hcju/isabelleps huggingface dataset:
that: fixes A\<^sub>F :: "name list"   and A\<^sub>G :: "name list"   and \<Psi>\<^sub>F :: "'b"   and \<Psi>\<^sub>G :: "'b" assumes "A\<^sub>F\<^sub>G = A\<^sub>F @ A\<^sub>G"   and "\<Psi>\<^sub>F\<^sub>G = \<Psi>\<^sub>F \<otimes> \<Psi>\<^sub>G"   and "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"   and "G = \<langle>A\<^sub>G, \<Psi>\<^sub>G\<rangle>"   and "A\<^sub>F \<sharp>* \<Psi>\<^sub>G"   and "A\<^sub>G \<sharp>* \<Psi>\<^sub>F" shows "thesis"
monom_mult_zero_left: fixes t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.monom_mult (0::'c) t p = 0"
cil_mon_pl: fixes w :: "'d list"   and w1 :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w2 :: "'d list" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "mon_pl (map \<alpha> w) = mon_pl (map \<alpha> w1) \<union> mon_pl (map \<alpha> w2)"
eq_poly_def: fixes p :: "('b monom \<times> 'c) list"   and q :: "('b monom \<times> 'c) list" shows "p =p q \<equiv> \<forall>\<alpha>. eval_poly \<alpha> p = eval_poly \<alpha> q"
4: shows "bounded (range (\<lambda>k. integral UNIV (indicat_real (\<Union> (S ` {..k})))))"
factor_composition: fixes K :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and g :: "'c"   and h :: "'c" assumes "normal_subgroup K G composition unit"   and "g \<in> G"   and "h \<in> G" shows "monoid_congruence.quotient_composition G composition (normal_subgroup.Congruence K G composition unit) (coset_notation.Left_Coset composition g K) (coset_notation.Left_Coset composition h K) = coset_notation.Left_Coset composition (composition g h) K"
tagrel_sub: fixes sub :: "'b run"   and r :: "'b run"   and c\<^sub>1 :: "clock"   and c\<^sub>2 :: "clock"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
bhv_hn_ex: fixes tid :: "'nid"   and n :: "nat"   and t :: "nat \<Rightarrow> cnf" assumes "honest tid"   and "\<exists>n'\<ge>n. \<parallel>tid\<parallel>\<^bsub>t n'\<^esub>"   and "\<exists>n'. latestAct_cond tid t n n'"   and "\<exists>b\<in>pin (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>). length (bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>)) < length b" shows "\<not> mining (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>) \<and> bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>) = MAX (pin (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>)) \<or> mining (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>) \<and> (\<exists>b. bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>) = MAX (pin (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>)) @ [b])"
val_Zp_dist_def: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" shows "val_Zp_dist a b \<equiv> val_Zp (a \<ominus> b)"
aa': shows "snd (TSdet init h (as @ x # bs @ [x]) (Suc (length as + length bs))) = rev (take (Suc (length as + length bs)) ((as @ x # bs) @ [x])) @ h"
sets_D: shows "sets (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x)))) = sets (stream_space (Pi\<^sub>M UNIV (\<lambda>s. count_space UNIV)))"
id_1: fixes dir :: "('i, 'b) Direction"   and x :: "nat"   and c :: "'b"   and s :: "('i, 'b) state" assumes "dir = Positive \<or> dir = Negative" shows "\<nabla> (update\<B>\<I> (UBI_upd dir) i x c s) = \<nabla> s"
\<C>_def: shows "\<C>\<^sub>A \<equiv> \<Union> (range \<C>_vars\<^sub>A)"
f_fvi: shows "Formula.fvi_trm b f \<subseteq> Formula.fvi b \<phi>"
sup2: fixes B :: "'a"   and k :: "'a"   and b :: "'a"   and a :: "'a" assumes "B \<equiv> k \<squnion> b" shows "a \<squnion> B \<equiv> k \<squnion> (a \<squnion> b)"
