hcju/isabelleps huggingface dataset:
node_net_state_trans: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and \<zeta>\<^sub>i :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool"   and a :: "'f node_action"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state" assumes "(\<sigma>, s) \<in> oreachable (\<langle>i : \<zeta>\<^sub>i : R\<^sub>i\<rangle>\<^sub>o) S U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans (\<langle>i : \<zeta>\<^sub>i : R\<^sub>i\<rangle>\<^sub>o)" obtains \<zeta> :: "'e" and R :: "nat set" and \<zeta>' :: "'e" and R' :: "nat set" where "s = NodeS i \<zeta> R" and "s' = NodeS i \<zeta>' R'"
directly_attracted_contains_no_deadends: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player"   and W :: "'c set" assumes "ParityGame G"   and "v \<in> ParityGame.directly_attracted G p W" shows "\<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)"
extend_indets_times: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "extend_indets (p * q) = extend_indets p * extend_indets q"
bij_combine: fixes A :: "'b set"   and B :: "'b set"   and C :: "'c set"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "A \<subseteq> B"   and "C \<subseteq> D"   and "bij_betw f A C"   and "bij_betw g (B - A) (D - C)" obtains h :: "'b \<Rightarrow> 'c" where "bij_betw h B D" and "\<And>x. x \<in> A \<Longrightarrow> h x = f x" and "\<And>x. x \<in> B - A \<Longrightarrow> h x = g x"
suc_underS: fixes r :: "('d \<times> 'd) set"   and a :: "'d" assumes "wo_rel r"   and "a \<in> Field r" shows "a = suc r (underS r a)"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "min a b \<le> a"
sum_split_nat_ivl: fixes a :: "nat"   and j :: "nat"   and b :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "a \<le> Suc j"   and "j \<le> b" shows "sum f {a..j} + sum f {Suc j..b} = sum f {a..b}"
simps: fixes f :: "fid_t"   and n :: "nat" shows "fid (f $ n) = f"
simps: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'c"   and S :: "'b SDG_node set"   and n :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "(a \<in> SDG.relevant_vars sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S n) = (\<exists>as n' V. a = V \<and> CFG.intra_path sourcenode targetnode kind valid_edge (parent_node n) as (parent_node n') \<and> n' \<in> SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S \<and> SDG.SDG_Use sourcenode targetnode kind valid_edge get_proc procs Use ParamUses V n' \<and> (\<forall>n''. SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n'' \<and> parent_node n'' \<in> set (CFG.sourcenodes sourcenode as) \<longrightarrow> \<not> SDG.SDG_Def sourcenode targetnode kind valid_edge get_proc procs Def ParamDefs V n''))"
Infinitesimal_hypreal_sqrt: fixes x :: "real star" assumes "0 \<le> x"   and "x \<in> Infinitesimal" shows "(*f* sqrt) x \<in> Infinitesimal"
INF_lower: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and i :: "'i"   and A :: "'i set"   and f :: "'i \<Rightarrow> 'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (Inf (f ` A)) (f i)"
predecessors_def: fixes g :: "'a"   and n :: "'node" shows "gp2.predecessors g n \<equiv> map getFrom (gp2.inEdges g n)"
simps_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "(case None of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
residual_network_sel_2: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" shows "capacity (residual_network f) (x, y) = (if edge \<Delta> x y then capacity \<Delta> (x, y) - f (x, y) else if y = source \<Delta> then 0 else f (y, x))"
set_fold1_def: shows "set_fold1 \<equiv> map_fun semilattice_set_apply id semilattice_set.F"
zip_eq: fixes u :: "'d list"   and v :: "'e list"   and r :: "'d list"   and s :: "'e list" assumes "length u = length v"   and "length r = length s" shows "(u || v = r || s) = (u = r \<and> v = s)"
preserves_reflects_arr: fixes f :: "'c local.term set" shows "\<F>C.arr (\<F>C.\<ll> f) = \<F>C.arr f"
