hcju/isabelleps huggingface dataset:
init_reds_sync_unit: fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v' :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and e' :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v',s\<^sub>1,b\<^sub>1\<rangle>"   and "init_exp_of e\<^sub>0 = \<lfloor>unit\<rfloor>"   and "INIT_class e\<^sub>0 = \<lfloor>C\<rfloor>"   and "\<not> sub_RI e'" shows "P \<turnstile> \<langle>init_switch e\<^sub>0 e',s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>1,icheck P (the (INIT_class e\<^sub>0)) e'\<rangle>"
Cod_Trg: fixes t :: "'c local.term" assumes "Arr t" shows "Cod (Trg t) = Trg t"
echelon_form_condition2_explicit: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'c" assumes "echelon_form A"   and "i < j"   and "\<not> is_zero_row i A"   and "\<not> is_zero_row j A" shows "(LEAST n. A $ i $ n \<noteq> (0::'a)) < (LEAST n. A $ j $ n \<noteq> (0::'a))"
R_step: fixes A :: "'a"   and B :: "'b" assumes "A \<le> lfp f"   and "B \<le> lfp g"   and "R A B" shows "R (f A) (g B)"
path_join_def: fixes G :: "graph"   and p :: "nat list"   and q :: "nat list" shows "path_join G p q = p @ tl q"
init_exp_of_neq': fixes e :: "'a exp"   and e' :: "'a exp" assumes "init_exp_of e = \<lfloor>e'\<rfloor>" shows "e \<noteq> e'"
coprime_h_c_poly: fixes c1 :: "'b mod_ring"   and c2 :: "'b mod_ring"   and h :: "'b mod_ring poly" assumes "c1 \<noteq> c2" shows "comm_monoid_mult_class.coprime (h - [:c1:]) (h - [:c2:])"
NDVD_mono: fixes x :: "real"   and bs :: "real list"   and i :: "int" assumes "0 \<le> x"   and "x < 1" shows "\<forall>a n s. Inum (x # bs) a = Inum (x # bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x # bs) (NDVD i n s) = Ifm (x # bs) (NDvd i a)"
CatchSteps: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cfg\<^sub>1 :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate"   and cfg\<^sub>2 :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate"   and c\<^sub>1' :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "\<Gamma> \<turnstile> cfg\<^sub>1 \<rightarrow>\<^sup>* cfg\<^sub>2"   and "cfg\<^sub>1 = (c\<^sub>1, s)"   and "cfg\<^sub>2 = (c\<^sub>1', s')" shows "\<Gamma> \<turnstile> (Catch c\<^sub>1 c\<^sub>2, s) \<rightarrow>\<^sup>* (Catch c\<^sub>1' c\<^sub>2, s')"
foldri_simp_2: fixes l :: "'e list"   and x :: "'e"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldri (l @ [x]) c f \<sigma> = (if c \<sigma> then foldri l c f (f x \<sigma>) else \<sigma>)"
sqrt_def: shows "sqrt = ma_sqrt (ma_of_rat (rat_of_nat fact))"
assms: shows "\<forall>i. 0 < i \<and> i < Suc n \<longrightarrow> s ! (x + i) = []"
add_mset_subseteq_single_iff: fixes a :: "'a"   and M :: "'a multiset"   and b :: "'a" shows "(add_mset a M \<subseteq># {#b#}) = (M = {#} \<and> a = b)"
DomainI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "r a b" shows "Domainp r a"
Ord_Vset_in_Vset_succI: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "Vset \<alpha> \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
advantage_multi_def: fixes init :: "('a \<times> 'b \<times> 'c) spmf"   and "oracle" :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf"   and eval :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'f \<Rightarrow> bool spmf"   and \<A> :: "'b \<Rightarrow> (unit, 'd + 'f, 'e + unit) gpv" shows "guessing_many_one.advantage_multi init oracle eval \<A> = spmf (guessing_many_one.game_multi init oracle eval \<A>) True"
