hcju/isabelleps huggingface dataset:
strong_induct: fixes x1 :: "trm"   and x2 :: "trm"   and P :: "'a \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<longrightarrow>\<^sub>l x2"   and "\<And>x M a b y. \<lbrakk>x \<sharp> y; a \<sharp> y; x \<sharp> M; a \<sharp> b; fic M a\<rbrakk> \<Longrightarrow> P y (Cut <a>.M x.Ax x b) (M[a\<turnstile>c>b])"   and "\<And>a M x y ya. \<lbrakk>x \<sharp> ya; a \<sharp> ya; a \<sharp> M; x \<sharp> y; fin M x\<rbrakk> \<Longrightarrow> P ya (Cut <a>.Ax y a x.M) (M[x\<turnstile>n>y])"   and "\<And>y M N x a b ya. \<lbrakk>b \<sharp> ya; y \<sharp> ya; x \<sharp> ya; a \<sharp> ya; b \<noteq> a; y \<noteq> x; y \<sharp> (M, N); x \<sharp> (N, y); a \<sharp> (M, N, b); b \<sharp> M; y \<noteq> x; b \<noteq> a\<rbrakk> \<Longrightarrow> P ya (Cut <a>.NotR x.M a y.NotL <b>.N y) (Cut <b>.N x.M)"   and "\<And>b a1 M1 a2 M2 N y x ya. \<lbrakk>x \<sharp> ya; y \<sharp> ya; a2 \<sharp> ya; a1 \<sharp> ya; b \<sharp> ya; x \<noteq> y; a2 \<noteq> a1; a2 \<noteq> b; a1 \<noteq> b; b \<sharp> ([a1].M1, [a2].M2, N, a1, a2); y \<sharp> ([x].N, M1, M2, x); x \<sharp> (M1, M2); a1 \<sharp> (M2, N); a2 \<sharp> (M1, N); a1 \<noteq> a2\<rbrakk> \<Longrightarrow> P ya (Cut <b>.AndR <a1>.M1 <a2>.M2 b y.AndL1 x.N y) (Cut <a1>.M1 x.N)"   and "\<And>b a1 M1 a2 M2 N y x ya. \<lbrakk>x \<sharp> ya; y \<sharp> ya; a2 \<sharp> ya; a1 \<sharp> ya; b \<sharp> ya; x \<noteq> y; a2 \<noteq> a1; a2 \<noteq> b; a1 \<noteq> b; b \<sharp> ([a1].M1, [a2].M2, N, a1, a2); y \<sharp> ([x].N, M1, M2, x); x \<sharp> (M1, M2); a1 \<sharp> (M2, N); a2 \<sharp> (M1, N); a1 \<noteq> a2\<rbrakk> \<Longrightarrow> P ya (Cut <b>.AndR <a1>.M1 <a2>.M2 b y.AndL2 x.N y) (Cut <a2>.M2 x.N)"   and "\<And>b a M N1 N2 y x1 x2 ya. \<lbrakk>x2 \<sharp> ya; x1 \<sharp> ya; y \<sharp> ya; a \<sharp> ya; b \<sharp> ya; x2 \<noteq> x1; x2 \<noteq> y; x1 \<noteq> y; a \<noteq> b; b \<sharp> ([a].M, N1, N2, a); y \<sharp> ([x1].N1, [x2].N2, M, x1, x2); x1 \<sharp> (M, N2); x2 \<sharp> (M, N1); a \<sharp> (N1, N2); x1 \<noteq> x2\<rbrakk> \<Longrightarrow> P ya (Cut <b>.OrR1 <a>.M b y.OrL x1.N1 x2.N2 y) (Cut <a>.M x1.N1)"   and "\<And>b a M N1 N2 y x1 x2 ya. \<lbrakk>x2 \<sharp> ya; x1 \<sharp> ya; y \<sharp> ya; a \<sharp> ya; b \<sharp> ya; x2 \<noteq> x1; x2 \<noteq> y; x1 \<noteq> y; a \<noteq> b; b \<sharp> ([a].M, N1, N2, a); y \<sharp> ([x1].N1, [x2].N2, M, x1, x2); x1 \<sharp> (M, N2); x2 \<sharp> (M, N1); a \<sharp> (N1, N2); x1 \<noteq> x2\<rbrakk> \<Longrightarrow> P ya (Cut <b>.OrR2 <a>.M b y.OrL x1.N1 x2.N2 y) (Cut <a>.M x2.N2)"   and "\<And>z N y Pa x M b a c ya. \<lbrakk>y \<sharp> ya; c \<sharp> ya; z \<sharp> ya; a \<sharp> ya; x \<sharp> ya; b \<sharp> ya; y \<noteq> z; y \<noteq> x; c \<noteq> a; c \<noteq> b; z \<noteq> x; a \<noteq> b; z \<sharp> (N, [y].Pa, [x].M, y, x); b \<sharp> ([a].M, [c].N, Pa, c, a); x \<sharp> (N, [y].Pa, y); c \<sharp> (Pa, [a].M, b, a); a \<sharp> ([c].N, Pa); y \<sharp> (N, [x].M)\<rbrakk> \<Longrightarrow> P ya (Cut <b>.ImpR x.<a>.M b z.ImpL <c>.N y.Pa z) (Cut <a>.Cut <c>.N x.M y.Pa)" shows "P z x1 x2"