map_indets_map_indets: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'e"   and p :: "('f \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "map_indets f (map_indets g p) = map_indets (f \<circ> g) p"
set_insert: fixes x :: "'f"   and A :: "'f set" assumes "x \<in> A" obtains B :: "'f set" where "A = insert x B" and "x \<notin> B"
thm_noncont_e_e_2: fixes v :: "i" shows "[Impossible (L\<^sup>-) in v]"
SetDC_def: fixes f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and S :: "'a \<Rightarrow> 'b set" shows "SetDC f S \<equiv> \<lambda>ab P s. INF a\<in>S s. f a ab P s"
resolvents_preserve_equivalence: fixes S :: "'at Literal set set"   and A :: "'at" shows "equivalent S (S \<union> all_resolvents_upon S A)"
assms_2: shows "a \<notin> ptrm_fvs B"
fn_snd_conv: fixes f :: "'b \<Rightarrow> 'c" shows "(\<lambda>x. f (snd x)) = (\<lambda>(uu_, b). f b)"
False: shows "\<not> (p \<noteq> 0 \<and> q \<noteq> 0 \<and> degree p < degree q)"
natural_transformation_axioms: shows "natural_transformation A C (K \<circ> F) (K \<circ> H) Ko\<tau>'_Ko\<tau>.map"
assms: fixes b :: "'a" assumes "b \<in> CBasis" shows "isCont (\<lambda>x. \<langle>b, f x\<rangle> *\<^sub>C b) x"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "C.arr (\<I>\<^sub>C \<otimes>\<^sub>C f)"
map_to_set_upd: fixes m :: "'d \<Rightarrow> 'e option"   and k :: "'d"   and v :: "'e" shows "map_to_set (m(k \<mapsto> v)) = insert (k, v) (map_to_set m - {(k, v') |v'. True})"
sys_block_sizes_in: fixes bl :: "'a set" assumes "bl \<in># \<B>'" shows "int (card bl) \<in> sys2.sys_block_sizes"
that: fixes i :: "nat" assumes "i < dim_row (row_echelon (Macaulay_mat ps))"   and "p = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i"   and "f i < dim_col (row_echelon (Macaulay_mat ps))" shows "thesis"
fs_def: shows "fs TYPE('a) TYPE('b) \<equiv> \<forall>x. finite (supp x)"
a_supdist: fixes x :: "'a"   and y :: "'a" shows "ad (x + y) \<le> ad x"
convergent_def: fixes X :: "nat \<Rightarrow> 'b" shows "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
Diff_iff: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" shows "(c \<in> A\<setminus>B) = (c \<in> A \<and> c \<notin> B)"
congruenceSubstDef_2: shows "congruence \<equiv> {(P, Q) |P Q. P \<leadsto><weakBisim> Q \<and> Q \<leadsto><weakBisim> P}"
assms_1: shows "A \<turnstile> .\<not> p"
wr: shows "well_ordered_set Px (\<sqsubseteq>)"
zdiv_numeral_Bit0: fixes v :: "num"   and w :: "num" shows "numeral (num.Bit0 v) div numeral (num.Bit0 w) = numeral v div numeral w"
ipd_pd_cd0: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n" assumes "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> 0" shows "ipd (\<pi> 0) pd\<rightarrow> \<pi> n"
roofed_circE: fixes x :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and T :: "'k set" assumes "x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> T" obtains "x \<in> RF\<^bsub>\<Gamma>\<^esub> T" and "\<not> essential \<Gamma> (B \<Gamma>) T x"
f_Exec_input_map: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and c :: "'d" shows "f_Exec_Comp trans_fun (map f xs) c = f_Exec_Comp (trans_fun \<circ> f) xs c"
fps_compose_linear: fixes f :: "'a fps"   and c :: "'a" shows "f oo fps_const c * fps_X = Abs_fps (\<lambda>n. c ^ n * f $ n)"
assms_2: shows "lookup p ` keys p \<subseteq> P[Y]"
left_def: shows "left \<equiv> (rep_msg ---> abs_msg) freeleft"
r_result1_some_phi: fixes t :: "nat"   and i :: "nat"   and x :: "nat"   and v :: "nat" assumes "eval r_result1 [t, i, x] \<down>= Suc v" shows "\<phi> i x \<down>= v"
assms_5: shows "V'' \<in> N (vs @ xsR) M1 V"
