hcju/isabelleps huggingface dataset:
strict_trans1: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "b \<le>lex a"   and "c <lex b" shows "c <lex a"
intro: fixes x :: "'d \<Longrightarrow> 'e" assumes "weak_lens x"   and "mwb_lens_axioms x" shows "mwb_lens x"
all: shows "\<forall>V\<in>fv e. \<Gamma> V = Some Low"
add_order_preserving_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and u :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq a b" shows "less_eq (plus u a) (plus u b)"
cCons_append_Cons_eq: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "x ## xs @ y # ys = x # xs @ y # ys"
GHK_def: shows "GHK = G\<lparr>carrier := H \<inter> K\<rparr>"
B: shows "Mapping.keys m = set_pmf (cond_pmf p A)"
finalStateCharacterization: fixes state0 :: "State"   and F0 :: "Literal list list"   and state :: "State"   and decisionVars :: "nat set" assumes "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "isFinalState state F0 decisionVars" shows "\<not> formulaFalse F0 (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state)) \<or> formulaFalse F0 (elements (getM state)) \<and> decisions (getM state) = []"
stream_rel_shift: fixes u\<^sub>1 :: "'b list"   and v\<^sub>1 :: "'c list"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and u\<^sub>2 :: "'b stream"   and v\<^sub>2 :: "'c stream" assumes "length u\<^sub>1 = length v\<^sub>1" shows "stream_all2 P (u\<^sub>1 @- u\<^sub>2) (v\<^sub>1 @- v\<^sub>2) = (list_all2 P u\<^sub>1 v\<^sub>1 \<and> stream_all2 P u\<^sub>2 v\<^sub>2)"
foo: shows "p * A $ a $ j + q * A $ b $ j = d \<and> d dvd A $ a $ j \<and> d dvd A $ b $ j \<and> (\<forall>d'. d' dvd A $ a $ j \<and> d' dvd A $ b $ j \<longrightarrow> d' dvd d) \<and> d * u = - A $ b $ j \<and> d * v = A $ a $ j"
proved_iff_proved_PfP: fixes \<alpha> :: "fm" shows "{} \<turnstile> \<alpha> = {} \<turnstile> PfP \<guillemotleft>\<alpha>\<guillemotright>"
grid_child: fixes d :: "nat"   and ds :: "nat set"   and p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and dir :: "dir" assumes "d \<in> ds"   and "p \<in> grid (child b dir d) ds" shows "p \<in> grid b ds"
absorb_iff2: fixes a :: "'a"   and b :: "'a" shows "less_eq a b = (sup a b = b)"
simps_8: fixes f :: "'f"   and c :: "'d \<Rightarrow> ('d, 'e, 'g) com" shows "mark_guards f (DynCom c) = DynCom (\<lambda>s. mark_guards f (c s))"
counit_in_hom_2: shows "\<guillemotleft>\<epsilon> : f \<star> g \<Rightarrow> trg f\<guillemotright>"
canc_unitr: fixes x :: "'a"   and e :: "'a" assumes "D x e"   and "x \<oplus> e = x" shows "e \<in> E"
extends_to_weak_composition_with_homs: shows "weak_composition_with_homs (\<cdot>) (\<star>) some_src some_trg"
neg_power_inverse: fixes g :: "'a"   and x :: "nat" assumes "g \<in> carrier \<G>"   and "x < order \<G>" shows "g [^] (order \<G> - x) = inv (g [^] x)"
cf'_alt: shows "cf' = cf.augment_cf (set p) (resCap p)"
delta_2: shows "delta \<longlonglongrightarrow> 0"
isotone_def: fixes f :: "'a \<Rightarrow> 'a" shows "isotone f \<equiv> \<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y"
vec_mat_Tensor_vector_id: fixes v :: "'a list list" shows "vec_mat_Tensor [id] v = v"
inorder_rotateR: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" assumes "l \<noteq> \<langle>\<rangle>" shows "Tree2.inorder (rotateR l a r) = Tree2.inorder l @ a # Tree2.inorder r"
eqvtI: fixes f :: "'a" assumes "\<And>p. p \<bullet> f \<equiv> f" shows "eqvt f"
open_prod_elim: fixes S :: "('b \<times> 'c) set"   and x :: "'b \<times> 'c" assumes "open S"   and "x \<in> S" obtains A :: "'b set" and B :: "'c set" where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
