hcju/isabelleps huggingface dataset:
prime_imp_prime_elem': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "ASSUMPTION (normalization_semidom.prime zero times one normalize x)" shows "comm_semiring_1.prime_elem times one zero x"
eval\<^sub>w_def: shows "eval\<^sub>w \<equiv> {(x, y). eval\<^sub>wp x y}"
simps_15: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b gexp"   and va :: "'b gexp" shows "GExp_Lexorder.less_gexp_aux (Gt e1 e2) (Nor v va) = True"
invmono: fixes I :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "\<turnstile> I \<longrightarrow> P"   and "|~ P \<and> _actrans N f \<longrightarrow> \<circle>P" shows "\<turnstile> I \<and> \<box>[N]_f \<longrightarrow> \<box>P"
CCexp_simps_8: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and a :: "Arity" shows "ccExp_syn a (scrut ? e1 : e2) = ccExp_syn 0 scrut \<squnion> (ccExp_syn a e1 \<squnion> ccExp_syn a e2) \<squnion> edom (Aexp_syn' 0 scrut) G\<times> (edom (Aexp_syn' a e1) \<union> edom (Aexp_syn' a e2))"
plus_game_assoc: fixes F :: "game"   and G :: "game"   and H :: "game" shows "plus_game (plus_game F G) H = plus_game F (plus_game G H)"
MkIde_Dom: fixes a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr a" shows "MkIde (Dom a) = local.dom a"
f: shows "f \<in> M \<rightarrow>\<^sub>M M1"
index_append_vec_1: fixes i :: "nat"   and v :: "'b vec"   and w :: "'b vec" assumes "i < dim_vec v + dim_vec w" shows "(v @\<^sub>v w) $ i = (if i < dim_vec v then v $ i else w $ (i - dim_vec v))"
prod_2_def: fixes p1 :: "'e \<Rightarrow> 'g decision option"   and p2 :: "'f \<Rightarrow> 'h decision option" shows "p1 \<Otimes>\<^sub>2 p2 \<equiv> \<lambda>(x, y). case p1 x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor> | \<lfloor>deny d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor>"
radical_sqrt_correct_expr: fixes x :: "real" assumes "x \<in> radical_sqrt" shows "\<exists>e. \<lbrace>e\<rbrace> = x"
PO_m1_inv4_rfresh: shows "reach m1 \<subseteq> m1_inv4_rfresh"
fpxs_of_fls_shift: fixes n :: "int"   and f :: "'b fls" shows "fpxs_of_fls (fls_shift n f) = fpxs_shift (rat_of_int n) (fpxs_of_fls f)"
try_gpv_gen_lossless: fixes b :: "bool"   and gpv :: "('f, 'd, 'e) gpv"   and gpv' :: "('f, 'd, 'e) gpv" assumes "gen_lossless_gpv b \<I>_full gpv" shows "TRY gpv ELSE gpv' = gpv"
mset_set_set_mset: fixes M :: "'b multiset"   and A :: "'b set" assumes "M \<subseteq># mset_set A" shows "mset_set (set_mset M) = M"
simps_1: fixes r :: "'rule" shows "outPorts (Rule r) = outPortsRule r"
set_child_nodes_def: shows "Shadow_DOM.set_child_nodes \<equiv> l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes"
rat_number_collapse_6: fixes k :: "int" shows "Fract k 0 = 0"
simps_2: fixes d :: "nat"   and n :: "nat"   and j :: "nat"   and i :: "nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "ladder_shift_j d ((n, j, i) # L) = (n, j - d, i) # ladder_shift_j d L"
uik: shows "unat i + unat k < 2 ^ LENGTH('a)"
basic_cong_2: fixes p :: "int" assumes "2 < p" shows "[1 \<noteq> - 1] (mod p)"
valid_cases: fixes X :: "msg" assumes "X \<in> valid" obtains A :: "agent" and B :: "agent" and M :: "msg" where "X = implInsec A__ B__ Ma__" and "Ma__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implConfid Aa__ Ba__ Mb__" and "Mb__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implAuth Ab__ Bb__ Mc__" and "Mc__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implSecure A B M" and "M \<in> payload"
distinct_5: shows "UMinusOp \<noteq> RoundOp"
ch2ch_snd: fixes Y :: "nat \<Rightarrow> 'a \<times> 'b" assumes "chain Y" shows "chain (\<lambda>i. snd (Y i))"
expand: fixes contT :: "('d, 'e) contT"   and contT' :: "('d, 'e) contT" assumes "run_cont contT = run_cont contT'" shows "contT = contT'"
verts3_K33: fixes G :: "'b pair_pre_digraph" assumes "K\<^bsub>3,3\<^esub> (with_proj G)" shows "verts3 (with_proj G) = verts (with_proj G)"
