hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat list"   and a5 :: "exp" assumes "mbody(a1,a2,a3) = a4 . a5" obtains CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and CDef :: "classDef" and m :: "nat" and mDef :: "methodDef" and xs :: "nat list" and e :: "exp" where "a1 = CT__" and "a2 = m__" and "a3 = C__" and "a4 = xs__" and "a5 = e__" and "CT__ C__ = Some CDef__" and "lookup (cMethods CDef__) (\<lambda>md. mName md = m__) = Some mDef" and "varDefs_names (mParams mDef) = xs__" and "mBody mDef = e__"   | CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and CDef :: "classDef" and m :: "nat" and D :: "nat" and xs :: "nat list" and e :: "exp" where "a1 = CT" and "a2 = m" and "a3 = C" and "a4 = xs" and "a5 = e" and "CT C = Some CDef" and "lookup (cMethods CDef) (\<lambda>md. mName md = m) = None" and "cSuper CDef = D" and "mbody(CT,m,D) = xs . e"
AXc: fixes c1 :: "'a"   and c2 :: "'a"   and X :: "'a \<Rightarrow> nat"   and A1 :: "'a \<Rightarrow> nat"   and A2 :: "'a \<Rightarrow> nat" assumes "c1 \<noteq> c2"   and "c1 \<in># X"   and "c2 \<in># X" shows "A1 \<union># A2 \<union># X -# c2 -# c1 = A2 \<union># A1 \<union># X -# c1 -# c2"
surj_on_implies_set_mset_eq: fixes A :: "'c set"   and F :: "('c \<Rightarrow> 'd) set"   and B :: "'d set" assumes "finite A"   and "F \<in> (A \<rightarrow>\<^sub>E B) // domain_permutation A B"   and "univ (\<lambda>f. f ` A = B) F" shows "set_mset (msubset_of A F) = B"
orient_lit_semantics_pos: fixes I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"   and l :: "'a literal"   and u :: "'a trm"   and v :: "'a trm"   and \<eta> :: "('a \<times> 'a trm) list"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "fo_interpretation I"   and "orient_lit_inst l u v pos \<eta>"   and "validate_ground_lit I (subst_lit l \<sigma>)" shows "I (u \<lhd> \<sigma>) (v \<lhd> \<sigma>)"
f4: shows "\<not> P (last xs, j) \<longrightarrow> successive P (arcs i (last xs) (butlast xs))"
finite_PlusD_1: fixes A :: "'a set"   and B :: "'b set" assumes "finite (A <+> B)" shows "finite A"
join_assoc_lemma: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}"
obj_trg: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "D.obj (trg\<^sub>D \<mu>)"
put_is_create: fixes \<sigma> :: "'b"   and v :: "'a" shows "put \<sigma> v = create v"
T_size_tree_Node: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "T_size_tree \<langle>l, x, r\<rangle> = T_size_tree l + T_size_tree r + 1"
minus_int_code_2: fixes l :: "int" shows "0 - l = - l"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> s) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s \<and> s\<lparr>a\<rparr> = b)"
surjective: fixes r :: "'a gState\<^sub>I_scheme" shows "r = \<lparr>gState.vars = gState.vars r, channels = gState.channels r, timeout = timeout r, procs = procs r, handshake = handshake r, hsdata = hsdata r, exclusive = exclusive r, else = else r, \<dots> = gState\<^sub>I.more r\<rparr>"
a1: shows "(\<forall>va. virt_to_phys va (mmu s2) (mem s1) = virt_to_phys va (mmu s2) (mem s2)) \<and> (\<forall>pa. (\<exists>va b. virt_to_phys va (mmu s2) (mem s2) = Some (pa, b) \<and> mmu_readable (get_acc_flag b) 10) \<longrightarrow> mem_equal s1 s2 pa) \<and> mmu s1 = mmu s2 \<and> virt_to_phys va (mmu s2) ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) = Some (pa, b) \<and> mmu_readable (get_acc_flag b) 10"
of_nat_0: shows "of_nat 0 = (0::'b)"
assms_2: shows "p \<in> subterms (\<Gamma> pn)"
diff: fixes s :: "'a"   and S :: "'a set"   and t :: "'a" assumes "s \<in> group_closure S"   and "t \<in> group_closure S" shows "s - t \<in> group_closure S"
