hcju/isabelleps huggingface dataset:
left_lang_disjoint: fixes q1 :: "hf"   and q2 :: "hf" assumes "q1 \<noteq> q2" shows "Accessible.left_lang q1 \<inter> Accessible.left_lang q2 = {}"
unlex_inject: fixes x :: "'b lexlist"   and y :: "'b lexlist" shows "(unlex x = unlex y) = (x = y)"
intro: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice" assumes "\<And>(c::'b::refinement_lattice) (d\<^sub>0::'b::refinement_lattice) d\<^sub>1::'b::refinement_lattice. (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) c (d\<^sub>0 \<sqinter> d\<^sub>1) \<sqsubseteq> seq c d\<^sub>0 \<sqinter> seq c d\<^sub>1" shows "seq_distrib_left_axioms (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"
select_convs_1: fixes verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and player0 :: "'c set"   and priority :: "'c \<Rightarrow> nat"   and more :: "'d" shows "V0\<^bsub>\<lparr>verts = verts, arcs = arcs, player0 = player0, priority = priority, \<dots> = more\<rparr>\<^esub> = player0"
coprime_mult_self_left_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one (times c a) (times c b) = (dvd.dvd times c one \<and> algebraic_semidom.coprime times one a b)"
backward_simulation_composition: fixes step1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and step2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and final1 :: "'d \<Rightarrow> bool"   and final2 :: "'e \<Rightarrow> bool"   and order1 :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and match1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and step3 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and final3 :: "'g \<Rightarrow> bool"   and order2 :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and match2 :: "'h \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool" assumes "backward_simulation step1 step2 final1 final2 order1 match1"   and "backward_simulation step2 step3 final2 final3 order2 match2" shows "backward_simulation step1 step3 final1 final3 (lex_prodp order1\<^sup>+\<^sup>+ order2) (rel_comp match1 match2)"
map_key_id: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" shows "Poly_Mapping.map_key (\<lambda>x. x) p = p"
simps_6: fixes r :: "'b mrexp3"   and uy :: "'b set" shows "strip3 (Star3 r uy) = Star (strip3 r)"
exhaust: fixes y :: "sethi_type" obtains "y = SETHI"
\<gamma>_uminus_rep: fixes i :: "int"   and p :: "int extended \<times> int extended" assumes "i \<in> \<gamma>_rep p" shows "- i \<in> \<gamma>_rep (uminus_rep p)"
assms: shows "run_one_step d i (s, vs, ves, $b_e) = (s', vs', RSNormal es')"
ln_upper_3_neg: fixes x :: "real" assumes "0 < x"   and "x \<le> 1" shows "ln x \<le> ln_upper_3 x"
diff_add_cancel_ennreal: fixes a :: "ennreal"   and b :: "ennreal" assumes "a \<le> b" shows "b - a + a = b"
rel_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and x :: "'c set \<times> 'd"   and y :: "'c set \<times> 'd" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I" shows "direct_lim.rel \<FF> \<rho> I x y \<equiv> (fst x \<in> I \<and> fst y \<in> I) \<and> (snd x \<in> \<FF> (fst x) \<and> snd y \<in> \<FF> (fst y)) \<and> (\<exists>W. W \<in> I \<and> W \<subseteq> fst x \<inter> fst y \<and> \<rho> (fst x) W (snd x) = \<rho> (fst y) W (snd y))"
integrableD_2: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity>"
dist_triangle: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "class.metric_space dist uniformity open" shows "dist x z \<le> dist x y + dist y z"
sg_mult_closed: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "x \<in> H"   and "y \<in> H" shows "x \<cdot>\<^bsub>G\<^esub> y \<in> H"
Nneq1: shows "N \<noteq> {\<one>\<^bsub>H\<^esub>}"
lengthEqvt: fixes p :: "(name \<times> name) list"   and xs :: "'a list" shows "p \<bullet> length xs = length (p \<bullet> xs)"