p_decomp: shows "p = xs @ e # ys @ e # zs"
balL_def: fixes AB :: "('c \<times> nat) tree"   and c :: "'c"   and C :: "('c \<times> nat) tree" shows "balL AB c C = (if ht AB = ht C + 2 then case AB of \<langle>A, (a, x), B\<rangle> \<Rightarrow> if ht B \<le> ht A then node A a (node B c C) else case B of \<langle>B\<^sub>1, (b, x), B\<^sub>2\<rangle> \<Rightarrow> node (node A a B\<^sub>1) b (node B\<^sub>2 c C) else node AB c C)"
count_roots_above: fixes ps :: "real poly list"   and p :: "real poly"   and a :: "real" assumes "sturm_seq ps p"   and "p \<noteq> 0" shows "sign_changes ps a - sign_changes_inf ps = card {x. a < x \<and> poly p x = 0}"
has_integral_integral_lborel: fixes f :: "'b \<Rightarrow> 'c" assumes "integrable lborel f" shows "(f has_integral integral\<^sup>L lborel f) UNIV"
mkM_simps_5: fixes T :: "'s Trace set" shows "valuation (Clock.mkM T) = envVal \<circ> tLast"
XsubY: shows "bangRel weakBisim \<subseteq> bisim O bangRel weakBisim O bisim"
wf_subint1: fixes G :: "prog" assumes "ws_prog G" shows "wf ((subint1 G)\<inverse>)"
assms_3: shows "rank_of (insert x X) = rank_of X"
finite_lists_length_le: fixes A :: "'d set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
steps_Stuck: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com" shows "\<Gamma> \<turnstile> (c, Stuck) \<rightarrow>\<^sup>* (Skip, Stuck)"
solve_equation: fixes H :: "'a set"   and x :: "'a"   and y :: "'a" assumes "subgroup H (G\<lparr>carrier := S\<rparr>)"   and "x \<in> H"   and "y \<in> H" shows "\<exists>h\<in>H. y = h \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x"
WHILEI_unfold: fixes I :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres"   and x :: "'b" shows "WHILE\<^bsup>I\<^esup> b f x = (if I x then if b x then f x \<bind> WHILE\<^bsup>I\<^esup> b f else RETURN x else FAIL)"
trg_cod: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.trg (S.cod \<mu>) = S.trg \<mu>"
length_filter_le: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "|filter P xs| \<le> |xs|"
utp_theory_lattice_def: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel" shows "utp_theory_lattice hcond \<equiv> utp_theory hcond \<and> utp_theory_lattice_axioms hcond"
mset_filter: fixes P :: "'k \<Rightarrow> bool"   and xs :: "'k list" shows "mset (filter P xs) = filter_mset P (mset xs)"
star_rtc_least: fixes x :: "('a \<times> ('b \<times> 'a) list) set"   and y :: "('a \<times> ('b \<times> 'a) list) set" assumes "t_one \<union> x \<union> t_prod y y \<subseteq> y" shows "t_star x \<subseteq> y"
preserved_def: fixes f :: "('aq, 'ar, 'as) prog"   and h :: "'aq"   and h' :: "'aq" shows "preserved f h h' = (\<forall>x. h \<turnstile> f \<rightarrow>\<^sub>r x = h' \<turnstile> f \<rightarrow>\<^sub>r x)"
insort_insert_key_triv: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "f x \<in> f ` set xs" shows "insort_insert_key f x xs = xs"