distinct_5: fixes x21 :: "dB"   and x22 :: "dB"   and x3 :: "dB" shows "x21 \<degree> x22 \<noteq> Abs x3"
p0: fixes p :: "nat"   and pm :: "nat" assumes "fixed_root p pm" shows "p \<noteq> 0"
simps_3: fixes uu :: "nat \<times> nat" shows "project_operation (SAT_PUSH' uu) = PUSH"
non_volatile_writes_unshared_append: fixes \<S> :: "nat \<Rightarrow> bool option"   and xs :: "'b memref list"   and ys :: "'b memref list" shows "non_volatile_writes_unshared \<S> (xs @ ys) = (non_volatile_writes_unshared \<S> xs \<and> non_volatile_writes_unshared (share xs \<S>) ys)"
cmp_in_hom'_1: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : map\<^sub>0 (src\<^sub>C \<nu>) \<rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>C \<mu>)\<guillemotright>"
satisfies_bounded_nFAll: fixes \<AA> :: "'i"   and b :: "'k"   and \<phi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b nFAll b \<phi>) = (\<AA> \<Turnstile>\<^sub>b FAll b \<phi>)"
isin_case: fixes b :: "bool"   and m :: "'b \<Rightarrow> 'b trie option"   and xs :: "'b list" shows "Trie_Fun.isin (trie.Nd b m) xs = (case xs of [] \<Rightarrow> b | x # ys \<Rightarrow> case m x of None \<Rightarrow> False | Some t \<Rightarrow> Trie_Fun.isin t ys)"
interleaves_all_nil: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list" shows "xs \<simeq> {xs, [], P}"
spmf_conv_measure_spmf: fixes p :: "'h spmf"   and x :: "'h" shows "spmf p x = Sigma_Algebra.measure (measure_spmf p) {x}"
simps_2: fixes l :: "('b \<times> 'c) tree"   and m :: "'b"   and uu :: "'c"   and r :: "('b \<times> 'c) tree" shows "Leftist_Heap.linorder_class.heap \<langle>l, (m, uu), r\<rangle> = (Ball (Tree2.set_tree l \<union> Tree2.set_tree r) ((\<le>) m) \<and> Leftist_Heap.linorder_class.heap l \<and> Leftist_Heap.linorder_class.heap r)"
jonsson3a: shows "(\<forall>x. x \<rhd> (1::'a) \<rhd> (1::'a) = x) = (\<forall>x. (1::'a) \<lhd> ((1::'a) \<lhd> x) = x)"
e2xs_xs: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2xs (encode_config ((f, xs, ls) # ss, rv)) = Partial_Recursive.list_encode xs"
preserves_seq: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.seq g f" shows "seq (fst g \<star> snd g) (fst f \<star> snd f)"
cont_compose: fixes c :: "'c \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'c" assumes "cont c"   and "cont f" shows "cont (\<lambda>x. c (f x))"
that: shows "c \<subseteq>\<^sub>g Seq c1 c2"
distinct_11: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x13 :: "floatarith"   and x14 :: "form"   and x71 :: "form"   and x72 :: "form" shows "Bound x11 x12 x13 x14 \<noteq> Disj x71 x72"
max_def: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" shows "ordered_powerprod_lin.max a b = (if a \<preceq> b then b else a)"
psubstt_lift_1: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b FOL_Fitting.term" shows "psubstt f (liftt t) = liftt (psubstt f t)"
unionFinestPart: fixes X :: "'c set set" shows "{Y |Y. \<exists>x. Y \<in> finestpart x \<and> x \<in> X} = \<Union> (finestpart ` X)"
wf_mbuf2_add: fixes i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and buf :: "event_data option list set list \<times> event_data option list set list"   and ja' :: "nat"   and xs :: "event_data option list set list"   and jb' :: "nat"   and ys :: "event_data option list set list" assumes "wf_mbuf2 i ja jb P Q buf"   and "list_all2 P [ja..<ja'] xs"   and "list_all2 Q [jb..<jb'] ys"   and "ja \<le> ja'"   and "jb \<le> jb'" shows "wf_mbuf2 i ja' jb' P Q (mbuf2_add xs ys buf)"
min: shows "min (X1 * Y1) (min (X1 * Y2) (min (X2 * Y1) (X2 * Y2))) \<le> x * y"
