is_empty_def: fixes A :: "'a Predicate.pred" shows "Predicate.is_empty A = (A = bot)"
imp_eqvt: fixes pi :: "('a \<times> 'a) list"   and A :: "bool"   and B :: "bool" shows "pi \<bullet> (A \<longrightarrow> B) = (pi \<bullet> A \<longrightarrow> pi \<bullet> B)"
linear: fixes f :: "'e \<Rightarrow> 'f" assumes "bounded_linear f" shows "linear f"
FI_p_nomatch: fixes m :: "(assn \<times> assn) list"   and ps :: "assn"   and qs :: "assn"   and q :: "assn"   and p :: "assn"   and up :: "assn"   and uq :: "assn"   and f :: "assn" assumes "FI m ps (qs * q) (p * up) uq f" shows "FI m (ps * p) (qs * q) up uq f"
zero_ow_def: fixes U :: "'b set"   and zero :: "'b" shows "zero_ow U zero \<equiv> neutral_ow U zero"
set_locs_def: fixes s :: "state"   and s' :: "state" shows "set_locs s s' \<equiv> s'\<lparr>locals := locals s\<rparr>"
invar_children: fixes r :: "nat"   and v :: "'b"   and ts :: "'b tree list" assumes "invar_tree (Node r v ts)" shows "invar (rev ts)"
comm_monoid_axioms: shows "comm_monoid (+) (0::'d)"
predE: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" assumes "Measurable.pred M P" shows "{x \<in> space M. P x} \<in> sets M"
intro: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" assumes "ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd Ch"   and "ContractsWithLAD Xd Xh Pd Ch" shows "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch"
float_distinct_7: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_infinity a \<and> is_zero a)"
Sec2_deriv: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and q :: "L" assumes "(p, G, c, H) \<in> HS" shows "\<exists>A. X \<rhd> c : Sec (Q p H) (EQ G q) (EQ H q) A"
assms: shows "\<forall>v\<in>set vs. \<turnstile> v : r"
chine_in_hom: shows "\<guillemotleft>\<mu>\<nu>_\<pi>.chine : \<mu>\<nu>_\<pi>.apex \<rightarrow>\<^sub>C \<mu>\<nu>_\<pi>.cod.apex\<guillemotright>"
ltype_simps_3: fixes T :: "Arraytype"   and a :: "nat" shows "ltype (arrLenLoc T a) = IntgT"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq x x" shows "reflexive A less_eq"
simps_4: fixes la :: "'a list"   and acc2 :: "'a list list"   and l :: "'a list" shows "merge_list (la # acc2) [l] = merge_list [] (l # la # acc2)"
invertibleI: fixes u :: "'a"   and v :: "'a" assumes "u \<cdot> v = \<one>"   and "v \<cdot> u = \<one>"   and "u \<in> inverse ` H"   and "v \<in> inverse ` H" shows "inv.invertible u"
seqE: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.VxV.seq g f" obtains "C.seq (fst g) (fst f)" and "C.seq (snd g) (snd f)"
set_sel_2: fixes a :: "'g list"   and x :: "'g" assumes "a \<noteq> []"   and "x \<in> set (tl a)" shows "x \<in> set a"
between_scaleR_lift: fixes a :: "real"   and v :: "'a"   and b :: "real"   and c :: "real" shows "between (a *\<^sub>R v, b *\<^sub>R v) (c *\<^sub>R v) = (v = (0::'a) \<or> between (a, b) c)"
eval_change: fixes f :: "nat \<Rightarrow> nat"   and i :: "char list \<Rightarrow> tv"   and p :: "fm" assumes "inj f" shows "eval (change_int f i) p = change_tv f (eval i p)"
rel_F_Grp_weak: fixes l1 :: "'n \<Rightarrow> 'x"   and l2 :: "'o \<Rightarrow> 'y"   and l3 :: "'p \<Rightarrow> 'z"   and co1 :: "'q \<Rightarrow> 'aa"   and co2 :: "'r \<Rightarrow> 'ab"   and co3 :: "'s \<Rightarrow> 'ac"   and contra1 :: "'ad \<Rightarrow> 't"   and contra2 :: "'ae \<Rightarrow> 'u"   and contra3 :: "'af \<Rightarrow> 'v" shows "rel_F (Grp UNIV l1) (Grp UNIV l2) (Grp UNIV l3) (Grp UNIV co1) (Grp UNIV co2) (Grp UNIV co3) (Grp UNIV contra1)\<inverse>\<inverse> (Grp UNIV contra2)\<inverse>\<inverse> (Grp UNIV contra3)\<inverse>\<inverse> = Grp UNIV (map_F l1 l2 l3 co1 co2 co3 contra1 contra2 contra3)"