zero_le_power2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a 2)"
l_set_child_nodes_get_child_nodes_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_child_nodes_get_child_nodes_axioms type_wf known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs \<equiv> (\<forall>h ptr children h'. type_wf h \<longrightarrow> known_ptr ptr \<longrightarrow> h \<turnstile> set_child_nodes ptr children \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children) \<and> (\<forall>ptr ptr' w h h' r. ptr \<noteq> ptr' \<longrightarrow> w \<in> set_child_nodes_locs ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_child_nodes_locs ptr' \<longrightarrow> r h h')"
Healthy_def: fixes P :: "'a upred"   and H :: "'a upred \<Rightarrow> 'a upred" shows "P is H \<equiv> H P = P"
that: fixes S :: "'a set" assumes "S \<subseteq> {x \<in> space M. P x}"   and "S \<in> events"   and "prob S = 1" shows "thesis"
Lim_def: fixes f :: "'c \<Rightarrow> ('b, 'c) Old_Datatype.node set" shows "Old_Datatype.Lim f \<equiv> \<Union> {z. \<exists>x. z = Push_Node (Inl x) ` f x}"
qOp_alpha_iff: fixes delta :: "'j"   and inp :: "'f \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qTerm option"   and binp :: "'g \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qAbs option"   and X' :: "('f, 'g, 'h, 'i, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "(qOp delta inp binp #= X') = (\<exists>inp' binp'. X' = qOp delta inp' binp' \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
l2_inv3_trans: shows "{l2_inv3 \<inter> l2_inv2} TS.trans l2 {> l2_inv3}"
\<tau>: shows "\<forall>x\<in>subst_domain \<theta>. \<theta> x = (\<delta> \<circ>\<^sub>s \<tau>) x"
R'_walk_no_Q: fixes v :: "'a" assumes "v \<in> set R'_walk"   and "v \<noteq> z" shows "\<not> Q.hitting_paths v"
set_times_intro: fixes a :: "'c"   and C :: "'c set"   and b :: "'c"   and D :: "'c set" assumes "a \<in> C"   and "b \<in> D" shows "a * b \<in> C * D"
k: shows "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
axioms_3: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and S' :: "'d set"   and is_open' :: "'d set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphism S is_open S' is_open' f" shows "continuous_map S' is_open' S is_open (inverse_map f S S')"
bottom_left_bottom: fixes a :: "'a" shows "\<^bold>\<bottom> \<^bold>* a = \<^bold>\<bottom>"
execn_strip_guards_to_execn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> =n\<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' \<in> Fault ` (- F) \<longrightarrow> t' = t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
associatedI2: fixes u :: "'a"   and a :: "'a"   and b :: "'a" assumes "u \<in> Units G"   and "a = b \<otimes> u"   and "b \<in> carrier G" shows "a \<sim> b"
r_one: fixes x :: "'b \<Rightarrow> 'e" assumes "x \<in> carrier (DirProds Hs J)" shows "x \<otimes>\<^bsub>DirProds Hs J\<^esub> \<one>\<^bsub>DirProds Hs J\<^esub> = x"
cond_spmf_fst_eq_return_None: fixes p :: "('i \<times> 'h) spmf"   and x :: "'i" shows "(cond_spmf_fst p x = return_pmf None) = (x \<notin> fst ` set_spmf p)"
simps_5: fixes ux :: "'b tree list" shows "cost Merge ux = 1"
inject: fixes x :: "'d"   and ya :: "'d" shows "(return_id x = return_id ya) = (x = ya)"
map_cong0: fixes x :: "'a tree"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>z. z \<in> set_tree x \<Longrightarrow> f z = g z" shows "map_tree f x = map_tree g x"
i_hcomplex_of_hypreal: fixes r :: "real star" shows "iii * hcomplex_of_hypreal r = HComplex 0 r"
openin_pullback_topology: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d topology"   and S :: "'c set" shows "openin (pullback_topology A f T) S = (\<exists>U. openin T U \<and> S = f -` U \<inter> A)"
replacement_raw: fixes X :: "V set"   and f :: "V \<Rightarrow> V" assumes "X \<in> range elts" shows "f ` X \<in> range elts"