Cod_comp: fixes Obj :: "'b set"   and Hom :: "'b \<Rightarrow> 'b \<Rightarrow> 'c set"   and Id :: "'b \<Rightarrow> 'c"   and Comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "('b, 'c) Maps.arr"   and g :: "('b, 'c) Maps.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) g"   and "concrete_category.Dom g = Maps.Cod f" shows "Maps.Cod (concrete_category.COMP Obj Hom Comp g f) = Maps.Cod g"
residual_block_num: shows "int (size \<B>\<^sup>R) = \<b> - 1"
P1_S1_def: fixes \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'd) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'd \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'd) spmf) \<times> ('d \<Rightarrow> 'f spmf)"   and M :: "'grp \<times> 'grp"   and z :: "'e" shows "P1_S1 \<A> M z = (let (\<A>1, \<A>2, \<A>3) = \<A> in sample_uniform (order \<G>) \<bind> (\<lambda>r. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>0. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>1. let h0 = \<^bold>g [^] \<alpha>0; h1 = \<^bold>g [^] \<alpha>1; a = \<^bold>g [^] r; b0 = h0 [^] r; b1 = h1 [^] r \<otimes> \<^bold>g in \<A>1 M h0 h1 a b0 b1 z \<bind> (\<lambda>((in1, in2, in3), s). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in assert_spmf ((in1, in2, in3) = (h, a, b)) \<bind> (\<lambda>_. \<A>2 h0 h1 a b0 b1 M s \<bind> (\<lambda>(((w0, z0), w1, z1), s'). let x0 = z0 \<otimes> inv w0 [^] \<alpha>0; x1 = z1 \<otimes> inv w1 [^] \<alpha>1 in return_spmf ((x0, x1), s'))))))))"
cinner_ket_left: fixes i :: "'c"   and \<psi> :: "'c ell2" shows "ket i \<bullet>\<^sub>C \<psi> = Rep_ell2 \<psi> i"
fmsubset_alt_def: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "(m \<subseteq>\<^sub>f n) = fmpred (\<lambda>k v. fmlookup n k = Some v) m"
dg_is_arrE: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" obtains "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" and "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" and "\<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a" and "\<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
interval_inner_leI_2: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and i :: "'a" assumes "x \<in> {a..b}"   and "(0::'a) \<le> i" shows "x \<bullet> i \<le> b \<bullet> i"
ord_inj_Srestrict_isom: fixes D :: "'c Order"   and E :: "'d Order"   and f :: "'c \<Rightarrow> 'd"   and T :: "'c set" assumes "Order D"   and "Order E"   and "ord_inj D E f"   and "T \<subseteq> carrier D" shows "ord_isom (SIod D T) (SIod E (f ` T)) (restrict f T)"
assms_7: shows "bounded (m - 1) r\<^sub>0 j"
assms_1: shows "c \<le> a \<or> c \<le> b"
sums: shows "(\<lambda>n. 2 * ((x - 1) / (x + 1)) ^ (2 * n + 1) / real (2 * n + 1)) sums ln x"
pmdl_closed_monom_mult: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set"   and c :: "'c"   and t :: "'b" assumes "p \<in> MPoly_Type_Class.punit.pmdl B" shows "MPoly_Type_Class.punit.monom_mult c t p \<in> MPoly_Type_Class.punit.pmdl B"
assms_1: shows "\<forall>x\<in>space M. 0 < f x"
net_ips_is_dom_netmap: fixes s :: "'d net_state" shows "net_ips s = dom (netmap s)"
plus_in_intervalI: fixes x :: "'b"   and X :: "'b interval"   and y :: "'b"   and Y :: "'b interval" assumes "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "x + y \<in>\<^sub>i X + Y"
subgraph_union_induced: fixes H\<^sub>1 :: "nat set \<times> nat set set"   and S :: "nat set"   and H\<^sub>2 :: "nat set \<times> nat set set"   and T :: "nat set"   and G :: "nat set \<times> nat set set" assumes "uverts H\<^sub>1 \<subseteq> S"   and "uverts H\<^sub>2 \<subseteq> T"   and "uwellformed H\<^sub>1"   and "uwellformed H\<^sub>2" shows "(subgraph H\<^sub>1 (induced_subgraph S G) \<and> subgraph H\<^sub>2 (induced_subgraph T G)) = subgraph (uverts H\<^sub>1 \<union> uverts H\<^sub>2, uedges H\<^sub>1 \<union> uedges H\<^sub>2) (induced_subgraph (S \<union> T) G)"