derivable_well_related: assumes "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"   and "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x" shows "well_related_set (Collect derivable) (\<sqsubseteq>)"
drop_minus_lem: fixes n :: "nat"   and xs :: "'b list"   and n' :: "nat" assumes "n \<le> length xs"   and "0 < n"   and "n' = length xs" shows "drop (n' - n) xs = rev xs ! (n - 1) # drop (Suc (n' - n)) xs"
\<sigma>_img_ground: shows "ground (subst_range \<sigma>)"
shortest_is_augmenting: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list" assumes "NFlow c s t f"   and "Graph.isShortestPath (residualGraph c f) s p t" shows "NPreflow.isAugmentingPath c s t f p"
worklist_aux_def: fixes succs :: "'d \<Rightarrow> 'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" shows "worklist_aux succs f = while_option (\<lambda>(ws, s). ws \<noteq> []) (\<lambda>(ws, s). case ws of x # ws' \<Rightarrow> (succs s x @ ws', f x s))"
action_def: fixes G :: "('a, 'b) monoid_scheme"   and act :: "'c \<Rightarrow> 'a \<Rightarrow> 'c" shows "action G act \<equiv> Group.group G \<and> action_axioms G act"
homologous_rel_eq: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "(homologous_rel p X S a = homologous_rel p X S b) = homologous_rel p X S a b"
right_adjoint_simps_3: fixes f :: "'a" assumes "C.is_left_adjoint f" shows "trg\<^sub>C f\<^sup>*\<^sup>C = src\<^sub>C f"
rep_eq: fixes x :: "uint8" shows "Rep_uint8 (- x) = - Rep_uint8 x"
is_obj_of_type_def: fixes n :: "object"   and T :: "PDDL_STRIPS_Semantics.type" shows "is_obj_of_type n T \<equiv> case objT n of None \<Rightarrow> False | Some oT \<Rightarrow> of_type oT T"
af\<^sub>G_sat_core: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl"   and w :: "'b set list" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P \<phi>" shows "\<G> \<Turnstile>\<^sub>P af\<^sub>G \<phi> w"
le_union1: fixes v1 :: "val"   and v2 :: "val"   and v12 :: "val" assumes "v1 \<squnion> v2 = Some v12" shows "v1 \<sqsubseteq> v12"
ring_simprules_5: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
min_mult_distrib_right: fixes x :: "'b"   and y :: "'b"   and p :: "'b" shows "min x y * p = (if (0::'b) \<le> p then min (x * p) (y * p) else max (x * p) (y * p))"
LESS_SUC_2: fixes k :: "'k"   and k' :: "'k"   and l :: "nat"   and idx :: "'n" assumes "k \<noteq> k'" shows "LESS k l (SUC k' idx) = LESS k l idx"
word_test_bit_def: fixes a :: "'c word" shows "(!!) a = bit (uint a)"
intro: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f"   and "multiplicative_function g" shows "multiplicative_dirichlet_prod f g"
mwb_lens_def: fixes x :: "'c \<Longrightarrow> 'd" shows "mwb_lens x \<equiv> weak_lens x \<and> mwb_lens_axioms x"
thetaAtm_def: fixes atm :: "'d" shows "thetaAtm atm \<equiv> {(Atm atm, Atm atm)}"
A_rw: shows "{0..CARD('a)} = insert CARD('a) (insert 0 ({0..CARD('a)} - {0} - {CARD('a)}))"
QR_lemma_14: shows "(- 1) ^ (m + n) = (- 1) ^ r"
intro: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" assumes "non_neg_mat A" shows "pf_nonneg_mat A"