nwprops: shows "qty_sum pf1 pf2 x n w \<noteq> 0"
smc_SemiCAT_is_arrE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<^bsub>smc_SemiCAT \<alpha>\<^esub> \<BB>" obtains "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
simps_2: fixes n :: "nat" shows "k (Suc n) = k n"
H_def: shows "[\<^bold>\<not>(\<lbrace>x\<^sup>P,H\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,H\<rbrace>) in v]"
hinsert_def: fixes a :: "hf"   and b :: "hf" shows "b \<triangleleft> a = HF (insert a (hfset b))"
finite_swap_ne: fixes f :: "'c \<Rightarrow> 'd"   and c :: "'d"   and b :: "'c"   and i :: "'c" assumes "finite {x. f x \<noteq> c}" shows "finite {x. f (swap b i x) \<noteq> c}"
errMOD_igSwapIGOpSTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igWlsAbsIsInBar Delta MOD"   and "FixSyn.igOpIPresIGWls Delta MOD"   and "igSwapIPresIGWls MOD"   and "FixSyn.igSwapAbsIPresIGWlsAbs Delta MOD"   and "igWlsDisj MOD"   and "FixSyn.igWlsAbsDisj Delta MOD"   and "FixSyn.igSwapIGOp Delta MOD" shows "igSwapIGOpSTR (FixSyn.errMOD Delta MOD)"
in_segment_2: fixes x :: "'c"   and a :: "'c"   and b :: "'c" shows "(x \<in> open_segment a b) = (a \<noteq> b \<and> (\<exists>u>0. u < 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b))"
degeneralize_nodes_finite: fixes A :: "('d, 'c) dgba" shows "finite (DBA.nodes (degeneralize A)) = finite (DGBA.nodes A)"
complete_square: fixes a :: "real"   and x :: "real"   and b :: "real"   and c :: "real" assumes "a \<noteq> 0" shows "(a * x\<^sup>2 + b * x + c = 0) = ((2 * a * x + b)\<^sup>2 = discrim a b c)"
perp_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "A B Perp D C"
asym: fixes a :: "'c"   and b :: "'c" assumes "a < b"   and "b < a" shows "False"
sym_group_card_carrier: fixes n :: "nat" shows "card (carrier (sym_group n)) = fact n"
sym_conv_converse_eq: fixes r :: "('e \<times> 'e) set" shows "sym r = (r\<inverse> = r)"
Cn_disj: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<or>. q) = (Cn {q} \<subseteq> Cn (A \<union> {.\<not> p}))"
allowed_redex_def: fixes t :: "'c trm"   and C :: "'c eclause"   and \<sigma> :: "('c \<times> 'c trm) list" shows "allowed_redex t C \<sigma> = (\<not> (\<exists>s\<in>trms_ecl C. occurs_in (t \<lhd> \<sigma>) (s \<lhd> \<sigma>)))"
in_aux0_le_\<tau>: fixes t :: "nat"   and X :: "(mregex, event_data option list set) mapping" assumes "(t, X) \<in> set aux0" shows "t \<le> \<tau> \<sigma> ne"
well_defined_Tensor: fixes M1 :: "int poly fract list list"   and M2 :: "int poly fract list list" assumes "mat (rat_poly.row_length M1) (length M1) M1 \<and> mat (rat_poly.row_length M2) (length M2) M2" shows "mat (rat_poly.row_length M1 *n rat_poly.row_length M2) (length M1 *n length M2) (M1 \<otimes> M2)"
compact_insert: fixes s :: "'a set"   and x :: "'a" assumes "compact s" shows "compact (insert x s)"
inverse_le_iff_le: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "(0::'a) < b" shows "(inverse a \<le> inverse b) = (b \<le> a)"
eq_trans: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<chi> :: "'a ltln" assumes "\<phi> \<sim>\<^sub>L \<psi>"   and "\<psi> \<sim>\<^sub>L \<chi>" shows "\<phi> \<sim>\<^sub>L \<chi>"
\<A>_def: shows "\<A> \<equiv> (\<lambda>N. \<Inter> (from_nat_into \<B> ` N)) ` Collect finite"
add_increasing: fixes x :: "'a"   and y :: "'a" shows "x \<le> x + y"
unit_factor_dvd: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)" shows "unit_factor a dvd b"
a_closure: fixes x :: "'a" shows "d (ad x) = ad x"
pcr_cr_eq: shows "pcr_fset (=) = cr_fset"
subc_new: fixes c :: "'a"   and p :: "('a, 'b) form"   and s :: "'a FOL_Fitting.term" assumes "new c p" shows "subc c s p = p"
assms_1: fixes n :: "nat" shows "b n \<in> s"
adjacent_Cons_implies_adjacent: fixes a :: "nat"   and b :: "nat"   and t :: "'b tree"   and ts :: "'b tree list" assumes "adjacent (a, b) (t # ts)" shows "adjacent (a, l t - Suc 0) ts"
