hcju/isabelleps huggingface dataset:
fg_elem_single_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and z :: "'e"   and n :: "int" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> = (if 0 = n then z else if 0 < n then THE x. x \<in> add_set f {a} (nat (n - 1)) else THE x. x \<in> add_set f {\<^sub>i- a} (nat (- n - 1)))"
finite_code_4: fixes xs :: "'b list" shows "finite (Set_Monad xs) = True"
WPC_through_make_step_2: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and X :: "nat \<Rightarrow> nat \<times> ('d \<times> nat \<times> nat) set"   and selector :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option" assumes "set_of_graph_rules Rs"   and "graph (graph_of (X 0))"   and "\<forall>i. X (Suc i) = make_step selector (X i)"   and "valid_selector Rs selector" shows "chain (\<lambda>i. graph_of (X i))"
pcr_linear_poly_def: shows "pcr_linear_poly \<equiv> rel_fun (=) (=) OO cr_linear_poly"
assms: shows "Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> Q'"
simps_2: fixes Ci :: "'a literal list \<times> nat"   and Dj :: "'a literal list \<times> nat"   and Djs :: "('a literal list \<times> nat) list" shows "select_min_weight_clause Ci (Dj # Djs) = select_min_weight_clause (if weight (apfst mset Dj) < weight (apfst mset Ci) then Dj else Ci) Djs"
Gramian_matrix_def: fixes G :: "'a vec list"   and k :: "nat" shows "gso.Gramian_matrix G k = (let M = mat k n (\<lambda>(i, j). G ! i $ j) in M * M\<^sup>T)"
assms_2: shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary C. valid_path \<gamma>"
cnj_add_mult_eq_Re: fixes z :: "complex"   and w :: "complex" shows "z * cnj w + cnj z * w = cor (2 * Re (z * cnj w))"
measurable_on_combine: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "f measurable_on S"   and "g measurable_on S"   and "continuous_on UNIV (\<lambda>x. h (fst x) (snd x))"   and "h (0::'b) (0::'c) = (0::'d)" shows "(\<lambda>x. h (f x) (g x)) measurable_on S"
finite_dimensional_vector_space_euclidean: shows "finite_dimensional_vector_space (*\<^sub>R) Basis"
steps_Fault: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and f :: "'c" shows "\<Gamma>\<turnstile> (c, Fault f) \<rightarrow>\<^sup>* (Skip, Fault f)"
FL_valid_Act_strong: fixes X :: "'b"   and P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and x :: "('c, 'pred, 'act, 'effect) formula" assumes "finite (supp X)" shows "P \<Turnstile> Act f \<alpha> x = (\<exists>\<alpha>' x' P'. Act f \<alpha> x = Act f \<alpha>' x' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> P' \<Turnstile> x' \<and> bn \<alpha>' \<sharp>* X)"
symrun_interp_fixpoint: fixes \<Gamma> :: "'a constr list" shows "\<Inter> (symbolic_run_interpretation_primitive ` set \<Gamma>) = \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
Le_le_LtD: fixes a :: "'c"   and b :: "'c" assumes "Le a \<le> Lt b" shows "a < b"
P\<^sub>1_B\<^sub>1_lower_bound_card: fixes P :: "'a set set"   and P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "bp P"   and "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 (S \<union> L)"   and "bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))" shows "card (P\<^sub>1 \<union> wrap B\<^sub>1) \<le> card P"
mutual_information_eq_entropy_conditional_entropy: fixes X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e" assumes "simple_function M X"   and "simple_function M Y" shows "\<I>(X ; Y) = \<H>(X) - \<H>(X | Y)"