monomials_lowerPoly: fixes b :: "nat"   and i :: "nat"   and mp :: "'a mpoly" shows "monomials (lowerPoly b i mp) = higherPowers b i ` (monomials mp \<inter> {x. \<forall>j\<in>{b..<b + i}. lookup x j = 0})"
exhaust: fixes y :: "('a, 'b, 'c) Stmt" obtains x11 :: "'a" and x12 :: "'b" where "y = x11 \<leftarrow> x12"   | "y = Skip"   | x31 :: "('a, 'b, 'c) Stmt" and x32 :: "'a ModeUpd" where "y = x31@[x32]"   | x41 :: "('a, 'b, 'c) Stmt" and x42 :: "('a, 'b, 'c) Stmt" where "y = x41 ;; x42"   | x51 :: "'c" and x52 :: "('a, 'b, 'c) Stmt" and x53 :: "('a, 'b, 'c) Stmt" where "y = Stmt.If x51 x52 x53"   | x61 :: "'c" and x62 :: "('a, 'b, 'c) Stmt" where "y = While x61 x62"   | x71 :: "'c" and x72 :: "('a, 'b, 'c) Stmt" where "y = Await x71 x72"   | "y = Stop"
as: shows "a $ 1 < b $ 1 \<and> a $ 2 < b $ 2"
negligible: fixes S :: "'i set" shows "Henstock_Kurzweil_Integration.negligible S = (\<forall>T. (indicat_real S has_integral 0) T)"
ins_list_split_right_general: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and ts :: "('b btree \<times> 'b) list"   and x :: "'b"   and ls :: "('b btree \<times> 'b) list"   and sub :: "'b btree"   and sep :: "'b"   and rs :: "('b btree \<times> 'b) list"   and zs :: "'b list" assumes "split split"   and "split ts x = (ls, (sub, sep) # rs)"   and "sorted_less (inorder_list ts)"   and "sep \<noteq> x" shows "ins_list x (inorder_list ((sub, sep) # rs) @ zs) = ins_list x (inorder sub) @ sep # inorder_list rs @ zs"
MapsToCompDef: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d"   and g :: "'e"   and Z :: "'d" assumes "f maps\<^bsub>C\<^esub> X to Y"   and "g maps\<^bsub>C\<^esub> Y to Z" shows "f \<approx>>\<^bsub>C\<^esub> g"
dominant_term: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis"   and "trimmed F" shows "eval F \<sim>[at_top] eval_monom (dominant_term F) basis"
compseries_is_td_gchain: fixes r :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "compseries G r f" shows "td_gchain G r f"
uint_div: fixes x :: "'a word"   and y :: "'a word" shows "uint (x div y) = uint x div uint y"
power_minus_Bit1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus x) (numeral (num.Bit1 k)) = uminus (power.power one times x (numeral (num.Bit1 k)))"
res: fixes C :: "'c literal set"   and S :: "'c literal set set" assumes "C \<in> res S" shows "S \<turnstile> C"
chamber_map: fixes C :: "'a set" assumes "folding_g.chamber C" shows "folding_g.chamber (f ` C)"
modify_tprog_fetch_state: fixes st :: "nat"   and tp :: "(action \<times> nat) list"   and b :: "nat" assumes "st \<le> length tp div 2"   and "0 < st"   and "b = 1 \<or> b = 0" shows "modify_tprog tp ! Suc (4 * (st - Suc 0) + 2 * b) = snd (tp ! (2 * (st - Suc 0) + b))"
exponent_def: fixes f :: "float" shows "exponent f = snd (SOME p. f = 0 \<and> fst p = 0 \<and> snd p = 0 \<or> f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> odd (fst p))"
division_ofD_2: fixes s :: "'a set set"   and i :: "'a set"   and K :: "'a set" assumes "s division_of i"   and "K \<in> s" shows "K \<subseteq> i"
ideal_zero: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "\<zero>\<^bsub>R\<^esub> \<in> I"
assms_2: fixes s :: "'a" assumes "P s"   and "b s" shows "P (c s)"
cyclic_on_inI: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and s :: "'b" assumes "cyclic_on f S"   and "s \<in> S" shows "f s \<in> S"
step_Stuck_prop: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and c' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Stuck" shows "s' = Stuck"
1: shows "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) (dbproj i D))\<rbrakk>\<^sub>d \<I>"
aboveS_above_trans: fixes r :: "('a \<times> 'a) set"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "trans r"   and "antisym r"   and "b \<in> aboveS r a"   and "c \<in> above r b" shows "c \<in> aboveS r a"