mono_implies_arr: fixes f :: "'a" assumes "local.mono f" shows "arr f"
absolutely_integrable_on_scaleR_iff: fixes S :: "'a set"   and c :: "real"   and f :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on S) = (c = 0 \<or> f absolutely_integrable_on S)"
correctCompositionVARSYSTEM_def: shows "correctCompositionVARSYSTEM \<equiv> \<forall>S. correctCompositionVAR S"
union: fixes A :: "'a word set"   and B :: "'a word set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "signed.Max (A \<union> B) = signed.max (signed.Max A) (signed.Max B)"
learn_bcE_2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option" assumes "learn_bc \<psi> U s"   and "f \<in> U" shows "\<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. \<psi> (the (s (f \<triangleright> n))) = f"
of_int_floor_le: fixes x :: "'b" shows "of_int \<lfloor>x\<rfloor> \<le> x"
Max_eqI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq y x"   and "x \<in> A" shows "linorder.Max less_eq A = x"
del_list_sorted1: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and x :: "'b" assumes "sorted (xs @ a # ys)"   and "a \<le> x" shows "del_list x (xs @ a # ys) = xs @ del_list x (a # ys)"
wf_tcsigs_iff_exe_wf_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss" shows "wf_tcsigs (set cs) (translate_ars arss) = exe_wf_tcsigs cs arss"
group_BijGroup: fixes S :: "'e set" shows "Group.group (BijGroup S)"
filter_sort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "filter P (linorder.sort_key less_eq f xs) = linorder.sort_key less_eq f (filter P xs)"
subsumes_def: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "subsumes C D = (\<exists>\<sigma>. C \<cdot> \<sigma> \<subseteq># D)"
item_rhs_finished_item: fixes \<alpha> :: "symbol list" shows "item_rhs (finished_item \<alpha>) = \<alpha>"
word_suicide_r: fixes X :: "'a list set"   and Y :: "'a list set" assumes "\<not> l_ewp X"   and "Y \<noteq> {}" shows "\<forall>w\<in>Y. \<exists>n. w \<notin> Y \<cdot> X\<^bsup>Suc n\<^esup>"
sup_inf_distrib2: fixes y :: "'b multiset"   and z :: "'b multiset"   and x :: "'b multiset" shows "y \<inter># z \<union># x = y \<union># x \<inter># (z \<union># x)"
LstSeq_insf: fixes s :: "hf"   and k :: "hf"   and z :: "hf"   and y :: "hf" assumes "LstSeq s k z" shows "LstSeq (insf s (succ k) y) (succ k) y"
interior_of_unique: fixes T :: "'c set"   and S :: "'c set"   and X :: "'c topology" assumes "T \<subseteq> S"   and "openin X T"   and "\<And>T'. \<lbrakk>T' \<subseteq> S; openin X T'\<rbrakk> \<Longrightarrow> T' \<subseteq> T" shows "X interior_of S = T"
uncurry_split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and p :: "'b \<times> 'c" shows "P (uncurry f p) = (\<forall>x1 x2. p = (x1, x2) \<longrightarrow> P (f x1 x2))"
monom_mult_zero_right: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and c :: "'e"   and t :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.monom_mult pair_of_term term_of_pair c t 0 = 0"
coerce_eq_prj: shows "COERCE(udom, 'a) = PRJ('a)"
inj_on_eq_iff: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d"   and y :: "'d" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A" shows "(f x = f y) = (x = y)"
A_updates_dma: shows "\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x \<noteq> dma mem\<^sub>1 x \<longrightarrow> \<not> var_asm_not_written mds x"
induct: fixes P :: "stack_elem list \<Rightarrow> bool"   and a0 :: "stack_elem list" assumes "P []"   and "\<And>x S. P S \<Longrightarrow> P (Arg x # S)"   and "\<And>e1 e2 S. P (Alts e1 e2 # S)"   and "\<And>x S. P (Upd x # S)"   and "\<And>x S. P (Dummy x # S)" shows "P a0"