subsumes_def: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'b Literal set"   and D :: "'b Literal set" assumes "propositional_atoms atom_ordering" shows "subsumes C D \<equiv> C \<subseteq> D"
summable: fixes z :: "complex" assumes "cmod z < K" shows "summable (\<lambda>n. (a gchoose n) * z ^ n)"
addfunsetI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and M :: "'b set" assumes "supp f \<subseteq> A"   and "range f \<subseteq> M"   and "\<forall>x\<in>A. \<forall>y\<in>A. f (x + y) = f x + f y" shows "f \<in> addfunset A M"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b llist" shows "<> \<noteq> x21 % x22"
cases: fixes x :: "exetheory" obtains cto :: "(String.literal \<times> typ) list" and tao :: "(String.literal \<times> nat) list" and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" and ax :: "Core.term list" where "x = ExeTheory (ExeSignature cto tao sa) ax"
wf_inst_effect: fixes P :: "ast_problem"   and Q :: "variable \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and f :: "variable \<Rightarrow> object"   and \<phi> :: "PDDL_STRIPS_Semantics.term ast_effect" assumes "\<And>x T. ast_domain.is_of_type (domain P) Q x T \<Longrightarrow> ast_domain.is_of_type (domain P) (ast_problem.objT P) (f x) T"   and "ast_domain.wf_effect (domain P) (ty_term Q (ast_domain.constT (domain P))) \<phi>" shows "ast_domain.wf_effect (domain P) (ast_problem.objT P) ((map_ast_effect \<circ> subst_term) f \<phi>)"
D: shows "offs_set_prev ns (x # xs) index key mi ma i = {}"
inj_on_image_mem_iff: fixes f :: "'e \<Rightarrow> 'f"   and B :: "'e set"   and a :: "'e"   and A :: "'e set" assumes "inj_on f B"   and "a \<in> B"   and "A \<subseteq> B" shows "(f a \<in> f ` A) = (a \<in> A)"
Inf_conj_distrib: fixes D :: "'a::refinement_lattice set"   and c :: "'a::refinement_lattice" assumes "(D::'a::refinement_lattice set) \<noteq> {}" shows "\<Sqinter> (D::'a::refinement_lattice set) \<iinter> (c::'a::refinement_lattice) = (\<Sqinter>d::'a::refinement_lattice\<in>D. d \<iinter> c)"
jkbpC_jkbpCn_jAction_eq: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> jkbpC\<^bsub>n\<^esub>" shows "jAction MC t = jAction MC\<^bsub>n\<^esub> t"
trms\<^sub>e\<^sub>s\<^sub>t_ik_assignment_rhsI: fixes t :: "('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list" assumes "t \<in> ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A" shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A)"
sum: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd"   and S :: "'f set" assumes "module_hom s1 s2 f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
gauss_lemma: fixes p :: "'b fract poly"   and q :: "'b fract poly" shows "content_ff (p * q) =dff content_ff p * content_ff q"
conga__lea: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F"
v: shows "vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t)"
sup_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "x + x \<cdot> y = x"
five_segment_with_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OFSC Bet Cong A B C D A' B' C' D'"   and "A \<noteq> B" shows "Cong C D C' D'"
substAbs_def: fixes xs :: "'varSort"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and x :: "'var" shows "substAbs xs X x \<equiv> psubstAbs (idEnv [x \<leftarrow> X]_xs)"