af_mono: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "af A P"   and "\<forall>x y. x \<in> A \<and> y \<in> A \<and> P x y \<longrightarrow> Q x y" shows "af A Q"
update_accessor_cong_assist_idI: shows "Record.iso_tuple_update_accessor_cong_assist id id"
solution_pell_mul: fixes D :: "nat"   and z1 :: "'a \<times> 'a"   and z2 :: "'a \<times> 'a" assumes "pell D"   and "pell.solution D z1"   and "pell.solution D z2" shows "pell.solution D (pell.pell_mul D z1 z2)"
drop_Cons_numeral: fixes v :: "num"   and x :: "'a"   and xs :: "'a list" shows "drop (numeral v) (x # xs) = drop (numeral v - 1) xs"
proj_lift_poly_fun_2_homogenous: fixes f :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and k :: "'k"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "f 0 0 = 0" shows "proj_poly k (lift_poly_fun_2 f p q) = f (proj_poly k p) (proj_poly k q)"
keys_flatten_subset: fixes p :: "'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c" shows "keys (flatten p) \<subseteq> (\<Union>t\<in>keys p. (+) t ` keys (lookup p t))"
graph_accs_2: fixes V :: "'b set"   and E :: "('b \<times> 'b) set" assumes "finite V"   and "finite E" shows "edges (graph V E) = E \<union> E\<inverse> - Id"
PPs_closed_single: fixes x :: "'b"   and X :: "'b set"   and e :: "nat" assumes "x \<in> X" shows "monomial e x \<in> .[X]"
progress_le: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> \<phi> j \<le> j"
invariant_reaches: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Graph_Invariant E P"   and "E\<^sup>*\<^sup>* a b"   and "P a" shows "P b"
RID\<^sub>LI_2: fixes s :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr"   and r :: "'b" assumes "s = (\<sigma>, \<tau>, e)"   and "r \<in> RID\<^sub>S \<tau>" shows "r \<in> RID\<^sub>L s"
atoms_equiv_bounds_extend: fixes dir :: "('a, 'b) Direction"   and c :: "'b"   and s :: "('a, 'b) state"   and x :: "nat"   and ats :: "'b atom set"   and i :: "'a" assumes "dir = Positive \<or> dir = Negative"   and "\<not> \<unrhd>\<^sub>u\<^sub>b (Direction.lt dir) c (Direction.UB dir s x)"   and "ats \<doteq> \<B> s" shows "(ats \<union> {Direction.LE dir x c}) \<doteq> \<B> (update\<B>\<I> (Direction.UBI_upd dir) i x c s)"
update_simps_15: fixes V :: "(nat, 'b) mapping"   and s :: "('a, 'b) state" shows "\<B>\<^sub>i\<^sub>u (\<V>_update V s) = \<B>\<^sub>i\<^sub>u s"
b: fixes s :: "('a, 'b) Term.term"   and P :: "'a \<Rightarrow> 'a list"   and f :: "'a"   and T :: "('a, 'b) Term.term list" assumes "s \<in> set (term_variants P (Fun f T))" shows "\<exists>S\<in>set (U P T). s = Fun f S \<or> (\<exists>g\<in>set (P f). s = Fun g S)"
num_params_tm_inc_err: fixes t :: "taylor_model"   and X :: "nat"   and i :: "float interval" assumes "num_params (tm_poly t) \<le> X" shows "num_params (tm_poly (tm_inc_err i t)) \<le> X"
iso_is_arr: fixes f :: "'a" assumes "Left.iso f" shows "Left.arr f"
hbt_split_max: fixes t :: "('b \<times> nat) tree" assumes "hbt t"   and "t \<noteq> \<langle>\<rangle>" shows "hbt (fst (split_max t)) \<and> height t \<in> {height (fst (split_max t)), height (fst (split_max t)) + 1}"
coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> c" shows "a \<sqinter> b \<le> c"