set_quicksort_acc: fixes ac :: "'a list"   and xs :: "'a list" shows "set (quicksort_acc ac xs) = set ac \<union> set xs"
scale_zero_right: fixes a :: "'a" shows "a * (0::'a) = (0::'a)"
2: shows "r !! i \<in> successors A ((p ## r) !! i)"
set_Keys_to_list: fixes ps :: "('a \<Rightarrow>\<^sub>0 'c) list" shows "set (punit.Keys_to_list ps) = Keys (set ps)"
LNil_eq_ltake_iff: fixes n :: "enat"   and xs :: "'c llist" shows "(LNil = ltake n xs) = (xs = LNil \<or> n = 0)"
conjunction_sup: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "conj_distrib (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) (d::'b::refinement_lattice) \<sqsubseteq> c \<squnion> d"
def_hashmap_size_sum_def: shows "def_hashmap_size = (\<lambda>_. def_hashmap_size TYPE('a) + def_hashmap_size TYPE('b))"
subst_preserves_beta2: fixes r :: "dB"   and s :: "dB"   and t :: "dB"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta> s" shows "t[r/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t[s/i]"
p: shows "b = a # b' \<and> branch b' (deeptree n)"
sum_nodes: fixes t :: "int tree"   and i :: "int"   and j :: "int"   and f :: "int \<Rightarrow> nat" assumes "inorder t = [i..j]" shows "(\<Sum>xy\<in>nodes t. f (fst xy)) = sum f {i..j}"
conj_idem: fixes P :: "'b upred" shows "(P \<and> P) = P"
Gauss_Jordan_column_k_iarrays_def: fixes A' :: "nat \<times> 'a iarray iarray"   and k :: "nat" shows "Gauss_Jordan_column_k_iarrays A' k = (let A = snd A'; i = fst A' in if vector_all_zero_from_index (i, column_iarray k A) \<or> i = nrows_iarray A then (i, A) else (Suc i, Gauss_Jordan_in_ij_iarrays A i k))"
isMinim_def: fixes A :: "'b set"   and b :: "'b" shows "s.isMinim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (b, a) \<in> s)"
B: shows "path_image p \<subseteq> cball 0 B"
two: fixes I :: "'c"   and a :: "'b"   and b :: "'b"   and J :: "'c"   and c :: "'b"   and d :: "'b" assumes "Q I a b"   and "Q J c d"   and "a = b \<and> b = c \<and> c = d \<or> a = b \<and> b \<noteq> c \<and> c = d" shows "P I J"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "methodDef"   and a3 :: "nat" assumes "a1 \<turnstile> a2 OK IN a3" obtains CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and CDef :: "classDef" and D :: "nat" and mDef :: "methodDef" and m :: "nat" and C0 :: "nat" and Cxs :: "varDef list" and e0 :: "exp" and Cs :: "nat list" and xs :: "nat list" and \<Gamma> :: "nat \<Rightarrow> nat option" and E0 :: "nat" where "a1 = CT" and "a2 = mDef" and "a3 = C" and "CT C = Some CDef" and "cName CDef = C" and "cSuper CDef = D" and "mName mDef = m" and "lookup (cMethods CDef) (\<lambda>md. mName md = m) = Some mDef" and "mReturn mDef = C0" and "mParams mDef = Cxs" and "mBody mDef = e0" and "varDefs_types Cxs = Cs" and "varDefs_names Cxs = xs" and "\<Gamma> = [xs [\<mapsto>] Cs, this \<mapsto> C]" and "CT;\<Gamma> \<turnstile> e0 : E0" and "CT \<turnstile> E0 <: C0" and "\<forall>Ds D0. mtype(CT,m,D) = Ds \<rightarrow> D0 \<longrightarrow> Cs = Ds \<and> C0 = D0"
indRelLSTPO_impl_SRel_and_TRel_weakly_reflect_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelLSTPO SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs SRel SWB"
mem_rel_interior_ball: fixes x :: "'b"   and S :: "'b set" shows "(x \<in> rel_interior S) = (x \<in> S \<and> (\<exists>e>0. ball x e \<inter> affine hull S \<subseteq> S))"
blinding_of_blindable_simps_4: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "('c, 'd) blindable\<^sub>m"   and x :: "'d blindable\<^sub>h" shows "blinding_of_blindable h bo z (Blinded x) = (z = Blinded x)"