set_of_real_interval_mono: fixes x :: "float interval"   and y :: "float interval" assumes "set_of x \<subseteq> set_of y" shows "set_of (real_interval x) \<subseteq> set_of (real_interval y)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> String.literal \<Rightarrow> 'b val list \<Rightarrow> 'b val \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> htype \<Rightarrow> 'a"   and f5 :: "'c \<Rightarrow> 'a"   and f6 :: "'c \<Rightarrow> 'a"   and f7 :: "'b \<Rightarrow> 'a"   and f8 :: "'b \<Rightarrow> 'a"   and f9 :: "'c \<Rightarrow> 'a"   and f10 :: "'c \<Rightarrow> 'a"   and obs_event :: "('b, 'c) obs_event" shows "P (case obs_event of ExternalCall x xa xb xc \<Rightarrow> f1 x xa xb xc | ReadMem x xa xb \<Rightarrow> f2 x xa xb | WriteMem x xa xb \<Rightarrow> f3 x xa xb | NewHeapElem x xa \<Rightarrow> f4 x xa | ThreadStart x \<Rightarrow> f5 x | ThreadJoin x \<Rightarrow> f6 x | SyncLock x \<Rightarrow> f7 x | SyncUnlock x \<Rightarrow> f8 x | ObsInterrupt x \<Rightarrow> f9 x | ObsInterrupted x \<Rightarrow> f10 x) = ((\<forall>x11 x12 x13 x14. obs_event = ExternalCall x11 x12 x13 x14 \<longrightarrow> P (f1 x11 x12 x13 x14)) \<and> (\<forall>x21 x22 x23. obs_event = ReadMem x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)) \<and> (\<forall>x31 x32 x33. obs_event = WriteMem x31 x32 x33 \<longrightarrow> P (f3 x31 x32 x33)) \<and> (\<forall>x41 x42. obs_event = NewHeapElem x41 x42 \<longrightarrow> P (f4 x41 x42)) \<and> (\<forall>x5. obs_event = ThreadStart x5 \<longrightarrow> P (f5 x5)) \<and> (\<forall>x6. obs_event = ThreadJoin x6 \<longrightarrow> P (f6 x6)) \<and> (\<forall>x7. obs_event = SyncLock x7 \<longrightarrow> P (f7 x7)) \<and> (\<forall>x8. obs_event = SyncUnlock x8 \<longrightarrow> P (f8 x8)) \<and> (\<forall>x9. obs_event = ObsInterrupt x9 \<longrightarrow> P (f9 x9)) \<and> (\<forall>x10. obs_event = ObsInterrupted x10 \<longrightarrow> P (f10 x10)))"
prim_vc_3: fixes t :: "'a"   and v :: "'a"   and g :: "'a"   and r :: "'a" assumes "prim_spanning_invariant t v g r"   and "v * - v\<^sup>T \<sqinter> g = bot" shows "spanning_tree t g r"
symcl_def: fixes R :: "('c \<times> 'c) set" shows "R\<^sup>s = R \<union> (\<lambda>(a, b). (b, a)) ` R"
FAccThrow\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
map_iterateoi_def: fixes \<alpha> :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and invar :: "'c \<Rightarrow> bool"   and iterateoi :: "'c \<Rightarrow> ('f \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f" shows "map_iterateoi \<alpha> invar iterateoi \<equiv> ordered_finite_map \<alpha> invar \<and> map_iterateoi_axioms \<alpha> invar iterateoi"
narrow2: fixes narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.wn narrow widen less_eq less"   and "less_eq y x" shows "less_eq (narrow x y) x"
simply_connected_punctured_universe_eq: fixes a :: "'a" shows "simply_connected (- {a}) = (3 \<le> DIM('a))"
negligible_linear_singular_image: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "linear f"   and "\<not> inj f" shows "negligible (f ` S)"