leadsp_leads_eq: fixes F :: "'c program" shows "leadsp F = (\<lambda>x xa. (x, xa) \<in> leads F)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_check_right_move (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> ml = Suc m \<and> mr = 0 \<and> (if lm1 = [] then l = Bk # Bk # Oc \<up> ml @ Bk # Bk # ires else l = Bk # Bk # Oc \<up> ml @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> r = <lm2> @ Bk \<up> rn)"
subst_atm_mset_comp_subst: fixes AA :: "'a multiset"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "AA \<cdot>am (\<tau> \<odot> \<sigma>) = AA \<cdot>am \<tau> \<cdot>am \<sigma>"
simps: fixes a1 :: "'b poly"   and a2 :: "'b poly"   and a3 :: "'b poly \<times> 'b poly" shows "eucl_rel_poly a1 a2 a3 = ((\<exists>x. a1 = x \<and> a2 = 0 \<and> a3 = (0, x)) \<or> (\<exists>y x q. a1 = x \<and> a2 = y \<and> a3 = (q, 0) \<and> y \<noteq> 0 \<and> x = q * y) \<or> (\<exists>y r x q. a1 = x \<and> a2 = y \<and> a3 = (q, r) \<and> y \<noteq> 0 \<and> degree r < degree y \<and> x = q * y + r))"
rel_generat_reflI: fixes x :: "('d, 'e, 'f) generat"   and A :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>y. y \<in> generat_pures x \<Longrightarrow> A y y"   and "\<And>out. out \<in> generat_outs x \<Longrightarrow> B out out"   and "\<And>cont. cont \<in> generat_conts x \<Longrightarrow> C cont cont" shows "rel_generat A B C x x"
prime_multiplicity_square_part: fixes p :: "'b"   and n :: "'b" assumes "prime p" shows "multiplicity p (square_part n) = multiplicity p n div 2"
Zp_nat_inc_res: fixes n :: "nat"   and k :: "nat" shows "([n] \<cdot> \<one>) k = int n mod p ^ k"
positive_mult: fixes x :: "rat"   and y :: "rat" assumes "positive x"   and "positive y" shows "positive (x * y)"
simps_8: fixes uv :: "typ"   and v :: "Core.term"   and vc :: "nat" shows "eta_reducible (Abs uv (v $ Bv (Suc vc))) = eta_reducible (v $ Bv (Suc vc))"
arrI: fixes f :: "('obj, 'obj) concrete_category.arr" assumes "f \<noteq> Null"   and "Dom f \<in> Obj"   and "Cod f \<in> Obj"   and "Map f \<in> (if Dom f = Cod f then {Dom f} else {})" shows "arr f"
0: shows "hfun_to_arr (ide_to_hf (local.dom f)) (ide_to_hf (cod f)) (arr_to_hfun f) = mkArr (UP ` hfset (ide_to_hf (local.dom f))) (UP ` hfset (ide_to_hf (cod f))) (\<lambda>x. UP (happ (arr_to_hfun f) (DOWN x)))"
assms_1: shows "(\<gamma> \<or>\<^sup>c \<gamma>') t n"
l12_9: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and C1 :: "'a"   and C2 :: "'a"   and A1 :: "'a"   and B1 :: "'a"   and B2 :: "'a"   and A2 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet C1 C2 A1 B1"   and "Tarski_neutral_dimensionless.Coplanar Bet C1 C2 A1 B2"   and "Tarski_neutral_dimensionless.Coplanar Bet C1 C2 A2 B1"   and "Tarski_neutral_dimensionless.Coplanar Bet C1 C2 A2 B2"   and "Tarski_neutral_dimensionless.Perp Bet Cong A1 A2 C1 C2"   and "Tarski_neutral_dimensionless.Perp Bet Cong B1 B2 C1 C2" shows "Tarski_neutral_dimensionless.Par Bet A1 A2 B1 B2"
thing: shows "\<forall>I J. (\<forall>\<nu> \<nu>' \<mu> V. Iagree I J (SIGP \<alpha>) \<longrightarrow> Vagree \<nu> \<nu>' V \<longrightarrow> FVP \<alpha> \<subseteq> V \<longrightarrow> (\<nu>, \<mu>) \<in> prog_sem I \<alpha> \<longrightarrow> (\<exists>\<mu>'. (\<nu>', \<mu>') \<in> prog_sem J \<alpha> \<and> Vagree \<mu> \<mu>' (MBV \<alpha> \<union> V))) \<and> ode_sem_equiv \<alpha> I"
prems_1: shows "b \<in>\<^sub>\<circ> Y"
2: shows "FvarsT \<chi> - set us = FvarsT r - snd ` set txs"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "inf b (inf a c) = inf a (inf b c)"