last_ps_sgn_const: fixes x :: "real"   and y :: "real" shows "sgn (poly (last ps) x) = sgn (poly (last ps) y)"
m_comm: fixes x :: "complex vec"   and y :: "complex vec" assumes "x \<in> carrier_vec d"   and "y \<in> carrier_vec d" shows "x + y = y + x"
Un_commute: fixes A :: "'f set"   and B :: "'f set" shows "A \<union> B = B \<union> A"
elem_in_repeat_in_original: fixes a :: "'b"   and n :: "nat"   and A :: "'b multiset" assumes "a \<in># repeat_mset n A" shows "a \<in># A"
divide_cancel_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a c = divide b c) = (c = zero \<or> a = b)"
segment_reverse: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C" shows "\<exists>B'. Bet A B' C \<and> Cong C B' A B"
dim_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplexIsomorphism X Y f"   and "SimplicialComplex.maxsimp X C" shows "card (f ` C) = card C"
hyps: fixes I :: "nat set"   and J :: "nat set" assumes "card I = k"   and "card J = k"   and "I \<subseteq> {0..<n}"   and "J \<subseteq> {0..<m}"   and "0 < k" shows "Determinant.det (submatrix A I J) = (0::'a) \<or> (\<exists>S\<subseteq>{0..<min n m}. card S = k \<and> S = I \<and> (Determinant.det (submatrix A I J) = (\<Prod>i\<in>S. A $$ (i, i)) \<or> Determinant.det (submatrix A I J) = - (\<Prod>i\<in>S. A $$ (i, i))))"
index_col_mat_of_cols_list: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and l :: "complex list list" assumes "i < n"   and "j < length l" shows "Matrix.col (Tensor.mat_of_cols_list n l) j $ i = l ! j ! i"
mem_equal_def: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" shows "mem_equal s1 s2 pa \<equiv> mem s1 8 (pa AND 68719476732) = mem s2 8 (pa AND 68719476732) \<and> mem s1 8 ((pa AND 68719476732) + 1) = mem s2 8 ((pa AND 68719476732) + 1) \<and> mem s1 8 ((pa AND 68719476732) + 2) = mem s2 8 ((pa AND 68719476732) + 2) \<and> mem s1 8 ((pa AND 68719476732) + 3) = mem s2 8 ((pa AND 68719476732) + 3) \<and> mem s1 9 (pa AND 68719476732) = mem s2 9 (pa AND 68719476732) \<and> mem s1 9 ((pa AND 68719476732) + 1) = mem s2 9 ((pa AND 68719476732) + 1) \<and> mem s1 9 ((pa AND 68719476732) + 2) = mem s2 9 ((pa AND 68719476732) + 2) \<and> mem s1 9 ((pa AND 68719476732) + 3) = mem s2 9 ((pa AND 68719476732) + 3) \<and> mem s1 10 (pa AND 68719476732) = mem s2 10 (pa AND 68719476732) \<and> mem s1 10 ((pa AND 68719476732) + 1) = mem s2 10 ((pa AND 68719476732) + 1) \<and> mem s1 10 ((pa AND 68719476732) + 2) = mem s2 10 ((pa AND 68719476732) + 2) \<and> mem s1 10 ((pa AND 68719476732) + 3) = mem s2 10 ((pa AND 68719476732) + 3) \<and> mem s1 11 (pa AND 68719476732) = mem s2 11 (pa AND 68719476732) \<and> mem s1 11 ((pa AND 68719476732) + 1) = mem s2 11 ((pa AND 68719476732) + 1) \<and> mem s1 11 ((pa AND 68719476732) + 2) = mem s2 11 ((pa AND 68719476732) + 2) \<and> mem s1 11 ((pa AND 68719476732) + 3) = mem s2 11 ((pa AND 68719476732) + 3)"
split_scaleR_neg_le: fixes a :: "real"   and x :: "'b" assumes "0 \<le> a \<and> x \<le> (0::'b) \<or> a \<le> 0 \<and> (0::'b) \<le> x" shows "a *\<^sub>R x \<le> (0::'b)"
hyps_1: shows "P \<Longrightarrow>\<^sub>\<tau>\<^sub>l P'"
transitionE_1: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "\<exists>P'' P'''. P \<Longrightarrow>\<^sub>\<tau> P'' \<and> P'' \<longmapsto> \<alpha> \<prec> P''' \<and> P''' \<Longrightarrow>\<^sub>\<tau> P'"
ide_char: fixes a :: "'b" shows "hhom\<^sub>D.ide a = (hhom\<^sub>D.arr a \<and> D.ide a)"
open_Vf: fixes w :: "'a" assumes "w \<in> A" shows "open (Vf w)"