all_edges_disjoint: fixes S :: "nat set"   and T :: "nat set" assumes "S \<inter> T = {}" shows "all_edges S \<inter> all_edges T = {}"
iteration_idep: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and strong_iteration :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.dra plus times one zero less_eq less star strong_iteration"   and "times x y = zero" shows "times x (strong_iteration y) = x"
simpneq_def: fixes t :: "tm" shows "simpneq t = (let (c, s) = split0 (simptm t) in if c = 0\<^sub>p then neq s else NEq (CNP 0 c s))"
simps_2: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "color"   and l :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and r :: "('b, 'c) RBT_Impl.rbt" shows "ord.rbt_sorted less (Branch c l k v r) = (ord.rbt_less less k l \<and> ord.rbt_greater less k r \<and> ord.rbt_sorted less l \<and> ord.rbt_sorted less r)"
boundedI: fixes a :: "'a upred"   and b :: "'a upred"   and c :: "'a upred" assumes "b \<sqsubseteq> a"   and "c \<sqsubseteq> a" shows "(b \<and> c) \<sqsubseteq> a"
top_power_ennreal: fixes n :: "nat" shows "top ^ n = (if n = 0 then 1 else top)"
outstanding_refs_subsets_5: fixes sb :: "'b memref list" shows "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb \<subseteq> outstanding_refs (Not \<circ> is_volatile) sb"
assms_1: shows "g : a \<mapsto>\<^bsub>\<AA>\<^esub> c"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c" assumes "functor A B F"   and "functor B C G" shows "composite_functor A B C F G"
ide_char: fixes a :: "'a2" shows "A2.ide a = (A2.arr a \<and> A2.dom a = a \<and> A2.cod a = a)"
simps_3: fixes f :: "'d \<Rightarrow> 'c"   and c1 :: "'d acom"   and c2 :: "'d acom" shows "ACom.map_acom f (c1;; c2) = ACom.map_acom f c1;; ACom.map_acom f c2"
l_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ** y = x ** z) = (y = z)"
ring_n_pd: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "Ring (O\<^bsub>K P n\<^esub>)"
permutes_inj_on: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and A :: "'a set" assumes "f permutes S" shows "inj_on f A"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) tllist \<Rightarrow> 'a"   and tllist :: "('c, 'b) tllist" shows "P (case tllist of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = (\<not> ((\<exists>x1. tllist = TNil x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. tllist = TCons x21 x22 \<and> \<not> P (f2 x21 x22))))"
gt_emb_arg: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "App s t >\<^sub>t t"
c_insert_def: shows "c_insert = (\<lambda>x u. if c_in x u = 1 then u else u + 2 ^ x)"
fermat_theorem_power_poly: fixes a :: "'b mod_ring"   and n :: "nat" shows "[:a:] ^ CARD('b) ^ n = [:a:]"
U0_not_in_FIN: shows "U\<^sub>0 \<notin> FIN"
IH: shows "lookup (except p {w} * monomial c t) s = (lookup (except p {w}) (s - t) * c when t adds s)"
u_simps_3: shows "trg s\<^sub>0 = src u"
eval_def: fixes R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and phi :: "'c \<Rightarrow> 'd"   and s :: "'d" shows "eval R S phi s = (\<lambda>p\<in>carrier (UP R). \<Oplus>\<^bsub>S\<^esub>i\<in>{..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