reparam_weak_eq_line_integrals_basis_2: fixes \<gamma>1 :: "real \<Rightarrow> 'a"   and \<gamma>2 :: "real \<Rightarrow> 'a"   and basis :: "'a set"   and F :: "'a \<Rightarrow> 'a" assumes "reparam_weak \<gamma>1 \<gamma>2"   and "\<gamma>2 C1_differentiable_on {0..1}"   and "\<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"   and "finite basis" shows "line_integral_exists F basis \<gamma>1"
u1: shows "a1 = (b1 @ [fst q]) @ u"
listrel_rtrancl_refl: fixes xs :: "'g list"   and r :: "('g \<times> 'g) set" shows "(xs, xs) \<in> listrel (r\<^sup>*)"
push_bit_double: fixes n :: "nat"   and a :: "'a" shows "push_bit n (a * (2::'a)) = push_bit n a * (2::'a)"
unitor_coincidence_2: shows "MC.runit MC.unity = \<iota>"
Gramian_determinant_Ints: fixes k :: "nat" assumes "k \<le> length fs"   and "\<And>i j. \<lbrakk>i < n; j < length fs\<rbrakk> \<Longrightarrow> fs ! j $ i \<in> \<int>" shows "Gramian_determinant fs k \<in> \<int>"
G_increasing: fixes m :: "nat"   and M :: "nat set"   and l :: "nat" assumes "\<Phi> m M" shows "case G l m M of (n, N, j) \<Rightarrow> m < n \<and> N \<subseteq> M \<and> n \<in> M"
cxt_inv_skip: fixes E :: "('Var, 'AExp, 'BExp) Stmt list"   and c :: "('Var, 'AExp, 'BExp) Stmt" assumes "cxt_to_stmt E c = Skip" shows "c = Skip \<and> E = []"
matches_MatchNotAnd_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and m' :: "'c match_expr"   and a :: "action"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule (MatchAnd (MatchNot m) m') a], Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[], Undecided\<rangle> \<Rightarrow> t"
integral_component_le: fixes k :: "'b"   and f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "k \<in> Basis"   and "f integrable_on S"   and "g integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<bullet> k \<le> g x \<bullet> k" shows "integral S f \<bullet> k \<le> integral S g \<bullet> k"
hensels_lemma: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp"   and "pderiv f \<bullet> a \<noteq> \<zero>"   and "f \<bullet> a \<noteq> \<zero>"   and "eint 2 * val_Zp (pderiv f \<bullet> a) < val_Zp (f \<bullet> a)" obtains \<alpha> :: "nat \<Rightarrow> int" where "f \<bullet> \<alpha> = \<zero>" and "\<alpha> \<in> carrier Zp" and "val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> \<alpha>)" and "val_Zp (a \<ominus> \<alpha>) = val_Zp (local.divide (f \<bullet> a) (pderiv f \<bullet> a))" and "val_Zp (pderiv f \<bullet> \<alpha>) = val_Zp (pderiv f \<bullet> a)"
subls_def2: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" shows "L \<cdot>\<^sub>l\<^sub>s \<sigma> = {l \<cdot>\<^sub>l \<sigma> |l. l \<in> L}"
asm: shows "\<forall>s\<in>RED \<tau>. t[x::=s] \<in> RED \<sigma>"
l_minus: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F" shows "\<ominus>\<^bsub>F\<^esub> x \<otimes>\<^bsub>F\<^esub> y = \<ominus>\<^bsub>F\<^esub> (x \<otimes>\<^bsub>F\<^esub> y)"
f_def: fixes t :: "real" shows "f t = (if 0 < t then exp (- inverse t) else 0)"
in_nv2T: fixes x :: "Preliminaries.var"   and T :: "'a trm" shows "(x \<in> nv2T T) = (T = Var x)"
baseClasses_def: fixes Bs :: "base list" shows "baseClasses Bs \<equiv> set (map getbase Bs)"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_finite: fixes x :: "('d, 'c) stateful_strand_step" shows "finite (fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p x)"
prems: shows " t \<longrightarrow>\<^sub>1 t2"
simps_3: fixes r :: "ValD \<rightarrow> ValD" shows "ValD_copy_rec\<cdot>r\<cdot>ValFF = ValFF"
lang_lderiv: fixes n :: "nat"   and r :: "'b rexp"   and w :: "'a" assumes "local.wf n r"   and "w \<in> \<Sigma> n" shows "lang n (lderiv w r) = lQuot w (lang n r)"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "ide a"   and "seq f a" shows "f \<cdot> a = f"