a_inv_J: shows "inv a \<in> J \<and> inv a \<noteq> \<one>"
inducts: fixes x :: "'a"   and g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> terminates_on g"   and "\<And>s. \<not> fst g s \<Longrightarrow> P s"   and "\<And>s. \<lbrakk>fst g s; snd (snd g s) \<in> terminates_on g; P (snd (snd g s))\<rbrakk> \<Longrightarrow> P s" shows "P x"
Const\<^sub>0_minus: fixes x :: "'b" shows "Const\<^sub>0 (- x) = - Const\<^sub>0 x"
divisors_fun_def: fixes df :: "'a \<Rightarrow> 'a list" shows "divisors_fun df \<equiv> (\<forall>x. x \<noteq> (0::'a) \<longrightarrow> set (df x) = {d. d dvd x}) \<and> (\<forall>x. distinct (df x))"
filtrationE2: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and s :: "'c"   and t :: "'c" assumes "Filtration.filtration M F"   and "s \<le> t" shows "subalgebra (F t) (F s)"
dot_rowvector_columnvector: fixes A :: "(('a, 'd) vec, 'c) vec"   and v :: "('a, 'd) vec" shows "columnvector (A *v v) = A ** columnvector v"
con_rews_2: fixes cfun :: "'c \<rightarrow> ('b, 'c) Step"   and s :: "'c" shows "(Stream\<cdot>cfun\<cdot>s = \<bottom>) = (s = \<bottom>)"
simps_1: fixes t :: "'c bro" shows "tree (N1 t) = t"
vars_mset_singleton: fixes p :: "('b, 'a) Term.term \<times> ('c, 'a) Term.term" shows "vars_mset {#p#} = fv (fst p) \<union> fv (snd p)"
setbydecision_def: fixes iface :: "iface"   and rs :: "'b common_primitive rule list"   and dec :: "final_decision" shows "setbydecision TYPE('c) iface rs dec = {ip. \<exists>p. approximating_bigstep_fun (common_matcher, in_doubt_allow) (p\<lparr>p_iiface := iface_sel iface, p_src := ip\<rparr>) rs Undecided = Decision dec}"
principal: fixes R' :: "((C, BD) sequent list \<times> (C, BD) sequent) set"   and A :: "(C, BD) form" assumes "R' = Ax \<union> g3up \<union> g3mod1 \<union> g3mod2" shows "\<forall>r\<in>R'. rightPrincipal r (\<box> A) R' \<longrightarrow> ( \<Empt> \<Rightarrow>* \<LM> A  \<RM>) \<in> set (fst r)"
equiv_1: fixes bs :: "('i \<times> 'a atom) set"   and s' :: "('i, 'a) state" shows "bs \<Turnstile>\<^sub>i \<B>\<I> s' = Q bs (<) \<B>\<^sub>i\<^sub>u \<B>\<^sub>i\<^sub>l \<B>\<^sub>u \<B>\<^sub>l \<B>\<^sub>i\<^sub>u_update \<I>\<^sub>u \<I>\<^sub>l Leq Geq s'"
B: shows "tfr\<^sub>s\<^sub>e\<^sub>t B"
prj_in_hom_4: shows "\<guillemotleft>doms.Prj\<^sub>1 : Chn (local.dom \<mu> \<star> local.dom \<nu> \<star> local.dom \<pi>) \<rightarrow>\<^sub>C doms.\<mu>.apex\<guillemotright>"
rGamma_complex_of_real: fixes x :: "real" shows "rGamma (complex_of_real x) = complex_of_real (rGamma x)"
not_zero_a: shows "\<not> is_zero_row_upt_k a k A"
subtopology_topspace: fixes U :: "'a topology" shows "subtopology U (topspace U) = U"
Mutator_def: shows "Mutator \<equiv> \<lbrace>\<acute>Mut_init \<and> \<acute>z\<rbrace>  WHILE True INV \<lbrace>\<acute>Mut_init \<and> \<acute>z\<rbrace>  DO Redirect_Edge;; Color_Target OD"
ex_un_null: shows "\<exists>!n. \<forall>f. Left_a.comp n f = n \<and> Left_a.comp f n = n"
bdd_above_Iic: fixes b :: "'b multiset" shows "subset_mset.bdd_above (subset_mset.atMost b)"
simps: fixes callee :: "'b \<Rightarrow> 'f \<Rightarrow> ('e \<times> 'b, 'c, 'd) gpv"   and gpv :: "('a, 'f, 'e) gpv"   and s :: "'b" shows "inline1 callee gpv s = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1 callee (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))"