g'': shows "g'' = snd (snd (splitFace g' v a f' [countVertices g'..<countVertices g' + n]))"
project_Stable_D: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set" assumes "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> Stable A" shows "extend h F \<squnion> G \<in> Stable (extend_set h A)"
prod_filter_INF1: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b filter"   and B :: "'c filter" assumes "I \<noteq> {}" shows "Inf (A ` I) \<times>\<^sub>F B = (INF i\<in>I. A i \<times>\<^sub>F B)"
monic_factorization_axioms: shows "monic_factorization as p"
differentiable_on_id: fixes S :: "'a set" shows "id differentiable_on S"
naturally_isomorphic_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd" shows "naturally_isomorphic A B F G = (\<exists>\<tau>. natural_isomorphism A B F G \<tau>)"
module_pair_on_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" shows "module_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 \<equiv> module_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 \<and> module_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
quicksort_conv_sort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "ord.quicksort less xs = linorder.sort_key less_eq (\<lambda>x. x) xs"
ce_rel_lm_13: fixes x :: "nat"   and y :: "nat"   and A :: "nat set" assumes "(x, y) \<in> ce_set_to_rel A" shows "c_pair x y \<in> A"
borel_measurable_integrable: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "integrable M f" shows "f \<in> borel_measurable M"
assms: shows "prime P \<and> odd P \<and> 0 < Q \<and> is_qfN (P * Q) 3 \<and> \<not> is_qfN P 3"
calc3: shows "\<i> ^ Suc (Suc n) / fact (Suc n) = \<i> ^ Suc n / fact n * (\<i> / of_nat (Suc n))"
Sucl: shows "Suc (length (paid_A ! n) - Suc m) = length (paid_A ! n) - m"
o_inv_distrib: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c" assumes "bij f"   and "bij g" shows "inv (f \<circ> g) = inv g \<circ> inv f"
nat_filtration_vimage_finite: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "finite (f ` space M)"
char_of_def: fixes n :: "'a" shows "char_of n = Char (odd n) (bit n 1) (bit n 2) (bit n 3) (bit n 4) (bit n 5) (bit n 6) (bit n 7)"
pre_subdivFace'_def: fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and ram1 :: "nat"   and n :: "nat"   and vOptionList :: "nat option list" shows "pre_subdivFace' g f v' ram1 n vOptionList \<equiv> \<not> final f \<and> v' \<in> \<V> f \<and> ram1 \<in> \<V> f \<and> v' \<notin> set (removeNones vOptionList) \<and> distinct (vertices f) \<and> (filter (\<lambda>v. v \<in> set (removeNones vOptionList)) (verticesFrom f v') = removeNones vOptionList \<and> before (verticesFrom f v') ram1 (hd (removeNones vOptionList)) \<and> last vOptionList = Some (last (verticesFrom f v')) \<and> vOptionList \<noteq> [] \<and> (v' = ram1 \<and> 0 < n \<or> v' = ram1 \<and> hd vOptionList \<noteq> Some (last (verticesFrom f v')) \<or> v' \<noteq> ram1) \<and> \<not> invalidVertexList g f vOptionList \<and> (n = 0 \<and> hd vOptionList \<noteq> None \<longrightarrow> \<not> is_duplicateEdge g f ram1 (the (hd vOptionList))) \<or> vOptionList = [] \<and> v' \<noteq> ram1)"
le_less_linear: fixes y :: "'d"   and x :: "'d" shows "y \<le> x \<or> x < y"
zero_less: fixes x :: "nat list" assumes "nonzero x" shows "zeroes (length x) <\<^sub>v x"
invertible_mult: fixes A :: "(('b, 'c) vec, 'd) vec"   and B :: "(('b, 'e) vec, 'c) vec" assumes "invertible A"   and "invertible B" shows "invertible (A ** B)"