qOp_alpha_iff: fixes delta :: "'j"   and inp :: "'f \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qTerm option"   and binp :: "'g \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qAbs option"   and X' :: "('f, 'g, 'h, 'i, 'j) qTerm" shows "(qOp delta inp binp #= X') = (\<exists>inp' binp'. X' = qOp delta inp' binp' \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
assms_1: shows "p \<cdot> x \<le> x \<cdot> r"
initQ_def: shows "initQ \<equiv> (\<lambda>_. \<infinity>)(r := 0)"
lower_on_lit: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and L :: "'b literal"   and x :: "'b" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "basic_superposition.lower_on trm_ord \<sigma> \<eta> (vars_of_lit L)"   and "(Var x \<lhd> \<sigma>, Var x \<lhd> \<eta>) \<in> trm_ord"   and "x \<in> vars_of_lit L" shows "(subst_lit L \<sigma>, subst_lit L \<eta>) \<in> basic_superposition.lit_ord trm_ord"
vec_norm_geq_0: fixes v :: "complex vec" shows "0 \<le> vec_norm v"
rep_eq: fixes x :: "('d, 'e) slist" shows "slist_apply x = (\<lambda>i. the_default (0::'e) (map_of (Rep_slist x) i))"
times_set_def: fixes x :: "'b set"   and y :: "'b set" shows "x \<oplus> y = x \<union> y"
assms: shows "\<G> \<subseteq> \<^bold>G \<phi>"
Assign_complete: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x1 :: "char list"   and x2 :: "aexp"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and t :: "char list \<Rightarrow> int"   and t' :: "char list \<Rightarrow> int" assumes "ir_valid P (x1 ::= x2) c' Q"   and "Q t t'" shows "\<exists>s'. (\<exists>v. P (t(x1 := v)) s' \<and> t x1 = aval x2 (t(x1 := v))) \<and> (c', s') \<Rightarrow> t'"
healthy_intros_18: fixes S :: "'b \<Rightarrow> 'a set"   and prog :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>x s. x \<in> S s \<Longrightarrow> nearly_healthy (wlp (prog x))"   and "\<And>s. \<exists>x. x \<in> S s" shows "nearly_healthy (wlp (SetDC prog S))"
min_maxsimpchainD_maxsimpchain: fixes xs :: "'b set list" assumes "codomain.min_gallery xs" shows "codomain.gallery xs"
rep_eq: fixes x :: "'b"   and xa :: "'b" shows "map_option bounds_of_interval (Interval' x xa) = (if x \<le> xa then Some (x, xa) else None)"
cancel_at_def: fixes i :: "nat"   and l :: "(bool \<times> 'd) list" shows "cancel_at i l = take i l @ drop (2 + i) l"
semiring_simprules_7: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> y = y \<oplus> x"
fourth_point: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "Col A B P"   and "Bet A B C" shows "Bet P A B \<or> Bet A P B \<or> Bet B P C \<or> Bet B C P"
scaleQ_scaleQ: fixes a :: "'e"   and b :: "'e"   and x :: "'e['f, 'g]" shows "a *\<^sub>Q b *\<^sub>Q x = a \<cdot> b *\<^sub>Q x"
fmfilter_alt_defs_1: fixes a :: "'c" shows "fmdrop a = fmfilter (\<lambda>a'. a' \<noteq> a)"
simps_9: fixes P1 :: "pol"   and x :: "nat"   and P2 :: "pol"   and Q1 :: "pol"   and y :: "nat"   and Q2 :: "pol" shows "PX P1 x P2 \<langle>+\<rangle> PX Q1 y Q2 = (if x = y then mkPX (P1 \<langle>+\<rangle> Q1) x (P2 \<langle>+\<rangle> Q2) else if y < x then mkPX (PX P1 (x - y) (Pc 0) \<langle>+\<rangle> Q1) y (P2 \<langle>+\<rangle> Q2) else mkPX (PX Q1 (y - x) (Pc 0) \<langle>+\<rangle> P1) x (P2 \<langle>+\<rangle> Q2))"
incomplete_design_axioms_def: fixes point_set :: "'a set"   and u_block_size :: "int" shows "incomplete_design_axioms point_set u_block_size \<equiv> u_block_size < int (card point_set)"
EASSERT_simps_2: shows "EASSERT False = FAIL"
sorted_OAlist_tc_sorted_domain: fixes xs :: "('b, 'c) oalist_tc" shows "sorted_wrt (<) (OAlist_tc_sorted_domain xs)"
min_less_iff_conj: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and z :: "'f"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less" shows "less z (ord.min less_eq x y) = (less z x \<and> less z y)"
poly: fixes X :: "'a set" assumes "X \<in> \<F>'" shows "polytope X"
strict_sorted_interact_imp_concat: fixes as :: "'a list list"   and bs :: "'a list list" assumes "strict_sorted (interact as bs)" shows "strict_sorted (concat as) \<and> strict_sorted (concat bs)"