simps: fixes v :: "'a mod_ring poly"   and w :: "'a mod_ring poly"   and d :: "nat"   and res :: "(nat \<times> 'a mod_ring poly) list" shows "local.dist_degree_factorize_main v w d res = (if v = 1 then res else if degree v < d + d then (degree v, v) # res else let w = w ^ CARD('a) mod v; d = Suc d; gd = gcd (w - monom 1 1) v in if gd = 1 then local.dist_degree_factorize_main v w d res else let v' = v div gd in local.dist_degree_factorize_main v' (w mod v') d ((d, gd) # res))"
hC: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) C = Cv"
\<A>: shows "\<forall>l l'. l \<noteq> l' \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec"
weak_arrow_of_homs_axioms: shows "weak_arrow_of_homs (\<cdot>\<^sub>C) src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) src\<^sub>D trg\<^sub>D G"
listgcd_def: fixes xs :: "'a list" shows "listgcd xs = foldr gcd xs (0::'a)"
R_def: fixes h :: "'g"   and w :: "'h"   and e :: "'e" shows "\<Sigma>0.R h w e = init0 h w \<bind> (\<lambda>(r, a). response0 r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
depthEqvt: fixes p :: "(name \<times> name) list"   and P :: "pi" shows "depth (p \<bullet> P) = depth P"
\<phi>_zero: fixes x :: "real"   and i :: "int"   and l :: "nat" assumes "x \<notin> {real_of_int (i - 1) / 2 ^ (l + 1)<..<real_of_int (i + 1) / 2 ^ (l + 1)}" shows "\<phi> (l, i) x = 0"
desargues_prop_def: shows "desargues_prop \<equiv> \<forall>A B C A' B' C' P. are_perspective_from_point A B C A' B' C' P \<longrightarrow> are_perspective_from_line A B C A' B' C'"
sat_preserving_inference_system_axioms: shows "sat_preserving_inference_system \<Gamma>"
mn1f: shows "eval (Mn 1 f) [b] \<down>= (LEAST z. eval f [z, b] \<down>= 0)"
simps_2: fixes i :: "nat"   and a :: "'c" shows "list_insert [] i a = [a]"
leD: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and y :: "'k"   and x :: "'k" assumes "class.order less_eq less"   and "less_eq y x" shows "\<not> less x y"
S_contractE: fixes r :: "comb"   and P :: "bool" assumes "S \<rightarrow>\<^sup>1 r" shows "P"
no_quad: assumes "\<And>v u. \<lbrakk>v \<in> V; u \<in> V; v \<noteq> u\<rbrakk> \<Longrightarrow> \<exists>!n. adjacent v n \<and> adjacent u n" shows "\<nexists>v1 v2 v3 v4. v2 \<noteq> v4 \<and> v1 \<noteq> v3 \<and> adjacent v1 v2 \<and> adjacent v2 v3 \<and> adjacent v3 v4 \<and> adjacent v4 v1"
rep_eq: fixes x :: "'b \<Rightarrow> 'b" shows "Rep_blinop (Blinop x) = Blinfun x"
ordinal_exp_left_cancel_less: fixes w :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "oSuc 0 < w" shows "(w ** x < w ** y) = (x < y)"
dg_prod_is_arrD_7: fixes a :: "V"   and b :: "V"   and f :: "V"   and i :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod J \<AA>\<^esub> b"   and "i \<in>\<^sub>\<circ> J" shows "f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>"
assms_1: shows "{Deliver (i, Insert e k), Deliver (i', Insert e' (Some m))} \<subseteq> set (history j)"
IC2_1: shows "callee_invariant_on (callee_of_core core2) IC2 (\<I>E \<oplus>\<^sub>\<I> (\<I>CA \<oplus>\<^sub>\<I> \<I>CU))"
assms: shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, a) = (k, b)"
f_2: shows "\<D>\<^sub>\<circ> f = I"
move_graph_a_def: fixes n :: "char list"   and l :: "location"   and l' :: "location"   and g :: "igraph" shows "move_graph_a n l l' g \<equiv> Lgraph (gra g) (if n \<in> agra g l \<and> n \<notin> agra g l' then (agra g)(l := agra g l - {n}, l' := insert n (agra g l')) else agra g) (cgra g) (lgra g)"
cost_def: fixes R :: "'a set"   and i :: "nat" shows "cost R i = w i / real (card (S i \<inter> R))"