DSaxiom_def: shows "DSaxiom = ([[EvolveODE (OSing vid1 (f0 fid1)) (p1 vid2 vid1)]]p1 vid3 vid1 \<leftrightarrow> Forall vid2 (Geq (Var vid2) (Const 0) \<rightarrow> Forall vid3 ((Geq (Var vid3) (Const 0) && Geq (Var vid2) (Var vid3)) \<rightarrow> $\<phi> vid2 (singleton (Plus (Var vid1) (Times (f0 fid1) (Var vid3))))) \<rightarrow> [[vid1 := Plus (Var vid1) (Times (f0 fid1) (Var vid2))]]p1 vid3 vid1))"
ringE_2: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "Group.monoid R"
ENR_ANR: fixes S :: "'a set" shows "ENR S = (ANR S \<and> locally compact S)"
part_def: fixes J :: "'c set"   and P :: "'c set set" shows "part J P \<equiv> \<Union> P = J \<and> (\<forall>J1 J2. J1 \<in> P \<and> J2 \<in> P \<and> J1 \<noteq> J2 \<longrightarrow> J1 \<inter> J2 = {})"
of_int_ceiling_diff_one_le: fixes r :: "'a" shows "of_int \<lceil>r\<rceil> - (1::'a) \<le> r"
Rep_reg_lan_inject: fixes x :: "'a reg_lan"   and y :: "'a reg_lan" shows "(Rep_reg_lan x = Rep_reg_lan y) = (x = y)"
norm_unsigned_Cons1: fixes bs :: "bit list" shows "norm_unsigned (\<one> # bs) = \<one> # bs"
prog_sub_from_counter: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and prog :: "top0 list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_prog ck env s prog res" shows "\<forall>extra cnt cnt' s' r. clock s = extra + cnt \<and> clock s' = extra + cnt' \<and> res = (s', r) \<and> ck = True \<longrightarrow> evaluate_prog ck env (update_clock (\<lambda>_. cnt) s) prog (update_clock (\<lambda>_. cnt') s', r)"
carrier_subgroup_generated_by_singleton: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.group G"   and "x \<in> carrier G" shows "carrier (subgroup_generated G {x}) = range (([^]\<^bsub>G\<^esub>) x)"
convex_hull_caratheodory: fixes p :: "'c set" shows "convex hull p = {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('c) + 1 \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
subst_fm_Disjoint: fixes t :: "tm"   and u :: "tm"   and i :: "name"   and x :: "tm" shows "(Disjoint t u)(i::=x) = Disjoint (subst i x t) (subst i x u)"
bounded_iff: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" shows "(a \<subseteq># b \<inter># c) = (a \<subseteq># b \<and> a \<subseteq># c)"
times_uexpr_def: fixes u :: "('e, 'f) uexpr"   and v :: "('e, 'f) uexpr" shows "u * v = bop (*) u v"
Ri_least: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'c"   and y :: "'b"   and z :: "'b" assumes "mono f"   and "x \<le> f y" shows "Ri x y z \<le> f z"
induct: fixes g :: "('f, 'e) generator"   and P :: "'e \<Rightarrow> bool"   and a0 :: "'e" assumes "\<And>s. \<lbrakk>\<And>x2. generator g s = Skip x2 \<Longrightarrow> P x2; \<And>x31 x32. generator g s = Yield x31 x32 \<Longrightarrow> P x32\<rbrakk> \<Longrightarrow> P s" shows "P a0"
assert_pred_mono: fixes p :: "'e"   and q :: "'e" assumes "p \<sqsubseteq> q" shows "{. p .} \<sqsubseteq> {. q .}"
arr_dom: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.arr (hom'.dom f)"
restrict_spmf_empty: fixes p :: "'c spmf" shows "p \<upharpoonleft> {} = return_pmf None"
theI2: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
substT_tms: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "'b"   and t2 :: "'b"   and t :: "'b"   and x :: "'a" assumes "Syntax_Arith_aux var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "t \<in> trm"   and "x \<in> var" shows "substT (tms t1 t2) t x = tms (substT t1 t x) (substT t2 t x)"
IH: fixes S :: "'b literal set set"   and S' :: "'b literal set set" assumes "X S = Some S'" shows "\<forall>C\<in>S'. S \<turnstile> C"