bit_integer_def: shows "bit \<equiv> map_fun int_of_integer id bit"
strict_coboundedI1: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset" assumes "a \<subset># c" shows "a \<inter># b \<subset># c"
order_trans: fixes x :: "'c word"   and y :: "'c word"   and z :: "'c word" assumes "x \<le>s y"   and "y \<le>s z" shows "x \<le>s z"
rpo_optim: shows "rpo_optim ground_heads_var (>\<^sub>s) extf arity_sym arity_var"
split_u: shows "verticesFrom f v' = fst (splitAt u (verticesFrom f v')) @ u # snd (splitAt u (verticesFrom f v'))"
norm_6: shows "isnpolyh (CN c' n' p') n'"
IterateCallNE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and res :: "String.literal"   and res_t :: "'a OCL_Types.type option"   and res_init :: "'a expr"   and body :: "'a expr"   and call :: "'a call_expr" assumes "(\<Gamma>, \<tau>) \<turnstile>\<^sub>C Iterate its its_ty res res_t res_init body \<Rrightarrow> call" obtains res_init\<^sub>2 :: "'a expr" and \<sigma> :: "'a OCL_Types.type" and res_t\<^sub>2 :: "'a OCL_Types.type option" and body\<^sub>2 :: "'a expr" where "its_ty = Some \<sigma>__" and "call = Iterate its (Some \<sigma>__) res res_t\<^sub>2__ res_init\<^sub>2__ body\<^sub>2__" and "\<Gamma> \<turnstile> res_init \<Rrightarrow> res_init\<^sub>2__" and "\<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, \<sigma>__)) its) \<turnstile> expr.Let res res_t res_init body \<Rrightarrow> expr.Let res res_t\<^sub>2__ res_init\<^sub>2__ body\<^sub>2__"   | \<sigma> :: "'a OCL_Types.type" and res_init\<^sub>2 :: "'a expr" and res_t\<^sub>2 :: "'a OCL_Types.type option" and body\<^sub>2 :: "'a expr" where "its_ty = None" and "call = Iterate its (Some \<sigma>) res res_t\<^sub>2 res_init\<^sub>2 body\<^sub>2" and "element_type \<tau> \<sigma>" and "\<Gamma> \<turnstile> res_init \<Rrightarrow> res_init\<^sub>2" and "\<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, \<sigma>)) its) \<turnstile> expr.Let res res_t res_init body \<Rrightarrow> expr.Let res res_t\<^sub>2 res_init\<^sub>2 body\<^sub>2"
vars0_def: shows "st.vars0 = {k} \<union> (vars1 - {k})"
preserves_reflects_arr: fixes f :: "'a" shows "hom'.arr (\<psi>.map f) = hom'.arr f"
antisym: fixes b :: "'c list"   and a :: "'c list" assumes "prefix b a"   and "prefix a b" shows "a = b"
igWlsDisj_def: fixes MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igWlsDisj MOD \<equiv> \<forall>s s' X. igWls MOD s X \<and> igWls MOD s' X \<longrightarrow> s = s'"
inverse: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)"   and "f \<in> O[F](g)" shows "(\<lambda>x. inverse (g x)) \<in> O[F](\<lambda>x. inverse (f x))"
calculus_axioms_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" shows "calculus_axioms Bot Inf entails Red_I Red_F \<equiv> ((\<forall>N. Red_I N \<subseteq> Inf) \<and> (\<forall>B N. B \<in> Bot \<longrightarrow> entails N {B} \<longrightarrow> entails (N - Red_F N) {B}) \<and> (\<forall>N N'. N \<subseteq> N' \<longrightarrow> Red_F N \<subseteq> Red_F N')) \<and> ((\<forall>N N'. N \<subseteq> N' \<longrightarrow> Red_I N \<subseteq> Red_I N') \<and> (\<forall>N' N. N' \<subseteq> Red_F N \<longrightarrow> Red_F N \<subseteq> Red_F (N - N'))) \<and> (\<forall>N' N. N' \<subseteq> Red_F N \<longrightarrow> Red_I N \<subseteq> Red_I (N - N')) \<and> (\<forall>\<iota> N. \<iota> \<in> Inf \<longrightarrow> concl_of \<iota> \<in> N \<longrightarrow> \<iota> \<in> Red_I N)"
cond_fail: fixes r :: "'b"   and s :: "'a" assumes "\<not> C r s" shows "whileLoop C   B r s = condition (C r)   (B r >>= whileLoop C   B)   (return r) s"
simp: shows "sqrt_approx_main x = (if x * x - n < \<epsilon> then x else sqrt_approx_main ((n / x + x) / (2::'a)))"
assoc_simps\<^sub>A\<^sub>W\<^sub>C_3: 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.cod V (\<a> f g h) = H f (H g h)"