alphaAbs_trans: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A'' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs A \<or> qGoodAbs A' \<or> qGoodAbs A''"   and "A $= A'"   and "A' $= A''" shows "A $= A''"
induct: fixes P :: "'state \<Rightarrow> ('b, 'pred, 'act, 'effect) Tree \<Rightarrow> bool"   and a0 :: "'state"   and a1 :: "('b, 'pred, 'act, 'effect) Tree" assumes "\<And>P tset. (\<And>x. x \<in> set_bset tset \<Longrightarrow> P P x) \<Longrightarrow> P P (tConj tset)"   and "\<And>P t. P P t \<Longrightarrow> P P (tNot t)"   and "\<And>P f \<phi>. P P (tPred f \<phi>)"   and "\<And>P f \<alpha> t. (\<And>x xa xb. \<lbrakk>tAct f \<alpha> t =\<^sub>\<alpha> tAct f x xa; \<langle>f\<rangle>P \<rightarrow> \<langle>x,xb\<rangle>\<rbrakk> \<Longrightarrow> P xb xa) \<Longrightarrow> P P (tAct f \<alpha> t)" shows "P a0 a1"
count_of_subtract_entries_raw: fixes ys :: "('b \<times> nat) list"   and xs :: "('b \<times> nat) list"   and x :: "'b" assumes "distinct (map fst ys)" shows "count_of xs x - count_of ys x = count_of (subtract_entries_raw xs ys) x"
\<tau>exec_1_d_def: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list"   and \<sigma>' :: "'addr option \<times> 'heap \<times> 'addr frame list" shows "\<tau>exec_1_d P t \<sigma> \<sigma>' = (P,t \<turnstile> \<sigma> -\<lbrace>\<rbrace>-jvm\<rightarrow> \<sigma>' \<and> \<tau>Move2 P \<sigma> \<and> check P \<sigma>)"
less_trans: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k"   and z :: "'k" assumes "class.preorder less_eq less"   and "less x y"   and "less y z" shows "less x z"
inverse_float_intervalI: fixes x :: "real"   and X :: "float interval"   and p :: "nat" assumes "x \<in>\<^sub>r X" shows "inverse x \<in> set_of' (inverse_float_interval p X)"
max_norm_set_proptys: fixes A :: "(('c, 'e) vec, 'd) vec" shows "finite {\<bar>A $ i $ j\<bar> |i j. i \<in> UNIV \<and> j \<in> UNIV}"
per_nemp: fixes w :: "'b list"   and n :: "nat" assumes "periodN w n" shows "w \<noteq> \<epsilon>"
real_interval_avoid_countable_set: fixes a :: "real"   and b :: "real"   and A :: "real set" assumes "a < b"   and "countable A" shows "\<exists>x\<in>{a<..<b}. x \<notin> A"
circline_inversion_poincare_line: fixes H :: "circline" assumes "is_poincare_line H" shows "circline_inversion H = H"
vector_derivative_circlepath01: fixes x :: "real"   and z :: "complex"   and r :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "vector_derivative (circlepath z r) (at x within {0..1}) = complex_of_real (2 * pi) * \<i> * complex_of_real r * exp (2 * complex_of_real pi * \<i> * complex_of_real x)"
subst_atm_mset_list_def: fixes AAA :: "'a multiset list"   and \<sigma> :: "'s" shows "AAA \<cdot>aml \<sigma> = map (\<lambda>AA. AA \<cdot>am \<sigma>) AAA"