j: shows "on_triple f `` {e} \<subseteq> edges (S j)"
dual_space_linear_on: fixes E :: "'c \<Rightarrow> real"   and S :: "'c set" assumes "E \<in> dual_space S" shows "linear_fun_on S E"
converse_unif_subset_unifless: shows "unif\<inverse> \<subseteq> unifless"
interact_eq_Nil_iff: fixes xs :: "'a list list"   and ys :: "'a list list" assumes "xs \<in> lists (- {[]})"   and "ys \<in> lists (- {[]})" shows "(interact xs ys = []) = (xs = [] \<and> ys = [])"
rank_ge_imp_consistent: fixes A :: "(('c, 'd) vec, 'b) vec"   and b :: "('c, 'b) vec" assumes "(if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'c) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'c)) + 1 else 0) \<le> rank A" shows "consistent A b"
simps_6: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'd"   and x2 :: "'f" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
set_of_graph_rulesD_3: fixes Rs :: "(('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) set"   and R :: "('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph" assumes "set_of_graph_rules Rs"   and "R \<in> Rs" shows "subgraph (fst R) (snd R)"
derivesp_def: fixes g :: "(('c, 'd) rule \<Rightarrow> bool) \<times> 'c" shows "derivesp g = (\<lambda>lhs rhs. (lhs, rhs) \<in> derives (Collect (fst g), snd g))"
zero_le_log_cancel_iff: fixes a :: "real"   and x :: "real" assumes "1 < a"   and "0 < x" shows "(0 \<le> log a x) = (1 \<le> x)"
Abs_finfun_inverse: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> finfun" shows "finfun_apply (Abs_finfun y) = y"
cols_def: fixes A :: "'b mat" shows "cols A = map (col A) [0..<dim_col A]"
src_def: fixes \<mu> :: "'d" shows "D'.src \<mu> = (if D'.arr \<mu> then src\<^sub>D \<mu> else D'.null)"
prY_def: shows "prY = Sigma_Algebra.measure M (Y -` {Y w} \<inter> space M)"
HoVH_eq_ToCT: shows "H.HoVH = T.ToCT"
frees_const: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "frees (const name) = {||}"
map1_rel_witness_pmf: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c pmf"   and y :: "'d pmf" assumes "rel_pmf A x y" shows "map_pmf fst (rel_witness_pmf A (x, y)) = x"
left_neutral: fixes a :: "'c \<Rightarrow> bool" shows "(\<box> \<and>* a) = a"
closedin_compact_space: fixes X :: "'b topology"   and S :: "'b set" assumes "compact_space X"   and "closedin X S" shows "compactin X S"
mset_ran_by_sum: fixes a :: "int \<Rightarrow> 'a"   and r :: "int set" shows "mset_ran a r = (\<Sum>i\<in>r. {#a i#})"
A: fixes e :: "'e" shows "(case Inl e of Inl e \<Rightarrow> e) = e"
eqs: shows "{w \<in> space M. w !! 0} = pseudo_proj_True (Suc 0) -` {True ## sconst True}"
indspaceD: fixes f :: "('r, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('r, 'g) aezfun set"   and V :: "'v set" assumes "f \<in> indspace G FH V" shows "f \<in> addfunset FH V \<inter> smultfunset G FH"
not_in_preal_ub: fixes A :: "rat set"   and x :: "rat"   and y :: "rat" assumes "Dedekind_Real.cut A"   and "x \<notin> A"   and "y \<in> A"   and "0 < x" shows "y < x"
h5: shows "m = a'\<^sup>2 + b'\<^sup>2"
get_root_node_same_owner_document: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root" shows "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document = h \<turnstile> get_owner_document root \<rightarrow>\<^sub>r owner_document"
get_M_pure: fixes ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm"   and h :: "(_) heap" shows "pure (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter) h"
minus_frag_cmul: fixes k :: "int"   and x :: "'c \<Rightarrow>\<^sub>0 int" shows "- frag_cmul k x = frag_cmul (- k) x"
dg_Rel_incl_Rel_is_arr: fixes A :: "V"   and B :: "V" assumes "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>"   and "A \<subseteq>\<^sub>\<circ> B" shows "incl_Rel A B : A \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> B"