fold_max_rev_eq: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "fold max (tl xs) (hd xs) = fold max (tl (rev xs)) (hd (rev xs))"
simps_10: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "rec_latin_type f1 f2 f3 Row = f1"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>a e. P (Eq (CNP 0 a e))"   and "\<And>a e. P (Le (CNP 0 a e))"   and "\<And>a e. P (Lt (CNP 0 a e))"   and "\<And>a e. P (NEq (CNP 0 a e))"   and "P T"   and "P F"   and "\<And>va. P (Le (CP va))"   and "\<And>va. P (Le (tm.Bound va))"   and "\<And>va vb. P (Le (tm.Add va vb))"   and "\<And>va vb. P (Le (tm.Mul va vb))"   and "\<And>va. P (Le (tm.Neg va))"   and "\<And>va vb. P (Le (tm.Sub va vb))"   and "\<And>vd vb vc. P (Le (CNP (Suc vd) vb vc))"   and "\<And>va. P (Lt (CP va))"   and "\<And>va. P (Lt (tm.Bound va))"   and "\<And>va vb. P (Lt (tm.Add va vb))"   and "\<And>va vb. P (Lt (tm.Mul va vb))"   and "\<And>va. P (Lt (tm.Neg va))"   and "\<And>va vb. P (Lt (tm.Sub va vb))"   and "\<And>vd vb vc. P (Lt (CNP (Suc vd) vb vc))"   and "\<And>va. P (Eq (CP va))"   and "\<And>va. P (Eq (tm.Bound va))"   and "\<And>va vb. P (Eq (tm.Add va vb))"   and "\<And>va vb. P (Eq (tm.Mul va vb))"   and "\<And>va. P (Eq (tm.Neg va))"   and "\<And>va vb. P (Eq (tm.Sub va vb))"   and "\<And>vd vb vc. P (Eq (CNP (Suc vd) vb vc))"   and "\<And>va. P (NEq (CP va))"   and "\<And>va. P (NEq (tm.Bound va))"   and "\<And>va vb. P (NEq (tm.Add va vb))"   and "\<And>va vb. P (NEq (tm.Mul va vb))"   and "\<And>va. P (NEq (tm.Neg va))"   and "\<And>va vb. P (NEq (tm.Sub va vb))"   and "\<And>vd vb vc. P (NEq (CNP (Suc vd) vb vc))"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
lcs_adds: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and u :: "'a"   and t :: "'a" assumes "class.lcs_powerprod plus minus zero lcs"   and "plus.adds plus s u"   and "plus.adds plus t u" shows "plus.adds plus (lcs s t) u"
domA_map_ran: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "('b \<times> 'c) list" shows "domA (map_ran f m) = domA m"
assms_2: fixes x :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t x\<^esub>" shows "x \<le> i"
elims: fixes x :: "'a + 'b + 'c"   and y :: "('a + 'b) + 'c" assumes "lsumr x = y" obtains a :: "'a" where "x = Inl a" and "y = Inl (Inl a)"   | b :: "'b" where "x = Inr (Inl b)" and "y = Inl (Inr b)"   | c :: "'c" where "x = Inr (Inr c)" and "y = Inr c"
ctr_transfer_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(R ===> list_all2 R ===> list_all2 R) (#) (#)"
vertex_1_char_def: shows "vertex_1 \<equiv> CHR ''A''"
sub_list_id: fixes ps :: "('b, 'c) fm list" shows "sub_list id ps = ps"
inv_solve_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(a = \<ominus> b \<oplus> c) = (c = b \<oplus> a)"
trivial_homomorphism_kernel: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd" shows "trivial_homomorphism G H f = (f \<in> hom G H \<and> kernel G H f = carrier G)"
split_face_edges_f21_subset: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face f ram1 ram2 vs"   and "(f12, f21) = split_face f ram1 ram2 vs"   and "vs \<noteq> []" shows "{(last vs, ram2), (ram1, hd vs)} \<union> Edges vs \<subseteq> \<E> f21"
blocksReds: fixes Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "length Vs = length Ts"   and "length vs = length Ts"   and "distinct Vs"   and "P \<turnstile> \<langle>e,(h, l(Vs [\<mapsto>] vs), sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l', sh'),b'\<rangle>" shows "P \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h, l, sh),b\<rangle> \<rightarrow>* \<langle>blocks (Vs, Ts, map (the \<circ> l') Vs, e'),(h', override_on l' l (set Vs), sh'),b'\<rangle>"