cases: fixes x :: "('c, 'd) stateful_strand_step" obtains t :: "('c, 'd) Term.term" and s :: "('c, 'd) Term.term" where "x = insert\<langle>ta__,s__\<rangle>"   | t :: "('c, 'd) Term.term" and s :: "('c, 'd) Term.term" where "x = delete\<langle>tb__,sa__\<rangle>"   | uu_ :: "poscheckvariant" and t :: "('c, 'd) Term.term" and s :: "('c, 'd) Term.term" where "x = \<langle>uu_: t \<in> s\<rangle>"   | uv_ :: "'d list" and uw_ :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" and F' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" where "x = \<forall>uv_\<langle>\<or>\<noteq>: uw_ \<or>\<notin>: F'\<rangle>"   | v :: "('c, 'd) Term.term" where "x = send\<langle>v__\<rangle>"   | v :: "('c, 'd) Term.term" where "x = receive\<langle>va__\<rangle>"   | v :: "poscheckvariant" and va :: "('c, 'd) Term.term" and vb :: "('c, 'd) Term.term" where "x = \<langle>v: va \<doteq> vb\<rangle>"
select_convs_2: fixes verts :: "'a set"   and arcs :: "('a \<times> 'a) set"   and more :: "'b" shows "E\<^bsub>\<lparr>verts = verts, arcs = arcs, \<dots> = more\<rparr>\<^esub> = arcs"
assms_1: shows "WEAKEN_HNR_POST \<Gamma> \<Gamma>'' \<Gamma>'"
adjoint_pair_def: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" shows "adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory (\<bullet>) (\<star>) assoc unit src trg f g \<eta> \<epsilon>"
B_def: shows "B \<equiv> connected_component_set (- closure A) b"
le_emeasure_sup_measure'1: fixes B :: "'b measure"   and A :: "'b measure"   and X :: "'b set" assumes "sets B = sets A"   and "X \<in> sets A" shows "emeasure A X \<le> emeasure (sup_measure' A B) X"
mat_inv_r: fixes A :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det A \<noteq> 0" shows "A *\<^sub>m\<^sub>m mat_inv A = eye"
simps_4: fixes x :: "'sz"   and y :: "'sz"   and \<phi> :: "('sf, 'sc, 'sz) formula" shows "PUrename x y (? \<phi>) = ? (FUrename x y \<phi>)"
rcos_ring_hom: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "ideal I R" shows "(+>\<^bsub>R\<^esub>) I \<in> ring_hom R (R Quot I)"
set_pmf_eq: fixes M :: "'e pmf" shows "set_pmf M = {x. pmf M x \<noteq> 0}"
minus_one_word: shows "- 1 = 2 ^ LENGTH('a) - 1"
dependent_mono: fixes B :: "'d set"   and A :: "'d set" assumes "cdependent B"   and "B \<subseteq> A" shows "cdependent A"
oalist_inv_def: fixes xs :: "('a \<times> 'c) list \<times> 'o" shows "oalist_inv xs = oalist_inv_raw (snd xs) (fst xs)"
init_sim: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c) graph_rec_scheme"   and B :: "('b, 'd) graph_rec_scheme"   and a0 :: "'a" assumes "simulation R A B"   and "a0 \<in> g_V0 A" shows "\<exists>b0. b0 \<in> g_V0 B \<and> (a0, b0) \<in> R"
lconsts_renameLCs: fixes p :: "'var \<Rightarrow> 'var"   and f :: "'form" shows "lconsts (renameLCs p f) = p ` lconsts f"
Un_Int_eq_3: fixes S :: "'a set"   and T :: "'a set" shows "S \<inter> (S \<union> T) = S"
assms: shows "eq\<cdot>xs\<cdot>ys = TT"
empty_def: shows "FiniteGraph.empty \<equiv> \<lparr>nodes = {}, edges = {}\<rparr>"
mult_right_mono_le: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and z :: "'a" assumes "a \<le> b"   and "(0::'a) \<le> c"   and "b * c \<le> z" shows "a * c \<le> z"
X: shows "X \<in> sets (density M (marg_dens2 \<Y> x y \<rho>))"
simps_1: fixes x :: "'a" shows "FVT (Var x) = {Inl x}"