hyps: fixes p :: "(nat \<times> int) list"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "d \<le> dm"   and "p \<in> sparsegrid dm lm" shows "updown' dm lm d \<alpha> p = (\<Sum>p'\<in>lgrid (base {0..<d} p) {0..<d} lm. \<alpha> p' * (\<Prod>d' = 0..<d. l2_\<phi> (p' ! d') (p ! d')))"
hyps: assumes "sorted_wrt r xs" shows "sorted_wrt r (insort_wrt r x xs)"
D_GET: fixes y :: "Var"   and f :: "Field"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var" assumes "CONTEXT y = low"   and "GAMMA f = low" shows "(G, Get x y f, Sec (\<lambda>(s, t, \<beta>). \<exists>r l C Flds v. fst r y = RVal (Loc l) \<and> lookup (snd r) l = Some (C, Flds) \<and> lookup Flds f = Some v \<and> r \<equiv>\<^sub>\<beta> t \<and> s = (update (fst r) x v, snd r))) \<in> Deriv"
leq: fixes x :: "real"   and y :: "real" assumes "x \<in> C"   and "y \<in> C" shows "f' x * (y - x) \<le> f y - f x"
remove: fixes A :: "'g set"   and x :: "'g" assumes "finite A"   and "x \<in> A" shows "card A = Suc (card (A - {x}))"
total': fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "total_preorder_on carrier le"   and "\<not> le x y"   and "x \<in> carrier"   and "y \<in> carrier" shows "le y x"
tag: fixes i :: "'a set" assumes "i \<in> I" shows "pfn i tagged_division_of i"
heap_merge: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>"   and "heap l"   and "heap r" shows "heap (merge l r)"
wf_no_infinite_down_chainI: fixes r :: "('g \<times> 'g) set" assumes "\<And>f. (\<And>i. (f (Suc i), f i) \<in> r) \<Longrightarrow> False" shows "wf r"
submonoid_axioms: shows "submonoid (f ` A) B addB zeroB"
HStartBallot_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "Inv2a s"   and "HInv4b s q"   and "HInv4a s p" shows "HInv4b s' q"
delete_edge_def: fixes v :: "'b"   and v' :: "'b"   and G :: "'b list_graph" shows "FiniteListGraph.delete_edge v v' G = \<lparr>nodesL = nodesL G, edgesL = filter (\<lambda>(e1, e2). e1 \<noteq> v \<or> e2 \<noteq> v') (edgesL G)\<rparr>"
nmi: shows "\<not> Ifm vs (x # bs) (plusinf p)"
\<rr>_ide_simp: fixes f :: "'b" assumes "ide f" shows "\<rr> f = \<r>[f]"
induct: fixes P1 :: "'b exp \<Rightarrow> bool"   and P2 :: "'b exp list \<Rightarrow> bool"   and exp :: "'b exp" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 exp"
slice_take': fixes n :: "nat"   and w :: "'b word" shows "slice n w = of_bl (take (size w - n) (to_bl w))"
ground_resolution_with_selection_axioms: shows "ground_resolution_with_selection S"
thm_relation_negation_2_1: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and x :: "\<nu>" shows "[\<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,x\<^sup>P\<rparr> in v]"
simps_2: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and xs :: "nat list"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith_aux A (i # xs) bezout = diagonal_to_Smith_aux (Diagonal_to_Smith_row_i A i bezout) xs bezout"
frrestrictionI: fixes b :: "V"   and A :: "V"   and a :: "V"   and r :: "V" assumes "b \<in>\<^sub>\<circ> A"   and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<bullet> A"
distinct_9: fixes x2 :: "bool"   and x3 :: "int" shows "Bool x2 \<noteq> Intg x3"
gen_check_filter: fixes C :: "nat list \<Rightarrow> nat \<Rightarrow> bool"   and B :: "nat"   and as :: "nat list" assumes "bounded_gen_check C B" shows "bounded_gen_check.gen_check C as = filter (suffs C as) (alls B as)"
indRelTEQ_impl_TRel_is_weak_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_bisimulation (indRelTEQ TRel) (STCal Source Target)" shows "weak_reduction_bisimulation (TRel\<^sup>*) Target"
coboundedI1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "semilattice_order f less_eq less"   and "less_eq a c" shows "less_eq (f a b) c"