poly_deg_restrict_indets_le: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly_deg (restrict_indets p) \<le> poly_deg p"
simps_1: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'a" shows "foldl f a [] = a"
floor_fl_def: shows "floor_fl \<equiv> map_fun real_of_float float_of (\<lambda>x. real_of_int \<lfloor>x\<rfloor>)"
cl_def: fixes X :: "'a set" shows "cl X \<equiv> {x \<in> carrier. rank_of (insert x X) = rank_of X}"
update_element_type_det: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type"   and \<rho>\<^sub>1 :: "'b OCL_Types.type"   and \<rho>\<^sub>2 :: "'b OCL_Types.type" assumes "update_element_type \<tau> \<sigma> \<rho>\<^sub>1"   and "update_element_type \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
inside_no_overlap: fixes S :: "'a set" shows "inside S \<inter> S = {}"
fls_base_factor_to_fps_left_inverse: fixes f :: "'a fls"   and x :: "'a" shows "fls_base_factor_to_fps (fls_left_inverse f x) = fps_left_inverse (fls_base_factor_to_fps f) x"
fp_def: shows "fp = (\<lambda>w. f w powr of_int n)"
frange_iff_vdomain: fixes b :: "V"   and r :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> r) = (\<exists>a\<in>elts (\<D>\<^sub>\<bullet> r). [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r)"
doubleton_eq_iff: fixes a :: "V"   and b :: "V"   and c :: "V"   and d :: "V" shows "(set {a, b} = set {c, d}) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
trans2: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and p :: "'b"   and t :: "'b"   and r :: "'b"   and q :: "'b" assumes "class.interval meets \<I>"   and "meets p t"   and "meets t r"   and "meets r q" shows "\<not> meets p q"
HFun_Sigma_Iff: fixes z :: "name"   and r :: "tm"   and z' :: "name"   and x :: "name"   and y :: "name"   and x' :: "name"   and y' :: "name" assumes "atom z \<sharp> (r, z', x, y, x', y')"   and "atom z' \<sharp> (r, x, y, x', y')"   and "atom x \<sharp> (r, y, x', y')"   and "atom y \<sharp> (r, x', y')"   and "atom x' \<sharp> (r, y')"   and "atom y' \<sharp> r" shows "{} \<turnstile> HFun_Sigma r IFF All2 z r (All2 z' r (SyntaxN.Ex x (SyntaxN.Ex y (SyntaxN.Ex x' (SyntaxN.Ex y' (Var z EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y') AND OrdP (Var x) AND OrdP (Var x') AND (Var x NEQ Var x' OR Var y EQ Var y')))))))"
primerec_rec_eq_2: shows "primerec rec_eq (Suc (Suc 0))"
ide_dom: fixes f :: "'a" assumes "Right_a.arr f" shows "Right_a.ide (Right_a.dom f)"
chine_in_hom: shows "\<guillemotleft>\<eta>.chine : \<eta>.dom.apex \<rightarrow>\<^sub>C \<eta>.cod.apex\<guillemotright>"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>l1. P [] l1"   and "\<And>h t. P (h # t) []"   and "\<And>x l1 y l2. \<lbrakk>P l1 l2; P (x # l1) l2\<rbrakk> \<Longrightarrow> P (x # l1) (y # l2)" shows "P a0 a1"
f: shows "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> x"
ST: fixes x :: "'a" assumes "x \<in> W" shows "(x \<in> S) = (x \<in> T)"
last_in_set: fixes as :: "'x list" assumes "as \<noteq> []" shows "last as \<in> set as"
vc_complete: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} c { Q}" shows "\<exists>C. vc C Q \<and> (\<forall>s. P s \<longrightarrow> pre C Q s) \<and> strip C = c"
simps_1: fixes start :: "'b word"   and "end" :: "'b word" shows "wordinterval_to_set (WordInterval start end) = {start..end}"
iwlsFSb_termFSbImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e" assumes "iwlsFSb MOD" shows "termFSbImorph h hA MOD = ipresCons h hA MOD"