notin_alphabet_imp_sibling_id: fixes a :: "'a"   and t :: "'a tree" assumes "a \<notin> alphabet t" shows "sibling t a = a"
prems: shows "snd (usubstappp \<sigma> U \<alpha>^d) \<noteq> undefg"
sel_1: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "shd (siterate f x) = x"
intro: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "Type_Semigroups.semigroup f"   and "cancel_semigroup_axioms f" shows "cancel_semigroup f"
SN_rel_ext_step_mono: fixes P :: "('a \<times> 'a) set"   and P' :: "('a \<times> 'a) set"   and Pw :: "('a \<times> 'a) set"   and Pw' :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and Rw :: "('a \<times> 'a) set"   and Rw' :: "('a \<times> 'a) set"   and t :: "SN_rel_ext_type" assumes "P \<subseteq> P'"   and "Pw \<subseteq> Pw'"   and "R \<subseteq> R'"   and "Rw \<subseteq> Rw'" shows "SN_rel_ext_step P Pw R Rw t \<subseteq> SN_rel_ext_step P' Pw' R' Rw' t"
three_cases: fixes A :: "bool"   and B :: "bool"   and C :: "bool" assumes "A \<or> B \<or> C" obtains "A"   | "B"   | "C"
Complex_add_complex_of_real: fixes x :: "real"   and y :: "real"   and r :: "real" shows "Complex x y + complex_of_real r = Complex (x + r) y"
keys_diff: fixes a :: "'c \<Rightarrow>\<^sub>0 'd"   and b :: "'c \<Rightarrow>\<^sub>0 'd" shows "Poly_Mapping.keys (a - b) \<subseteq> Poly_Mapping.keys a \<union> Poly_Mapping.keys b"
ab: shows "(a, b) = bezout_coefficients y (prod_list (z # ys))"
mat_carrier: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "mat nr nc f \<in> carrier_mat nr nc"
\<i>'_def: fixes a :: "'a" shows "\<i>' a \<equiv> SOME \<phi>. local.iso \<phi> \<and> \<phi> \<in> hom (a \<star> a) a \<and> (obj a \<longrightarrow> \<phi> = \<i>[a])"
tarjan_back_def: fixes u :: "'v"   and v :: "'v"   and s :: "'v tarjan_state" shows "tarjan_back u v s = (if the_discovered s v < the_discovered s u \<and> v \<in> set (tj_stack s) then let ul' = min (the_lowlink s u) (the_discovered s v) in NOOP (s\<lparr>lowlink := lowlink s(u \<mapsto> ul')\<rparr>) else NOOP s)"
subspace_span: fixes S :: "'b set" shows "vs1.subspace (vs1.span S)"
simps_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
is_discovered_def: fixes u :: "'v"   and s :: "('v, 'es) state_scheme" shows "is_discovered u s \<equiv> u \<in> dom (discovered s)"
iter2: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> ; c\<^sup>\<omega> = c\<^sup>\<omega>"
assms_2: shows "isUnitClause uClause uLiteral (elements M)"
interchange: fixes h :: "'a"   and f :: "'a"   and k :: "'a"   and g :: "'a" assumes "seq h f"   and "seq k g" shows "(h \<otimes> k) \<cdot> (f \<otimes> g) = h \<cdot> f \<otimes> k \<cdot> g"
elims_1: fixes x :: "('c, 'd) fmap"   and xa :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and y :: "bool" assumes "sat_precond_as x xa = y" obtains s :: "('c, 'd) fmap" where "x = sb__" and "xa = []" and "y"   | s :: "('c, 'd) fmap" and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" where "x = s" and "xa = a # as" and "y = (fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as)"
E15: fixes F :: "bool" shows "\<turnstile> \<diamond>#F = #F"