bisim1CallParams': fixes P :: "'a prog"   and h :: "'heap"   and ps :: "'addr expr1 list"   and n :: "nat"   and ps' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and obj :: "'addr expr1"   and M :: "String.literal"   and v :: "'addr val" assumes "P,ps,n,h \<turnstile>' (ps', xs) [\<leftrightarrow>] (stk, loc, pc, xcp)"   and "ps \<noteq> []"   and "\<And>xs. P,obj,n,h \<turnstile>' (obj, xs) \<leftrightarrow> ([], xs, 0, None)" shows "P,obj\<bullet>M(ps),n,h \<turnstile>' (Val v\<bullet>M(ps'), xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 obj) + pc, xcp)"
dimacs_to_sat_plan: fixes M :: "nat \<Rightarrow> bool"   and h :: "nat"   and n_ops :: "nat"   and sat_plan_formula :: "sat_plan_variable formula" shows "M \<circ> cnf_to_dimacs.var_to_dimacs h n_ops \<Turnstile> sat_plan_formula = M \<Turnstile> map_formula (cnf_to_dimacs.var_to_dimacs h n_ops) sat_plan_formula"
induct: fixes c :: "('loc, 't) configuration"   and x1 :: "'loc"   and x2 :: "'loc"   and x3 :: "'t zmultiset"   and x4 :: "('loc \<times> 'sum \<times> 'loc) list"   and P :: "'loc \<Rightarrow> 'loc \<Rightarrow> 't zmultiset \<Rightarrow> ('loc \<times> 'sum \<times> 'loc) list \<Rightarrow> bool" assumes "loc_imps_fw c x1 x2 x3 x4"   and "\<And>loc. P loc loc (c_imp c loc) []"   and "\<And>loc1 loc2 M xs s loc3. \<lbrakk>loc_imps_fw c loc1 loc2 M xs; P loc1 loc2 M xs; s \<in>\<^sub>A summary loc2 loc3; distinct (xs @ [(loc2, s, loc3)])\<rbrakk> \<Longrightarrow> P loc1 loc3 ({#results_in t s. t \<in>#\<^sub>z M#} + c_imp c loc3) (xs @ [(loc2, s, loc3)])" shows "P x1 x2 x3 x4"
conc_assoc: fixes A :: "'b list set"   and B :: "'b list set"   and C :: "'b list set" shows "(A @@ B) @@ C = A @@ B @@ C"
rev_swap: fixes xs :: "'b list"   and ys :: "'b list" shows "(rev xs = ys) = (xs = rev ys)"
gallery_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "folding.gallery (x # xs)" shows "folding.gallery xs"
that: shows "map_\<I> g h \<I> \<turnstile>g gpv \<surd>"
eval: shows "prg Env\<turnstile>s0 \<midarrow>es\<doteq>\<succ>v\<rightarrow> s1"
homotopic_loops_def: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" shows "homotopic_loops s p q \<equiv> homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
hermitian_inner_prod_zero: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "hermitian A"   and "\<forall>v\<in>carrier_vec n. inner_prod v (A *\<^sub>v v) = 0" shows "A = 0\<^sub>m n n"
is_iso_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme" shows "G \<cong> H = (Group.iso G H \<noteq> {})"
simps: fixes a1 :: "'a trm"   and a2 :: "'a trm" shows "a1 \<rightarrow>\<beta>\<^sup>* a2 = ((\<exists>M. a1 = M \<and> a2 = M) \<or> (\<exists>M M' M''. a1 = M \<and> a2 = M'' \<and> M \<rightarrow>\<beta>\<^sup>* M' \<and> M' \<rightarrow>\<beta> M''))"
iso_assoc: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "C.iso \<a>\<^sub>C[f, g, h]"
fsegerberg: fixes y :: "'a"   and x :: "'a" shows "d y +  |x\<^sup>\<star>\<rangle> A x y =  |x\<^sup>\<star>\<rangle> y"
assms: shows "(\<lambda>n. f n / real n) \<longlonglongrightarrow> l"
semigroup_mult_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "semigroup_mult_hom hom \<equiv> times_hom hom"
index_zero_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat" assumes "i < nr"   and "j < nc" shows "0\<^sub>m nr nc $$ (i, j) = (0::'b)"
bot_empty_eq: shows "bot = (\<lambda>x. x \<in> {})"
inf1D2: fixes A :: "'g \<Rightarrow> bool"   and B :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "(A \<sqinter> B) x" shows "B x"
word_succ_rbl: fixes w :: "'a word"   and bl :: "bool list" assumes "to_bl w = bl" shows "to_bl (word_succ w) = rev (rbl_succ (rev bl))"