list_of_oalist_of_list: fixes xs :: "('a \<times> 'b) list \<times> 'o" shows "list_of_oalist (oalist_of_list xs) = sort_oalist_raw xs"
check_subseteq_def: fixes xs :: "'a list"   and ys :: "'a list" shows "check_subseteq xs ys = (try forallM (\<lambda>x. if x \<in> set ys then Inr () else Inl x) xs catch (\<lambda>x. Inl (snd x)))"
atomize_plus: fixes p :: "nat \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and q :: "nat \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "atomize_poly (p + q) = atomize_poly p + atomize_poly q"
distinct_3: shows "ConcatOp \<noteq> EqualsIgnoreCaseOp"
round_num_denom_d\<mu>_d: fixes j :: "nat"   and i :: "nat" assumes "j \<le> i"   and "i < length fs" shows "round_num_denom (fs.d\<mu> i j) (fs.d fs (Suc j)) = round (fs.gs.\<mu> i j)"
subst_fv_unfold: fixes t :: "('d, 'c) Term.term"   and s :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv (t \<cdot> s) = fv t - subst_domain s \<union> fv\<^sub>s\<^sub>e\<^sub>t (s ` (fv t \<inter> subst_domain s))"
partial_order_on_def: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
simps_1: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b" shows "RBT_Impl.map f Empty = Empty"
divides_pirreducible_condition: fixes K :: "'a set"   and q :: "'a list"   and p :: "'a list" assumes "pirreducible K q"   and "p \<in> carrier (K [X])"   and "p divides\<^bsub>K [X]\<^esub> q" shows "p \<in> Units (K [X]) \<or> p \<sim>\<^bsub>K [X]\<^esub> q"
model_to_dimacs_model_complete_max_var: fixes f :: "nat formula"   and dimacs_M :: "int list" assumes "\<forall>v\<in>atoms f. 0 < v"   and "is_cnf f"   and "dimacs_model dimacs_M (cnf_to_dimacs f)" shows "dimacs_model_to_abs dimacs_M (\<lambda>_. False) \<Turnstile> f"
dist_prod_def: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "dist x y = sqrt ((dist (fst x) (fst y))\<^sup>2 + (dist (snd x) (snd y))\<^sup>2)"
assms_1: shows "A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
simps_1: fixes s :: "'c stream" shows "[] @- s = s"
lmap_eq_LNil: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f llist" shows "(lmap f xs = LNil) = (xs = LNil)"
smooth_on_minus: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x - g x)"
to_bl_1: shows "to_bl 1 = replicate (LENGTH('b) - 1) False @ [True]"
"1_5": fixes n :: "nat" assumes "n < length fs" shows "f n = fs ! n"
unit: fixes H :: "'a set" shows "\<one> \<in> generate H"
parAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<parallel> Q) \<parallel> R \<sim>\<^sub>l P \<parallel> Q \<parallel> R"
assms_8: shows "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"
infi_def: fixes x :: "'a"   and y :: "'a" shows "infi x y = \<Sqinter> {x, y}"
single_one: shows "single (0::'b) (1::'c) = 1"
vcpower_index_vempty: fixes A :: "V" shows "A ^\<^sub>\<times> []\<^sub>\<circ> = set {[]\<^sub>\<circ>}"
subprob_measurableD_2: fixes N :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and S :: "'b measure"   and x :: "'a" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"   and "x \<in> space M" shows "sets (N x) = sets S"
Cod_norm: fixes t :: "'c local.term" assumes "Arr t" shows "Cod \<^bold>\<parallel>t\<^bold>\<parallel> = Cod t"
the_inv_into_into: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'c"   and B :: "'b set" assumes "inj_on f A"   and "x \<in> f ` A"   and "order.greater_eq B A" shows "the_inv_into A f x \<in> B"
group_l_invI: assumes "\<And>x. x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n)) \<Longrightarrow> \<exists>y\<in>carrier (relative_homology_group (int n) (nsphere n) (equator n)). y \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub>" shows "Group.group (relative_homology_group (int n) (nsphere n) (equator n))"