set_integrable_mult_right: fixes a :: "'a"   and M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "a \<noteq> (0::'a) \<Longrightarrow> set_integrable M A f" shows "set_integrable M A (\<lambda>t. a * f t)"
assoc: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "a \<^bold>*\<^sub>o\<^sub>w b \<^bold>*\<^sub>o\<^sub>w c = a \<^bold>*\<^sub>o\<^sub>w (b \<^bold>*\<^sub>o\<^sub>w c)"
h4: shows "satisfy_equation p (qs1 @ qs2) (get_subsets (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (get_signs (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
null_part_sets_1: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets M" shows "null_part M S \<in> sets M"
INT_D: fixes b :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and A :: "'b set"   and a :: "'b" assumes "b \<in> \<Inter> (B ` A)"   and "a \<in> A" shows "b \<in> B a"
P: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and m :: "nat"   and ps :: "char list \<Rightarrow> int option"   and n :: "nat" assumes "(P \<and>* $ m) (ps, n)" shows "P (ps, n - m)"
Half_def: fixes x :: "floatarith" shows "Half x = Mult (Num (Float 1 (- 1))) x"
distinct_87: fixes x4 :: "val"   and x91 :: "expr"   and x92 :: "char list"   and x93 :: "char list list"   and x94 :: "expr" shows "Val x4 \<noteq> x91\<bullet>x92{x93} := x94"
closed_compact_projection: fixes S :: "'a set"   and T :: "('a \<times> 'b) set" assumes "compact S"   and "closed T" shows "closed {y. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
dist_minus: fixes x :: "'c"   and y :: "'c" shows "dist (- x) (- y) = dist x y"
vars_coeff_extract_var: fixes p :: "'a mpoly"   and v :: "nat"   and j :: "nat \<Rightarrow>\<^sub>0 nat" shows "vars (MPoly_Type.coeff (extract_var p v) j) \<subseteq> {v}"
nat_pow_one: fixes n :: "nat" shows "\<one>\<^bsub>P\<^esub> [^]\<^bsub>P\<^esub> n = \<one>\<^bsub>P\<^esub>"
dim_UNIV: shows "sub.tangent_space.dim p (sub.tangent_space p) = card basis"
of_nat_dvd_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "dvd.dvd times (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m dvd n)"
path_Exit_source: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'"   and "n = (_Exit_)" shows "n' = (_Exit_) \<and> as = []"
h2: shows "satisfy_equation p (qs1 @ qs2) (subsets_smash (length qs1) subsets1 subsets2) (signs_smash signs1 signs2)"
diagonal_step_row_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and j :: "nat"   and c :: "'a"   and v :: "'a"   and a :: "'c" shows "diagonal_step_row A i j c v a = (\<chi>b. if a = from_nat i \<and> b = from_nat i then c else if a = from_nat j \<and> b = from_nat j then v * A $ from_nat j $ from_nat j else A $ a $ b)"
size_fsingleton: fixes f :: "'a fset" shows "(size f = 1) = (\<exists>e. f = {|e|})"
Sigma_empty1: fixes B :: "'b \<Rightarrow> 'c set" shows "Sigma {} B = {}"
image_image: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and A :: "'g set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
ruleset_def: shows "ruleset \<equiv> [''INPUT'' \<mapsto> [Rule (Match Established) Accept, Rule (MatchAnd (Match IsSSH) (Match New)) Accept]]"
simps_20: fixes x1 :: "'f option" shows "results (Resumption.resumption.Done x1) = set_option x1"
ineq: fixes k :: "int" assumes "1 \<le> k"   and "k \<le> int (n div 2)" shows "cmod (f k) \<le> real n / real_of_int (2 * k)"
wf_termi_call_steps: shows "wf termi_call_steps"
foo: fixes d :: "nat"   and v :: "'b vec" shows "monom (1::'b) d \<cdot>\<^sub>v coeff_lift_hom.vec_hom v = map_vec (\<lambda>a. monom a d) v"
contractible_imp_connected: fixes S :: "'b set" assumes "contractible S" shows "connected S"
right_total_conversep: fixes A :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "right_total A\<inverse>\<inverse> = left_total A"
combine_assoc: fixes c1 :: "'coll"   and c2 :: "'coll"   and c3 :: "'coll" shows "combine (combine c1 c2) c3 = combine c1 (combine c2 c3)"
Ng_def: shows "Ng \<equiv> \<lambda>y. norm (g (f y) - g (f x) - g' (f y - f x)) / norm (f y - f x)"