uminus1: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "ccw' 0 (- a) b = ccw' 0 b a"
length_neq_imp_not_raw_matches: fixes Ts :: "typ list"   and Us :: "typ list"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "length Ts \<noteq> length Us" shows "raw_matches Ts Us subs = None"
plus_in_interval: fixes x :: "'a"   and X :: "'a interval"   and y :: "'a"   and Y :: "'a interval" assumes "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "x + y \<in>\<^sub>i X + Y"
isomorphic_homology_groups_Euclidean_complements: fixes n :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and T :: "(nat \<Rightarrow> real) set"   and p :: "int" assumes "closedin (Euclidean_space n) S"   and "closedin (Euclidean_space n) T"   and "subtopology (Euclidean_space n) S homeomorphic_space subtopology (Euclidean_space n) T" shows "homology_group p (subtopology (Euclidean_space n) (topspace (Euclidean_space n) - S)) \<cong> homology_group p (subtopology (Euclidean_space n) (topspace (Euclidean_space n) - T))"
trg_cod: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "trg\<^sub>B (B.cod \<mu>) = trg\<^sub>B \<mu>"
mult_comp_trading_strat: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and qty :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "trading_strategy pf1"   and "borel_predict_stoch_proc F qty" shows "trading_strategy (qty_mult_comp pf1 qty)"
erf_real_altdef_nonneg: fixes x :: "real" assumes "0 \<le> x" shows "erf x = 2 / sqrt pi * integral {0..x} (\<lambda>t. exp (- t\<^sup>2))"
varnum_def: fixes X :: "'a set"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "varnum X t = (if keys t - X = {} then 0 else Suc (Max (elem_index ` (keys t - X))))"
morE2: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "('f, 'g, 'h) F1 \<Rightarrow> 'g"   and s2' :: "('f, 'g, 'h) F2 \<Rightarrow> 'h"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and z :: "('f, 'd, 'e) F2" assumes "mor B1 B2 s1 s2 B1' B2' s1' s2' f g"   and "z \<in> F2in UNIV B1 B2" shows "g (s2 z) = s2' (F2map id f g z)"
simps: fixes a1 :: "'a measure"   and a2 :: "'a measure" shows "(a1 \<le> a2) = ((\<exists>M N. a1 = M \<and> a2 = N \<and> space M \<subset> space N) \<or> (\<exists>M N. a1 = M \<and> a2 = N \<and> space M = space N \<and> sets M \<subset> sets N) \<or> (\<exists>M N. a1 = M \<and> a2 = N \<and> space M = space N \<and> sets M = sets N \<and> emeasure M \<le> emeasure N))"
sse_initial_def: fixes \<Sigma>i :: "'b set" shows "sse_initial \<Sigma>i \<equiv> (\<Sigma>i, \<Sigma>i)"
simps_3: fixes C :: "ty"   and D :: "ty"   and X :: "ctrm set" shows "NEGn (C AND D) X = AXIOMSn (C AND D) \<union> BINDINGn (C AND D) X \<union> ANDLEFT1 (C AND D) (\<parallel>(C)\<parallel>) \<union> ANDLEFT2 (C AND D) (\<parallel>(D)\<parallel>)"
finite_union_of_Union: fixes \<U> :: "'b set set"   and P :: "'b set \<Rightarrow> bool" assumes "finite \<U>"   and "\<And>S. S \<in> \<U> \<Longrightarrow> (finite union_of P) S" shows "(finite union_of P) (\<Union> \<U>)"
nn_integral_monotone_convergence_SUP_countable: fixes Y :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"   and B :: "'b set" assumes "Y \<noteq> {}"   and "Complete_Partial_Order.chain (\<le>) (f ` Y)"   and "countable B" shows "(\<Sum>\<^sup>+ x\<in>B. \<Squnion>i\<in>Y. f i x) = (\<Squnion>i\<in>Y. integral\<^sup>N (count_space B) (f i))"
PiF_def: fixes I :: "'b set set"   and M :: "'b \<Rightarrow> 'c measure" shows "Pi\<^sub>F I M \<equiv> sigma (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j)) {Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"