not_separated_by_wall_ex_foldpair: fixes C :: "'a set"   and D :: "'a set"   and H :: "'a set set set" assumes "chamber C"   and "chamber D"   and "H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}})"   and "\<not> separated_by H C D" shows "\<exists>(f, g)\<in>foldpairs. H = {f \<turnstile> \<C>, g \<turnstile> \<C>} \<and> C \<in> f \<turnstile> \<C> \<and> D \<in> f \<turnstile> \<C>"
is_functor: shows "functor J (\<cdot>) \<chi>.A.map"
substInp_preserves_good: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and inp :: "'h \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodInp inp" shows "goodInp (inp %[X / x]_xs)"
KBasic2_4: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>\<^bold>\<not>\<phi> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<diamond>\<phi> in v]"
le_imp_less_or_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<preceq> y" shows "x \<prec> y \<or> x = y"
borel_measurable_continuous_on: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and M :: "'e measure" assumes "continuous_on UNIV f"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f (g x)) \<in> borel_measurable M"
delete_not_domA: fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list" assumes "x \<notin> domA \<Gamma>" shows "delete x \<Gamma> = \<Gamma>"
mem\<^sub>A'_def: shows "mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C'"
map_add_restrict_right: fixes x :: "'c set"   and m' :: "'c \<Rightarrow> 'd option"   and m :: "'c \<Rightarrow> 'd option" assumes "x \<subseteq> dom m'" shows "(m ++ m') |` x = m' |` x"
insert_is_Un: fixes a :: "'b"   and A :: "'b set" shows "(a::'b) \<triangleright> (A::'b set) = {a} \<union> A"
moebius_pt_moebius_translation_inf: fixes v :: "complex" shows "moebius_pt (moebius_translation v) \<infinity>\<^sub>h = \<infinity>\<^sub>h"
munionM_return_empty2: fixes x :: "'m" shows "munionM x (return {#}) = x"
assms_1: shows "qGoodAbs A \<or> qGoodAbs A' \<or> qGoodAbs A''"
ide_prod: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "ide (HF'.prod a b)"
assms_3: shows "interior S \<inter> T \<noteq> {}"
bnds_mult_def: fixes prec :: "nat"   and a1 :: "float"   and a2 :: "float"   and b1 :: "float"   and b2 :: "float" shows "bnds_mult prec a1 a2 b1 b2 = (float_plus_down prec (nprt a1 * pprt b2) (float_plus_down prec (nprt a2 * nprt b2) (float_plus_down prec (pprt a1 * pprt b1) (pprt a2 * nprt b1))), float_plus_up prec (pprt a2 * pprt b2) (float_plus_up prec (pprt a1 * nprt b2) (float_plus_up prec (nprt a2 * pprt b1) (nprt a1 * nprt b1))))"
is_irreducible_mset_iff: fixes E :: "'a literal list"   and E' :: "'a literal list"   and Ds :: "'a literal list list"   and C :: "'a literal list" assumes "mset E = mset E'" shows "(reduce Ds C E = E) = (reduce Ds C E' = E')"
map_consI_1: fixes w :: "'b list"   and f :: "'c \<Rightarrow> 'b"   and ww :: "'c list"   and a :: "'c" assumes "w = map f ww" shows "f a # w = map f (a # ww)"
chain_imageI: fixes le_a :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and Y :: "'h set"   and le_b :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i" assumes "Complete_Partial_Order.chain le_a Y"   and "\<And>x y. \<lbrakk>x \<in> Y; y \<in> Y; le_a x y\<rbrakk> \<Longrightarrow> le_b (f x) (f y)" shows "Complete_Partial_Order.chain le_b (f ` Y)"
monom_mult_zero_right: fixes c :: "'d"   and t :: "'a" shows "monom_mult c t 0 = 0"
cases: fixes a :: "trm" assumes "valuep a" obtains T :: "POPLmark.type" and t :: "trm" where "a = (\<lambda>:Tb__. tb__)"   | T :: "POPLmark.type" and t :: "trm" where "a = (\<lambda><:T. t)"