weakSimI2: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>'' \<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; \<alpha> \<noteq> \<tau>\<rbrakk> \<Longrightarrow> \<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>'' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>'', P', Q') \<in> Rel)"   and "\<And>Q'. \<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q' \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
slp_of_fa_cnst_slp_of_fa_rev_cnst: fixes Cnst :: "'a"   and Cnst' :: "floatarith"   and M :: "('a, nat) mapping"   and slp :: "floatarith list" shows "slp_of_fa_rev_cnst Cnst Cnst' M slp (length slp) = (let (M, slp') = slp_of_fa_cnst Cnst Cnst' M (rev slp) in (M, rev slp', length slp'))"
P2: shows "\<lfloor>\<lambda>w. \<exists>x. (Necessary x \<^bold>\<and> Abstract x) w\<rfloor>"
sin_product_formula_complex: fixes z :: "complex" shows "(\<lambda>n. complex_of_real pi * z * (\<Prod>k = 1..n. 1 - z\<^sup>2 / (of_nat k)\<^sup>2)) \<longlonglongrightarrow> sin (complex_of_real pi * z)"
simps: fixes ass :: "'d \<Rightarrow> 'e \<Rightarrow> 'a + 'c"   and i :: "'d"   and cs :: "'b"   and s :: "'e" shows "assert_cs ass i (cs, s) = (case ass i s of Inl x \<Rightarrow> Inl x | Inr s' \<Rightarrow> Inr (cs, s'))"
induct: fixes S :: "'a set"   and t :: "real"   and x1 :: "'a"   and x2 :: "(real \<times> 'a) stream"   and P :: "'a \<Rightarrow> (real \<times> 'a) stream \<Rightarrow> bool" assumes "trace_in S t x1 x2"   and "\<And>t' s s' \<omega>. \<lbrakk>t < t'; s \<in> S\<rbrakk> \<Longrightarrow> P s ((t', s') ## \<omega>)"   and "\<And>t' s' \<omega> s. \<lbrakk>t' \<le> t; trace_in S t s' \<omega>; P s' \<omega>\<rbrakk> \<Longrightarrow> P s ((t', s') ## \<omega>)" shows "P x1 x2"
cases: fixes a1 :: "'a list"   and a2 :: "('b, 'c) sequent" assumes "(a1, a2) \<in> upRules" obtains c :: "('b, 'c) sequent" and R :: "'b" and Fs :: "('b, 'c) form list" and ps :: "'a list" where "a1 = ps" and "a2 = c" and "ModalSequents.mset c = \<LM> Compound R Fs  \<RM>" and "ps \<noteq> []"
sup_aci_1: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<union># y = y \<union># x"
MakeCatCompDef: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and g :: "'h" shows "f \<approx>>\<^bsub>MakeCat C\<^esub> g = f \<approx>>\<^bsub>C\<^esub> g"
splitAt_dist_ram20: fixes vs :: "'c list"   and a :: "'c list"   and ram1 :: "'c"   and b :: "'c list"   and ram2 :: "'c"   and c :: "'c list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "c = snd (splitAt ram2 vs)"
simps_4: fixes s :: "'a rexp"   and t :: "'a rexp" shows "rexp_ewp (s +\<^sub>r t) = (rexp_ewp s \<or> rexp_ewp t)"
pair_fin_digraph_axioms_def: fixes G :: "'b pair_pre_digraph" shows "pair_fin_digraph_axioms G \<equiv> finite (pverts G) \<and> finite (parcs G)"
simps_1: fixes e :: "(name, nat) finfun"   and t :: "trm"   and u :: "trm" shows "eval_fmla e (t EQ u) = (eval_trm e t = eval_trm e u)"
union_part_equivp: fixes R :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "part_equiv R" shows "part_equiv (per_union R a b)"
weak_supplementation: fixes x :: "i"   and y :: "i" assumes "EM"   and "PP x y" shows "\<exists>z. P z y \<and> \<not> O z x"
Constrains_weaken: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> A Co A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B Co B'"
msteps_msteps_stateless: fixes w :: "(char list \<times> 'b list) prefix"   and st :: "'b mstate" shows "fst (msteps w st) = msteps_stateless w st"
rep_eq: fixes x :: "perm"   and xa :: "'b set['c]" shows "set_bset (x \<bullet> xa) = x \<bullet> set_bset xa"
lnull_wit: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" shows "lnull (wit R xs) = lnull xs"
restriction_is_subrel: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "P || X \<subseteq> P"