z: shows "(r \<cdot> s) \<^sup>@ n \<cdot> r = z"
dim_image_eq: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and S :: "('a, 'b) vec set" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj_on f (vec.span S)" shows "vec.dim (f ` S) = vec.dim S"
le_less_trans2: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j < k"   and "i \<le> j" shows "i < k"
max2_equals2: fixes a :: "nat + 'a Bonk_Schramm_extension_unfolded set"   and b :: "nat + 'a Bonk_Schramm_extension_unfolded set" assumes "a \<in> Field wo_aux"   and "b \<in> Field wo_aux" shows "(max2 wo_aux a b = b) = ((a, b) \<in> wo_aux)"
remdups_adj_append: fixes xs\<^sub>1 :: "'e list"   and x :: "'e"   and xs\<^sub>2 :: "'e list" shows "remdups_adj (xs\<^sub>1 @ x # xs\<^sub>2) = remdups_adj (xs\<^sub>1 @ [x]) @ tl (remdups_adj (x # xs\<^sub>2))"
less_diff_conv: fixes i :: "nat"   and j :: "nat"   and k :: "nat" shows "(i < j - k) = (i + k < j)"
assms_2: fixes u :: "dB"   and ts :: "dB list" assumes "P u"   and "\<forall>t\<in>set ts. P t" shows "P (Abs u \<degree>\<degree> ts)"
ord_le_eq_subst: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd"   and c :: "'d" assumes "a \<le> b"   and "f b = c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
a'b': shows "a' \<in> S \<and> b' \<in> S"
bit_0_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit zero = bot"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) replete_setcat.comp \<phi>C \<phi>D F F.G F.\<phi> F.\<psi> (meta_adjunction.\<eta> (\<cdot>\<^sub>B) F F.\<phi>) (meta_adjunction.\<epsilon> (\<cdot>\<^sub>A) F F.G F.\<psi>) (transformation_by_components.map DopxC.comp replete_setcat.comp Hom_FopxC.map (meta_adjunction.\<Phi>o (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F F.G F.\<phi>)) (inverse_transformation.map DopxC.comp replete_setcat.comp (HomD.map \<circ> product_functor.map Dop.comp (\<cdot>\<^sub>A) Dop.comp (\<cdot>\<^sub>B) Dop.map F.G) (transformation_by_components.map DopxC.comp replete_setcat.comp Hom_FopxC.map (meta_adjunction.\<Phi>o (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F F.G F.\<phi>)))"
constant_functor_axioms: shows "constant_functor J (\<cdot>\<^sub>B) (F a)"
Read: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Read volatile a t # is, \<theta>, x, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is, \<theta>(t \<mapsto> m a), x, m, \<D>, \<O>, \<R>, \<S>)"
sctn_rel_comp: fixes A :: "('c \<times> 'e) set"   and B :: "('e \<times> 'd) set" shows "\<langle>A\<rangle>sctn_rel O \<langle>B\<rangle>sctn_rel = \<langle>A O B\<rangle>sctn_rel"
dyn_unique: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs' :: "char list list"   and Cs :: "char list list" assumes "P \<turnstile> C has least M = mthd via Cs'" shows "P \<turnstile> (C,Cs) selects M = mthd via Cs'"
gs_not_empty: fixes x :: "'a"   and i :: "nat" shows "(\<lambda>n. gs n x) ` {..i} \<noteq> {}"
indicator_inter_min: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" shows "indicator (A \<inter> B) x = min (indicator A x) (indicator B x)"