less_top_Top: fixes x :: "'a top" shows "(Top < x) = False"
goedel_after_prefixes: fixes m :: "nat"   and vs :: "nat list" shows "\<exists>n\<ge>m. \<phi> n = vs @ [n] \<odot> 0\<^sup>\<infinity>"
lem: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> F \<^bold>\<or> F"
simps_2: fixes kts :: "('c \<times> ('c, 'd) rbt) list"   and c :: "color"   and l :: "('c, 'd) rbt"   and k :: "'c"   and v :: "'d"   and r :: "('c, 'd) rbt" shows "rbt_keys_next (kts, Branch c l k v r) = rbt_keys_next ((k, r) # kts, l)"
connected_uncountable: fixes S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "connected S"   and "a \<in> S"   and "b \<in> S"   and "a \<noteq> b" shows "uncountable S"
bv_length_bv_mapzip: fixes f :: "bit \<Rightarrow> bit \<Rightarrow> bit"   and w1 :: "bit list"   and w2 :: "bit list" shows "length (bv_mapzip f w1 w2) = max (length w1) (length w2)"
poly_mapping_size_def: fixes m :: "'a \<Rightarrow>\<^sub>0 'b" shows "local.poly_mapping_size m = g (0::'b) + (\<Sum>k\<in>keys m. Suc (f k + g (lookup m k)))"
SigmaE2: fixes a :: "'g"   and b :: "'h"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" assumes "(a, b) \<in> Sigma A B" obtains "a \<in> A" and "b \<in> B a"
elims_2: fixes x :: "('a \<times> ('b \<times> 'c) \<times> 'a) set"   and xa :: "'a set" assumes "completely_specified_rel_over x xa" obtains rel :: "('a \<times> ('b \<times> 'c) \<times> 'a) set" and nods :: "'a set" where "x = rel" and "xa = nods" and "\<forall>s1\<in>nods. \<forall>x\<in>(fst \<circ> fst \<circ> snd) ` rel. \<exists>y\<in>(snd \<circ> fst \<circ> snd) ` rel. \<exists>s2. (s1, (x, y), s2) \<in> rel"
seq_1: fixes i :: "nat" shows "F i \<in> borel_measurable N"
A_interchange_def: shows "A_interchange = interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n)"
msubstlepos: fixes vs :: "'b list"   and c :: "poly"   and a :: "poly"   and b :: "tm"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "(0::'b) < \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"   and "islin (Le (CNP 0 a b))" shows "Ifm vs (x # bs) (msubstlepos c t a b) = Ifm vs (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> # bs) (Le (CNP 0 a b))"
assms_1: shows "\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)"
permutations_of_set_aux_correct: fixes A :: "'b set" shows "permutations_of_set_aux [] A = permutations_of_set A"
f_Exec_Stream_Acc_Output_eq_Acc_Trans_Fun_Step_Output: fixes k :: "nat"   and trans_fun :: "'d message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and localState :: "'e \<Rightarrow> 'f"   and pointwise_shrink :: "'e list \<Rightarrow> 'e"   and output_fun :: "'e \<Rightarrow> 'g message_af"   and xs :: "'d message_af list"   and c :: "'e" assumes "0 < k"   and "Deterministic_Trans_Fun trans_fun localState"   and "is_correct_localState_Pointwise_Output_Shrink pointwise_shrink localState"   and "is_Pointwise_Output_Shrink pointwise_shrink output_fun" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c = map output_fun (f_Exec_Comp_Stream (Acc_Trans_Fun_Step k trans_fun pointwise_shrink) xs c)"
sd_power_zero: fixes x :: "'a \<times> 'b" shows "sd_power x 0 = (1::'a, \<bottom>)"
simps_3: fixes \<Phi> :: "nat \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp" shows "\<Phi> \<Turnstile> b and b' = (\<Phi> \<Turnstile> b \<and> \<Phi> \<Turnstile> b')"