hyps: assumes "(f, g) \<in> fpsmodrel p" shows "(f ^ x, g ^ x) \<in> fpsmodrel p"
H_2: shows "AE x in MI. \<forall>n. vI n x = uI n x"
mult_le_one: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq a one"   and "less_eq zero b"   and "less_eq b one" shows "less_eq (times a b) one"
aristo_conversion2: fixes A :: "'a set"   and B :: "'a set" assumes "A I B" shows "B I A"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>" shows "thesis"
f: shows "f = (d + (2::'a) * b ^ 3 / ((27::'a) * a\<^sup>2) - b * c / ((3::'a) * a)) / a"
simps_3: fixes t :: "'a E.term"   and u :: "'a E.term" shows "\<lbrace>t \<^bold>\<star> u\<rbrace> = \<lbrace>t\<rbrace> \<star> \<lbrace>u\<rbrace>"
length_two_unique: assumes "length \<HH> = 2" shows "\<HH> = [{\<one>\<^bsub>G\<^esub>}, carrier G]"
dist_Pair_Pair: fixes a :: "'b"   and b :: "'c"   and c :: "'b"   and d :: "'c" shows "dist (a, b) (c, d) = sqrt ((dist a c)\<^sup>2 + (dist b d)\<^sup>2)"
col2_cop__cop: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "Coplanar A B C D"   and "C \<noteq> D"   and "Col C D E"   and "Col C D F" shows "Coplanar A B E F"
comm_groupE_2: fixes G :: "('b, 'c) monoid_scheme" assumes "comm_group G" shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
assms_1: shows "A \<in> sets (lborel_f (Suc n))"
trms\<^sub>s\<^sub>s\<^sub>t_fv_vars\<^sub>s\<^sub>s\<^sub>t_subset: fixes t :: "('c, 'd) Term.term"   and A :: "('c, 'd) stateful_strand_step list" assumes "t \<in> trms\<^sub>s\<^sub>s\<^sub>t A" shows "fv t \<subseteq> vars\<^sub>s\<^sub>s\<^sub>t A"
bisimilarity_implies_equivalence: fixes P :: "('state, 'effect) L_state"   and Q :: "('state, 'effect) L_state" assumes "P \<sim>\<cdot>\<^sub>L Q" shows "L_transform.logically_equivalent P Q"
basic_trans_rules_26: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b"   and "b \<le> c" shows "a \<le> c"
closest_supchamber_closest: fixes X :: "'b set set"   and C :: "'b set"   and F :: "'b set"   and D :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "order.greater_eq C F" shows "order.greater_eq (ChamberComplex.chamber_distance X C D) (ChamberComplex.chamber_distance X (ChamberComplex.closest_supchamber X F D) D)"
continuous_transform_within_openin: fixes a :: "'c"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "continuous (at a within T) f"   and "openin (top_of_set T) S"   and "a \<in> S"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "continuous (at a within T) g"
trcl_empty_cons: fixes c :: "'f"   and c' :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set" assumes "(c, [], c') \<in> trcl t" shows "c = c'"
select_convs_1: fixes edge_rev :: "'c \<Rightarrow> 'c"   and edge_succ :: "'c \<Rightarrow> 'c"   and more :: "'d" shows "edge_rev \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_rev"
inner_not_same_Basis: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and u :: "'e"   and v :: "'e" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis"   and "u \<noteq> v" shows "inner u v = 0"
Un_range_binary: fixes a :: "'b set"   and b :: "'b set" shows "a \<union> b = \<Union> (range (binary a b))"
fls_trivial_delta_eq_zero: fixes b :: "'a"   and a :: "int" assumes "b = (0::'a)" shows "Abs_fls (\<lambda>n. if n = a then b else (0::'a)) = 0"
res_add_eq: fixes m :: "int"   and x :: "int"   and y :: "int" assumes "residues m" shows "x \<oplus>\<^bsub>residue_ring m\<^esub> y = (x + y) mod m"
idx_sequence_interval_unique: fixes idx :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat"   and m :: "nat" assumes "idx_sequence idx"   and "n \<in> {idx k..<idx (Suc k)}"   and "n \<in> {idx m..<idx (Suc m)}" shows "k = m"
c_hd_def: shows "c_hd = (\<lambda>u. if u = 0 then 0 else hd (nat_to_list u))"