nat_pow_pow: fixes x :: "nat \<times> nat"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier S" shows "(x [^]\<^bsub>S\<^esub> n) [^]\<^bsub>S\<^esub> m = x [^]\<^bsub>S\<^esub> (n * m)"
asymptotics7: shows "\<forall>\<^sub>F x in at_top. hb / ln x powr (1 + e) < (1 - b) / 2"
continuous_on_of_real: fixes s :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. of_real (g x))"
hyps: shows "a<y>.(<\<nu>x>P) \<longmapsto> \<alpha> \<prec> P'"
group_comm_groupI: assumes "\<And>x y. \<lbrakk>x \<in> carrier \<G>; y \<in> carrier \<G>\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" shows "comm_group \<G>"
gfinprod_mono_neutral_cong_left: fixes A :: "'d set"   and B :: "'d set"   and h :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "A \<subseteq> B"   and "\<And>i. i \<in> B - A \<Longrightarrow> h i = \<one>"   and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"   and "h \<in> B \<rightarrow> carrier G" shows "gfinprod G g A = gfinprod G h B"
inj_on_map_the: fixes D :: "'a set"   and m :: "'a \<Rightarrow> 'b option" assumes "D \<subseteq> dom m"   and "inj_on m D" shows "inj_on (the \<circ> m) D"
map_replicate_const: fixes k :: "'a"   and lst :: "'b list" shows "map (\<lambda>x. k) lst = replicate (length lst) k"
matrix_mul_assoc: fixes A :: "(('a, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('a, 'e) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and C :: "(('a, 'b) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "A ** (B ** C) = A ** B ** C"
simps_2: fixes \<sigma> :: "'tp" shows "TE_resOf (Tag \<sigma>) = \<sigma>"
n_anti_iff: fixes x :: "'a"   and y :: "'a" shows "(!x \<le> !y) = (t y \<le> t x)"
prems_1: shows "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G (np(a)) w)"
IH_2: assumes "l \<turnstile> WHILE b DO c" shows "s\<^sub>2_ = s\<^sub>3_ (< l)"
order_antisym: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "order r"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> x" shows "x = y"
extremed_imp_ex_bound: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "extremed A r"   and "X \<subseteq> A" shows "\<exists>b\<in>A. bound X r b"
relation_of_spec_f_f: fixes y :: "('c, 'd) action"   and x :: "('c, 'd) action"   and a :: "('c, 'd) alpha_rp_scheme"   and b :: "('c, 'd) alpha_rp_scheme" assumes "\<forall>a b. (action.relation_of y \<longrightarrow> action.relation_of x) (a, b)"   and "action.relation_of y\<^sup>f\<^sub>f (a\<lparr>tr := []\<rparr>, b)" shows "action.relation_of x\<^sup>f\<^sub>f (a\<lparr>tr := []\<rparr>, b)"
I1: assumes "Can t" shows "\<lbrace>Inv t\<rbrace> = local.inv \<lbrace>t\<rbrace>"
assms_1: shows "\<forall>t\<in>T. g t \<noteq> 0"
fix_eq: fixes F :: "'e \<rightarrow> 'e" shows "fix\<cdot>F = F\<cdot>(fix\<cdot>F)"
Throw: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} e-\<succ> {\<lambda>Val:a:. abupd (throw a) .; Q\<leftarrow>\<diamondsuit>}" shows "G,A\<turnstile>{Normal P} .Throw e. {Q}"
B_eval: fixes g :: "dB"   and f :: "dB"   and x :: "dB" shows "\<B> \<degree> g \<degree> f \<degree> x \<rightarrow>\<^sub>\<beta>\<^sup>* g \<degree> (f \<degree> x)"
fix_clock_alt_def: fixes x1ba :: "'d state"   and xa :: "'b state \<times> 'c" shows "fix_clock x1ba xa = (case xa of (x3a, x2ba) \<Rightarrow> (update_clock (\<lambda>_. if clock x3a \<le> clock x1ba then clock x3a else clock x1ba) x3a, x2ba))"
construct_rhs_vector_def: fixes p :: "real poly"   and qs :: "real poly list"   and Is :: "nat list list" shows "v_vec p qs Is = vec_of_list (map (\<lambda>I. construct_NofI p (retrieve_polys qs I)) Is)"