scalars_zero_if_independent: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "finite A"   and "local.independent A"   and "(\<Sum>x\<in>A. f x *s x) = (0::'b)" shows "\<forall>x\<in>A. f x = (0::'a)"
zero_le_pprt: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq zero (lattice_ab_group_add.pprt zero sup a)"
type_less_eq_x_Required_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a basic_type"   and \<sigma> :: "'a basic_type" assumes "\<tau> = \<rho>[1]"   and "\<rho> \<le> \<sigma>" shows "\<tau> \<le> \<sigma>[1]"
uminus_add_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "- (A + B) = - B + - A"
refl_on_trancl: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "refl_on A r" shows "refl_on A (r\<^sup>+)"
vsv_appE: fixes a :: "V"   and b :: "V" assumes "f\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> I" obtains "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> f"
minimalize_offending_overapprox_sound: fixes G :: "'a graph"   and ff :: "('a \<times> 'a) list"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph G"   and "is_offending_flows (set ff) G nP"   and "set ff \<subseteq> edges G"   and "distinct ff" shows "is_offending_flows_min_set (set (minimalize_offending_overapprox ff [] G nP)) G nP"
simps_2: fixes shared :: "bool"   and r0 :: "nat"   and r1 :: "nat"   and m :: "(nat \<times> nat) convnet" shows "count_weights shared (Conv (r0, r1) m) = r0 * r1 + count_weights shared m"
pseudo_complement: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y = top) = (- y \<le> x)"
at_swap_simps_1: fixes a :: "'a"   and b :: "'a" assumes "at TYPE('a)" shows "[(a, b)] \<bullet> a = b"
strict_trans: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subset># b"   and "b \<subset># c" shows "a \<subset># c"
simps_5: fixes I :: "interp"   and \<alpha> :: "game"   and \<beta> :: "game" shows "game_sem I (\<alpha> ;; \<beta>) = (\<lambda>X. game_sem I \<alpha> (game_sem I \<beta> X))"
are_all_operator_effects_consistent_set: fixes ops :: "('a, 'b) strips_operator_scheme list" shows "are_all_operator_effects_consistent ops = (\<forall>op\<^sub>1\<in>set ops. \<forall>op\<^sub>2\<in>set ops. set (add_effects_of op\<^sub>1) \<inter> set (delete_effects_of op\<^sub>2) = {} \<and> set (delete_effects_of op\<^sub>1) \<inter> set (add_effects_of op\<^sub>2) = {})"
power2_eq_square: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.monoid_mult one times" shows "power.power one times a 2 = times a a"
unit_disc_fix_preserves_poincare_on_ray: fixes M :: "moebius"   and p :: "complex_homo"   and a :: "complex_homo"   and b :: "complex_homo" assumes "unit_disc_fix M"   and "p \<in> unit_disc"   and "a \<in> unit_disc"   and "b \<in> unit_disc" shows "poincare_on_ray (moebius_pt M p) (moebius_pt M a) (moebius_pt M b) = poincare_on_ray p a b"
seqI: fixes f :: "'a"   and g :: "'a" assumes "B.arr f"   and "B.arr g"   and "B.dom g = B.cod f" shows "B.seq g f"
intro: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a" assumes "vwb_lens x"   and "vwb_lens y"   and "x \<bowtie> y"   and "y \<bowtie> x" shows "myss x y"
eq_neutr_iff: fixes a :: "'a"   and b :: "'a" shows "(a + b = (0::'a)) = (a = (0::'a) \<and> b = (0::'a))"
rep_eq: fixes x :: "uint8"   and xa :: "nat"   and xb :: "bool" shows "Rep_uint8 (set_bit_class.set_bit x xa xb) = set_bit_class.set_bit (Rep_uint8 x) xa xb"