fps_lr_inverse_eq_0_iff_1: fixes f :: "'a fps"   and x :: "'a" shows "(fps_left_inverse f x = 0) = (x = (0::'a))"
diff_self: fixes a :: "'b" shows "a - a = (0::'b)"
bottom_lower: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<bottom> \<sqsubseteq> x"
g'_def: fixes i :: "'a"   and j :: "'a"   and H :: "'a" shows "g' i j H = integral {0..1} (\<lambda>x. (1 - x) * g H x i j)"
axioms_4: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "semiring_1_cancel_ow U plus minus zero one times" shows "monoid_mult_ow U one times"
subst_term_eqI: fixes \<sigma> :: "'e \<Rightarrow> ('d, 'f) Term.term"   and \<tau> :: "'e \<Rightarrow> ('d, 'f) Term.term" assumes "\<And>t. t \<cdot> \<sigma> = t \<cdot> \<tau>" shows "\<sigma> = \<tau>"
count_mset_0_iff: fixes xs :: "'b list"   and x :: "'b" shows "(count (mset xs) x = 0) = (x \<notin> set xs)"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) lit" assumes "\<not> U.satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>'__" and "xa = Pos a__" and "\<not> U.satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> U.satA \<xi> a"
intros_1: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "F \<in> \<Gamma>" shows "\<Gamma> \<turnstile> F"
density_context_axioms: shows "density_context V V' \<Gamma> (\<lambda>x. ennreal (extract_real (cexpr_sem x \<delta>)))"
dom_left_map_add: fixes x :: "'c"   and g :: "'c \<Rightarrow> 'd option"   and f :: "'c \<Rightarrow> 'd option" assumes "x \<in> dom g" shows "(f ++ g) x = g x"
map_converter_id: fixes conv :: "('i, 'j, 'k, 'l) converter" shows "map_converter id id id id conv = conv"
intdj: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> \<D>"   and "B \<in> \<D>" shows "A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}"
reduce_simple_call: fixes j :: "nat"   and es' :: "e list" shows "\<not> \<lparr>[$Call j]\<rparr> \<leadsto> \<lparr>es'\<rparr>"
distance_leq_d2_def: shows "distance_leq_d2 = (a\<^sub>e + a\<^sub>o) / (2 * a\<^sub>o\<^sup>2) * v\<^sub>o\<^sup>2 - v\<^sub>o * v\<^sub>e / a\<^sub>o"
ideal_asubg: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "R +> I"
ctr: fixes s1 :: "'b stream"   and s2 :: "'c stream" shows "s1 ||| s2 = (shd s1, shd s2) ## (stl s1 ||| stl s2)"
comm_semigroup_axioms: shows "comm_semigroup (\<oplus>\<^sub>a)"
C2_def: shows "C2 = \<lparr>carrier = UNIV, monoid.mult = (=), one = True\<rparr>"
le_less_trans: fixes y :: "'d"   and x :: "'d"   and z :: "'d" assumes "y \<le> x"   and "z < y" shows "z < x"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'c"   and f2 :: "'c"   and x :: "enat" shows "P (case x of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = ((\<forall>nat. x = enat nat \<longrightarrow> P (f1 nat)) \<and> (x = \<infinity> \<longrightarrow> P f2))"
simps_2: fixes t :: "('c, 'd) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (receive\<langle>t\<rangle>) = {t}"
split_beta: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and prod :: "'d \<times> 'e" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
forall\<^sub>\<o>_def: shows "forall\<^sub>\<o> \<equiv> map_fun (map_fun id eval\<o>) make\<o> (\<lambda>\<phi> s w. \<forall>x. \<phi> x s w)"
h2: shows "h \<turnstile> set_disconnected_nodes owner_document (child # disconnected_nodes_h) \<rightarrow>\<^sub>h h2"
outside_def: fixes S :: "'a set" shows "outside S \<equiv> - S \<inter> {x. \<not> bounded (connected_component_set (- S) x)}"
f9: shows "(0 \<le> - 1 * uint w) = (uint w \<le> 0)"
Impl_IUSpec: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> Implementation \<and> Hist rmhist \<longrightarrow> IUSpec memCh mm (resbar rmhist)"