conc_trans_additional_6: fixes A :: "'a nres"   and B :: "'a nres"   and C :: "'a nres" assumes "A \<le> B"   and "B \<le> \<Down> Id C" shows "A \<le> C"
resultant_hom: fixes p :: "'a poly poly"   and q :: "'a poly poly" shows "resultant (map_poly (map_poly hom) p) (map_poly (map_poly hom) q) = map_poly hom (resultant p q)"
psimps: fixes C :: "'a set"   and B :: "'a set" assumes "local.mk_linear_orders_dom (C, B)" shows "local.mk_linear_orders C B = (if f B = {} then {[]} else if f B \<subseteq> C then {b # cs |b cs. b \<in> f B \<and> cs \<in> local.mk_linear_orders {} (B - {b})} else let b = SOME x. x \<in> f B - C in {b # cs |cs. cs \<in> local.mk_linear_orders C (B - {b})})"
assoc_simps\<^sub>A\<^sub>W\<^sub>C_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<noteq> null"   and "g \<star> h \<noteq> null" shows "local.dom \<a>[f, g, h] = (f \<star> g) \<star> h"
stalk_is_ring: fixes V :: "'c set" assumes "is_open\<^sub>Y V"   and "f x \<in> V" shows "ring stfx.carrier_stalk stfx.add_stalk stfx.mult_stalk (stfx.zero_stalk V) (stfx.one_stalk V)"
simps_4: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and tst :: "'b"   and c :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_siso tval aval indis (while tst do c) = (PL_Indis.compatTst tval indis tst \<and> PL_Indis.SC_siso tval aval indis c)"
asciis_of_Literal: fixes b0 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and b3 :: "bool"   and b4 :: "bool"   and b5 :: "bool"   and b6 :: "bool"   and s :: "literal" shows "asciis_of_literal (Literal b0 b1 b2 b3 b4 b5 b6 s) = of_char (Char b0 b1 b2 b3 b4 b5 b6 False) # asciis_of_literal s"
to_nat_less_nrows: fixes a :: "'b"   and A :: "(('c, 'd) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "to_nat a < nrows A"
U_Finite: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and S :: "'b set"   and L :: "'b set" assumes "BinPacking U w c S L" shows "finite U"
frontier_of_def: fixes X :: "'b topology"   and S :: "'b set" shows "X frontier_of S \<equiv> X closure_of S - X interior_of S"
Inter_eq: fixes A :: "'a set set" shows "\<Inter> A = {x. \<forall>B\<in>A. x \<in> B}"
merge_def: fixes i :: "nat" shows "local.merge i = {uu_. \<exists>x y n j. uu_ = (x, y) \<and> j < i \<and> (token_run x n \<noteq> token_run y n \<and> rank y n \<noteq> None \<or> y = Suc n) \<and> token_run x (Suc n) = token_run y (Suc n) \<and> token_run x (Suc n) \<notin> F \<and> rank x n = Some j}"
presVar_def: fixes h :: "('c, 'd, 'varSort, 'var, 'e) qTerm set \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'varSort, 'i, 'j, 'var, 'f, 'k, 'l) Recursion.model_scheme" shows "presVar h MOD \<equiv> \<forall>xs x. h (Var xs x) = gVar MOD xs x"
completely_multiplicative_iff_inverse_moebius_mu: fixes f :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f" shows "completely_multiplicative_function f = (\<forall>n. dirichlet_inverse f (1::'b) n = moebius_mu n * f n)"
Class_undefined: fixes a :: "'a" assumes "a \<notin> S" shows "Class a = undefined"
funpow_Suc_right: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c" shows "f ^^ Suc n = f ^^ n \<circ> f"
path_imp_reversepath: fixes g :: "real \<Rightarrow> 'a" assumes "path g" shows "path (reversepath g)"
rel_funI: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "(A ===> B) f g"
distinct_12: fixes x41 :: "'g"   and x42 :: "('e, 'f, 'g) Tree"   and x3 :: "'f" shows "tAct x41 x42 \<noteq> tPred x3"
no_\<tau>moves1: fixes s1' :: "'s1" shows "\<not> s1 -\<tau>1\<rightarrow> s1'"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "Right.ide a"   and "Right.seq f a" shows "Right.comp f a = f"