noVal_def: fixes K :: "'state \<Rightarrow> bool"   and v :: "'value" shows "noVal K v \<equiv> \<forall>s a ou s'. reachNT s \<and> K s \<and> step s a = (ou, s') \<and> \<phi> (Trans s a ou s') \<longrightarrow> f (Trans s a ou s') \<noteq> v"
fail_env_def: fixes fail :: "'f" shows "fail_env fail = EnvT (\<lambda>r. fail)"
b_ge: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "0 < k"   and "j \<le> k"   and "i < j" shows "k - 1 \<le> \<Sqinter> (b k (j, i))"
integral_norm_bound: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" shows "norm (integral\<^sup>L M f) \<le> LINT x|M. norm (f x)"
log_base_powr: fixes a :: "real"   and b :: "real"   and x :: "real" assumes "a \<noteq> 0" shows "log (a powr b) x = log a x / b"
eq_LNil_conv_lnull: fixes w :: "'b llist" shows "(w = <>) = lnull w"
integrable_cong_AE_imp: fixes M :: "'c measure"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M g"   and "f \<in> borel_measurable M"   and "AE x in M. g x = f x" shows "integrable M f"
pmf_embed_pmf: fixes f :: "'h \<Rightarrow> real"   and x :: "'h" assumes "\<And>x. 0 \<le> f x"   and "(\<Sum>\<^sup>+ x. ennreal (f x)) = 1" shows "pmf (embed_pmf f) x = f x"
stepsST_STCal_steps: fixes P :: "('a, 'b) Proc"   and Source :: "'a processCalculus"   and Target :: "'b processCalculus"   and P' :: "('a, 'b) Proc" shows "P \<longmapsto>STCal Source Target* P' = ((\<exists>S S'. P = SourceTerm S \<and> P' = SourceTerm S' \<and> S \<longmapsto>Source* S') \<or> (\<exists>T T'. P = TargetTerm T \<and> P' = TargetTerm T' \<and> T \<longmapsto>Target* T'))"
tanh_altdef: fixes x :: "'a" shows "tanh x = (exp x - exp (- x)) / (exp x + exp (- x))"
iT_Minus_iWeakSince_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<B> t2 k \<ominus> I. Q t2) = (P (k - t1). t1 \<W> t2 I \<down>\<le> k. Q (k - t2))"
maximal_ideal_def: fixes R :: "('c, 'd) Ring_scheme"   and mx :: "'c set" shows "maximal_ideal R mx = (ideal R mx \<and> 1\<^sub>r\<^bsub>R\<^esub> \<notin> mx \<and> {J. ideal R J \<and> mx \<subseteq> J} = {mx, carrier R})"
weakStatImpWeakenStatImp: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<lessapprox><Rel> Q"   and "\<And>\<Psi>' R S \<Psi>''. \<lbrakk>(\<Psi>', R, S) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', R, S) \<in> Rel" shows "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<Rel> Q"
domain_eq: shows "Domainp pcr_mini_alg_unique = (\<lambda>r. ma_coeff r = 0 \<and> ma_base r = 0 \<or> ma_coeff r \<noteq> 0 \<and> prime_product (ma_base r))"
rMulLC: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r (y \<cdot>\<^sub>r z) = y \<cdot>\<^sub>r (x \<cdot>\<^sub>r z)"
dec_dom': fixes u :: "'b list"   and G :: "'b list set" assumes "u \<in> \<langle>G\<rangle>" shows "Dec G u \<in> lists G"
t3_space: fixes S :: "'a set"   and y :: "'a" assumes "closed S"   and "y \<notin> S" shows "\<exists>U V. open U \<and> open V \<and> y \<in> U \<and> S \<subseteq> V \<and> U \<inter> V = {}"
reduced_GB_lc: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and g :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.is_reduced_GB G"   and "g \<in> G" shows "punit.lc g = (1::'c)"
unitary_def: fixes M :: "complex mat" shows "unitary M \<equiv> M\<^sup>\<dagger> * M = 1\<^sub>m (dim_col M) \<and> M * M\<^sup>\<dagger> = 1\<^sub>m (dim_row M)"
rule0: fixes a :: "'g" shows "a \<equiv> a + (0::'g)"
"10_1": shows "set s\<^sub>2 \<subseteq> invisible"
mkarr_in_hom: fixes t :: "'c local.term" assumes "Arr t" shows "\<guillemotleft>mkarr t : mkarr (Dom t) \<rightarrow> mkarr (Cod t)\<guillemotright>"
distinct_71: fixes x41 :: "stmt"   and x42 :: "stmt"   and x101 :: "stmt"   and x102 :: "stmt" shows "x41;; x42 \<noteq> x101 Finally x102"