dom_sdom: fixes i :: "nat"   and j :: "nat" assumes "local.dom i j"   and "i \<noteq> j" shows "strict_dom i j"
function_factors_right: fixes g :: "'r \<Rightarrow> 's"   and f :: "'q \<Rightarrow> 's" shows "(\<forall>x. \<exists>y. g y = f x) = (\<exists>h. f = g \<circ> h)"
iMin_gr_iff: fixes I :: "'b set"   and n :: "'b" assumes "I \<noteq> {}" shows "(n < iMin I) = (\<forall>a\<in>I. n < a)"
seq_finite_conjunctive_axioms: shows "seq_finite_conjunctive (;) nil"
step_sim: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c) graph_rec_scheme"   and B :: "('b, 'd) graph_rec_scheme"   and a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "simulation R A B"   and "(a, a') \<in> g_E A"   and "(a, b) \<in> R" shows "\<exists>b'. (b, b') \<in> g_E B \<and> (a', b') \<in> R"
strict_trans1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "ordering less_eq less"   and "less_eq a b"   and "less b c" shows "less a c"
scale_left_diff_distrib: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and x :: "'d" assumes "module scale" shows "scale (a - b) x = scale a x - scale b x"
map_eq: shows "map2 (\<lambda>a b. Var (length slp + a - Suc b)) [0..<length fas] (rev (map (the \<circ> Mapping.lookup M) fas)) = map Var (map2 (\<lambda>a b. length slp + a - Suc b) [0..<length fas] (rev (map (the \<circ> Mapping.lookup M) fas)))"
j_def: fixes j' :: "nat"   and S :: "'a ltl set" assumes "j \<le> j'" shows "(\<forall>\<psi>. G \<psi> \<in> \<G> \<longrightarrow> S \<Turnstile>\<^sub>P G \<psi> \<and> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (\<F> \<psi> w \<G> j')) \<longrightarrow> S \<Turnstile>\<^sub>P af \<phi> (prefix j' w)"
cond_1: shows "\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c loc) \<and> t' \<le> t"
assms: shows "a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0"
antisym_conv2: fixes x :: "'a list"   and y :: "'a list" assumes "prefix x y" shows "(\<not> strict_prefix x y) = (x = y)"
ord_resolve_rename: fixes CAs :: "'a literal multiset list"   and n :: "nat"   and AAs :: "'a multiset list"   and As :: "'a list"   and DA :: "'a literal multiset"   and \<rho> :: "'s"   and \<rho>s :: "'s list"   and S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and \<sigma> :: "'s"   and E :: "'a literal multiset" assumes "length CAs = n"   and "length AAs = n"   and "length As = n"   and "\<forall>i<n. poss (AAs ! i) \<subseteq># CAs ! i"   and "negs (mset As) \<subseteq># DA"   and "\<rho> = hd (renamings_apart (DA # CAs))"   and "\<rho>s = tl (renamings_apart (DA # CAs))"   and "ord_resolve S (CAs \<cdot>\<cdot>cl \<rho>s) (DA \<cdot> \<rho>) (AAs \<cdot>\<cdot>aml \<rho>s) (As \<cdot>al \<rho>) \<sigma> E" shows "ord_resolve_rename S CAs DA AAs As \<sigma> E"
diff_add_inverse_Suc: fixes n :: "nat"   and m :: "nat" assumes "n < m" shows "n + (m - Suc n) = m - Suc 0"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (cod \<iota> \<otimes> a)"
ev_ev: fixes \<phi> :: "'b stream \<Rightarrow> bool" shows "ev (ev \<phi>) = ev \<phi>"
set_zip_leftD: fixes x :: "'b"   and y :: "'c"   and xs :: "'b list"   and ys :: "'c list" assumes "(x, y) \<in> set (zip xs ys)" shows "x \<in> set xs"
abs_eq: fixes y :: "'b \<Rightarrow> 'c option" shows "lookup (Mapping y) = y"
tm_bound_def: fixes taylor_model :: "taylor_model" shows "tm_bound taylor_model = (case taylor_model of TaylorModel x1 x2 \<Rightarrow> x2)"