semantics_ltln_ite_simps_2: fixes w :: "nat \<Rightarrow> 'a set"   and P :: "bool" shows "w \<Turnstile>\<^sub>n (if P then false\<^sub>n else true\<^sub>n) = (\<not> P)"
spr_sim_def: fixes jkbp :: "'a \<Rightarrow> ('a, 'b, 'c \<times> 'd) GC list"   and envInit :: "('a, 'e, 'f, 'c, 'd) BEState list"   and envVal :: "('a, 'e, 'f, 'c, 'd) BEState \<Rightarrow> 'b \<Rightarrow> bool"   and envObsC :: "'f \<Rightarrow> 'g"   and envActionES :: "'f \<Rightarrow> 'e \<times> ('a \<Rightarrow> 'c) \<Rightarrow> ('e \<times> 'h) list"   and envTransES :: "'e \<times> 'h \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> 'f \<Rightarrow> 'f" assumes "FiniteBroadcastEnvironment jkbp" shows "FiniteBroadcastEnvironment.spr_sim jkbp envInit envVal envObsC envActionES envTransES \<equiv> \<lambda>t. \<lparr>sprFst = tFirst t, sprLst = tLast t, sprCRel = FiniteBroadcastEnvironment.tObsC_abs jkbp envInit envVal envObsC envActionES envTransES t\<rparr>"
distinct_tl: fixes xs :: "'b list" assumes "distinct xs" shows "distinct (tl xs)"
ffilter_empty_if: fixes xs :: "'a fset"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>x|\<in>|xs. \<not> P x" shows "ffilter P xs = {||}"
sim: fixes p :: "'s1 spmf"   and q :: "'s2 spmf"   and a :: "'a" assumes "S p q"   and "a \<in> A" shows "p \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q \<bind> (\<lambda>s. map_spmf fst (callee2 s a))"
zero_less_power: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero a" shows "less zero (power.power one times a n)"
assms_2: shows "0\<^sub>h \<in> circline_set l"
disjoint_up_mod2: shows "upRules \<inter> modRules2 = {}"
P': shows "P' \<equiv> mult_row (interchange_rows P i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j)"
fref_def: fixes P :: "'k \<Rightarrow> bool"   and R :: "('i \<times> 'k) set"   and S :: "('j \<times> 'l) set" shows "[P]\<^sub>f R \<rightarrow> S \<equiv> {(f, g). \<forall>x y. P y \<and> (x, y) \<in> R \<longrightarrow> (f x, g y) \<in> S}"
pos_less_divideR_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "real"   and a :: "'a"   and b :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less a (scaleR (inverse c) b) = less (scaleR c a) b"
deterministic: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r x y"   and "r x z" shows "y = z"
fraction_scaleR_times: fixes u :: "num"   and v :: "num"   and w :: "num"   and a :: "'a" shows "(numeral u / numeral v) *\<^sub>R (numeral w * a) = (numeral u * numeral w / numeral v) *\<^sub>R a"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a XOR b) XOR c = a XOR b XOR c"
crossproduct_noteq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(a \<noteq> b \<and> c \<noteq> d) = (plus (times a c) (times b d) \<noteq> plus (times a d) (times b c))"
rep_cases: fixes y :: "'b \<Rightarrow> 'c option" assumes "y \<in> UNIV" obtains x :: "('b, 'c) mapping" where "y = rep x"
prognosis_env_cong: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and u :: "Arity"   and e :: "exp"   and S :: "stack_elem list" assumes "CardinalityPrognosisShape prognosis"   and "ae f|` domA \<Gamma> = ae' f|` domA \<Gamma>" shows "prognosis ae as u (\<Gamma>, e, S) = prognosis ae' as u (\<Gamma>, e, S)"
c123l: shows "set (labels_conv (fst (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3), snd (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3) @ snd (conv_mirror \<gamma>3))) \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step}"