piecewise_continuous_on_AE_boundedE: fixes a :: "real"   and I :: "real \<Rightarrow> real set" assumes "\<And>k. piecewise_continuous_on a k (I k) f" obtains B :: "real \<Rightarrow> real" where "\<And>k. AE x\<in>{a..k} in lebesgue. cmod (f x) \<le> B k"
induct: fixes P :: "aexp \<Rightarrow> aexp \<Rightarrow> bool"   and a0 :: "aexp"   and a1 :: "aexp" assumes "\<And>i\<^sub>1 i\<^sub>2. P (N i\<^sub>1) (N i\<^sub>2)"   and "\<And>i v. P (N i) (V v)"   and "\<And>i v va. P (N i) (Plus v va)"   and "\<And>v i. P (V v) (N i)"   and "\<And>v va i. P (Plus v va) (N i)"   and "\<And>v va. P (V v) (V va)"   and "\<And>v va vb. P (V v) (Plus va vb)"   and "\<And>v va vb. P (Plus v va) (V vb)"   and "\<And>v va vb vc. P (Plus v va) (Plus vb vc)"   and "\<And>va v. P (V va) (V v)"   and "\<And>va vb v. P (Plus va vb) (V v)"   and "\<And>vb v va. P (V vb) (Plus v va)"   and "\<And>vb vc v va. P (Plus vb vc) (Plus v va)" shows "P a0 a1"
cs_path_swap_le: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and n :: "nat"   and k :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> =\<^bsub>n\<^esub> \<pi>'"   and "k \<le> n" shows "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k"
ss: shows "{a. \<exists>b\<in>elts A. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r} \<subseteq> vfst ` elts r"
gen_entries_def: fixes kvts :: "(('c \<times> 'd) \<times> ('c, 'd) rbt) list"   and t :: "('c, 'd) rbt" shows "RBT_Impl.gen_entries kvts t = RBT_Impl.entries t @ concat (map (\<lambda>(kv, t). kv # RBT_Impl.entries t) kvts)"
rep_in_ARR: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f" shows "free_monoidal_category.rep C f \<in> f"
LIMSEQ_inverse_realpow_zero: fixes x :: "real" assumes "1 < x" shows "(\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
tt_less: fixes u :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" assumes "u \<in> keys p"   and "u \<prec> v" shows "punit.tt p \<prec> v"
2: fixes x :: "ennreal"   and y :: "ennreal"   and a :: "real" assumes "x = y + ennreal a * x"   and "x \<noteq> \<top>"   and "a \<in> {0<..<1}" shows "x = y / ennreal (1 - a)"
order_refl: fixes x :: "'a word" shows "(x::'a word) \<le>s x"
floor_log_nat_eq_powr_iff: fixes b :: "nat"   and k :: "nat"   and n :: "nat" assumes "2 \<le> b"   and "0 < k" shows "(\<lfloor>log (real b) (real k)\<rfloor> = int n) = (b ^ n \<le> k \<and> k < b ^ (n + 1))"
injective_conv_univalent: fixes x :: "'a" shows "injective x = coreflexive (x\<^sup>T[[x\<^sup>T]])"
blocking_on_allocation: fixes ds :: "'d set"   and X :: "'x set"   and h :: "'h"   and X'' :: "'x set" assumes "blocking_on ds X h X''" shows "inj_on Xd X''"
psubst_ty_lemma: fixes X :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list"   and T' :: "ty"   and T :: "ty" assumes "X \<sharp> \<theta>" shows "\<theta><T[X::=T']> = \<theta><T>[X::=\<theta><T'>]"
Weaken: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and \<Gamma>' :: "'b formula set" assumes "\<Gamma> \<turnstile> F"   and "\<Gamma> \<subseteq> \<Gamma>'" shows "\<Gamma>' \<turnstile> F"
cases: fixes a :: "fmla" assumes "a \<in> special_axioms" obtains A :: "fmla" and i :: "name" and x :: "trm" where "a = A(i::=x) IMP exi i A"
extf_ext_irrefl_before_trans: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and f :: "'b" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf" shows "ext_irrefl_before_trans (extf f)"
induced_arrowI_2: fixes \<chi>' :: "'b \<Rightarrow> 'a"   and a' :: "'a" assumes "\<chi>' \<in> D1.cones a'" shows "D1.cones_map (\<pi>1.induced_arrow a' \<chi>') \<pi>1 = \<chi>'"