runit'_naturality: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "(\<mu> \<star>\<^sub>D src\<^sub>D \<mu>) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[D.dom \<mu>] = \<r>\<^sub>D\<^sup>-\<^sup>1[D.cod \<mu>] \<cdot>\<^sub>D \<mu>"
map_acom_While: fixes f :: "'c \<Rightarrow> 'b"   and w :: "'c acom"   and I' :: "'b"   and b :: "bexp"   and c' :: "'b acom"   and P' :: "'b" shows "(ACom.map_acom f w = {I'} WHILE b DO c' {P'}) = (\<exists>I P c. w = {I} WHILE b DO c {P} \<and> ACom.map_acom f c = c' \<and> I' = f I \<and> P' = f P)"
wf: shows "wf_formula n (FOr \<phi> \<psi>)"
c'_at_n: fixes A :: "'a" assumes "A \<in> set As" shows "\<not> INTERP N \<Turnstile> C_of A"
sum_mset_mset_mono: fixes A :: "'b multiset"   and f :: "'b \<Rightarrow> 'c multiset"   and g :: "'b \<Rightarrow> 'c multiset" assumes "\<And>x. x \<in># A \<Longrightarrow> f x \<subseteq># g x" shows "\<Sum>\<^sub># (image_mset f A) \<subseteq># \<Sum>\<^sub># (image_mset g A)"
ex_new_if_finite: fixes A :: "'d set" assumes "infinite UNIV"   and "finite A" shows "\<exists>a. a \<notin> A"
prems: shows "\<exists>!x. x \<in> set (y # xs) \<and> P x"
assoc_in_vhom\<^sub>A\<^sub>W\<^sub>C: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and h :: "'c" assumes "associative_weak_composition V H \<a>"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "H f g \<noteq> partial_magma.null V"   and "H g h \<noteq> partial_magma.null V" shows "partial_magma.in_hom V (\<a> f g h) (H (H f g) h) (H f (H g h))"
b_in_verts: shows "\<forall>x\<in>set b. x \<in> set (g_V G)"
less_divide_eq_1_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less one (divide b a) = less a b"
Can_and_Nml_implies_Ide: fixes t :: "'a local.term" assumes "Can t"   and "Nml t" shows "Ide t"
split_paired_All: fixes P :: "'c \<times> 'd \<Rightarrow> bool" shows "(\<forall>x. P x) = (\<forall>a b. P (a, b))"
invG: shows "inv\<^bsub>G\<^esub> g \<in> carrier G"
take_cols_var_def: fixes A :: "'a mat"   and inds :: "nat list" shows "take_cols_var A inds = mat_of_cols (dim_row A) (map ((!) (cols A)) inds)"
morphic_prop_def: fixes R :: "('e, 'f) ring_scheme"   and P :: "'e \<Rightarrow> bool" shows "morphic_prop R P = (P \<one>\<^bsub>R\<^esub> \<and> (\<forall>r\<in>carrier R. P r) \<and> (\<forall>r1\<in>carrier R. \<forall>r2\<in>carrier R. P (r1 \<otimes>\<^bsub>R\<^esub> r2)) \<and> (\<forall>r1\<in>carrier R. \<forall>r2\<in>carrier R. P (r1 \<oplus>\<^bsub>R\<^esub> r2)))"
assms: shows "p \<in> setops\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
S_M_def: fixes C :: "'a literal multiset" shows "S_M C = (if C \<in> grounding_of_clss M then SOME C'. \<exists>D \<sigma>. D \<in> M \<and> C = D \<cdot> \<sigma> \<and> C' = S D \<cdot> \<sigma> \<and> is_ground_subst \<sigma> else S C)"
all_n_lists_def: fixes P :: "'b list \<Rightarrow> bool"   and n :: "nat" shows "all_n_lists P n = (\<forall>xs\<in>set (List.n_lists n enum). P xs)"
finite_field_factorization_main_uint64: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly list" assumes "p \<le> 4294967295"   and "finite_field_factorization_main p (finite_field_ops64 (uint64_of_int p)) f = (c, fs)"   and "square_free_m f" shows "unique_factorization_m f (c, mset fs) \<and> c \<in> {0..<p} \<and> (\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p})"
vector_derivative_minus_at: fixes f :: "real \<Rightarrow> 'a"   and a :: "real" assumes "f differentiable at a" shows "vector_derivative (\<lambda>x. - f x) (at a) = - vector_derivative f (at a)"
prems: shows "\<forall>x\<in>set (tl (m'' # ms)). x \<noteq> n"
SPEC_rule: fixes \<Phi> :: "'f \<Rightarrow> bool"   and \<Phi>' :: "'f \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Phi>' x" shows "SPEC \<Phi> \<le> SPEC \<Phi>'"
bidi_3: fixes a :: "'b" assumes "a \<in> arcs G" shows "edge_rev M (edge_rev M a) = a"