hcju/isabelleps huggingface dataset:
sum_of_2_squares_nat_necessary: fixes n :: "nat"   and p :: "nat" assumes "sum_of_2_squares_nat n"   and "0 < n"   and "prime p"   and "[p = 3] (mod 4)" shows "even (multiplicity p n)"
Arg2pi_lt_2pi: fixes z :: "complex" shows "Arg2pi z < 2 * pi"
merge_guards_guaranteeStrip: fixes f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "merge_guards (guaranteeStrip f g c) = (let c' = merge_guards c in if is_Guard c' then let (f', g', c') = dest_Guard c' in if f = f' then Guard f (g \<inter> g') c' else Guard f g (Guard f' g' c') else Guard f g c')"
choose_def: fixes v :: "'a" shows "choose v \<equiv> THE \<sigma>. minimal_good_strategy v \<sigma>"
sorted_wrt_true: fixes xs :: "'c list" shows "sorted_wrt (\<lambda>_ _. True) xs"
shiftr_not_mask_0: fixes n :: "nat"   and m :: "nat"   and w :: "'b word" assumes "LENGTH('b) \<le> n + m" shows "(w >> n) AND NOT (mask m) = 0"
bisimInputPushRes: fixes x :: "name"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> N" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P) \<sim> M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P"
constant_def: fixes f :: "'b \<Rightarrow> 'c" shows "constant f = (\<forall>x y. f x = f y)"
ord_equiv_trans: fixes E :: "('e, 'f) Order_scheme"   and F :: "('g, 'h) Order_scheme" assumes "Order E"   and "Order F"   and "ord_equiv D E"   and "ord_equiv E F" shows "ord_equiv D F"
catalan_aux_integral: shows "((\<lambda>x. sqrt ((4 - x) / x)) has_integral 2 * pi) {0..4}"
closed_buffon_set: fixes l :: "real"   and d :: "real" shows "closed (buffon_set l d)"
ag_nsum_1_nonzeroTr: fixes n :: "nat"   and l :: "nat" shows "\<forall>f. (\<forall>j\<le>n. f j \<in> carrier A) \<and> l \<le> n \<and> (\<forall>j\<in>{j. j \<le> n} - {l}. f j = \<zero>) \<longrightarrow> \<Sigma>\<^sub>e A f n = f l"
simps_2: fixes v :: "nat" shows "opposite (Neg v) = Pos v"
true_iff: fixes P :: "'c \<Rightarrow> bool" shows "(P \<longleftrightarrow> true) = P"
rComm1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
distinct_real_parts_3: fixes p :: "int poly" shows "distinct (map real_of_3 (real_parts_3 p))"
map_def: shows "\<tau>'.map = transformation_by_components.map A B G (\<lambda>a. \<tau>.B.inv (\<tau> a))"
ideal_deg_shifts: fixes d :: "nat"   and fs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "ideal (set (deg_shifts d fs)) = ideal (set fs)"
distance_witness: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H\<^esub> w"   and "\<And>xs'. v \<leadsto>xs'\<leadsto>\<^bsub>H\<^esub> w \<Longrightarrow> length xs \<le> length xs'" shows "H.distance v w = length xs"
r_hd: fixes e :: "nat" shows "eval r_hd [e] \<down>= e_hd e"
subset_ball: fixes d :: "real"   and e :: "real"   and x :: "'a" assumes "d \<le> e" shows "ball x d \<subseteq> ball x e"
is_class_def: fixes C :: "cname" shows "is_class C \<equiv> class C \<noteq> None"
showsp_rat_def: fixes p :: "nat"   and x :: "rat" shows "showsp_rat p x = (case quotient_of x of (d, n) \<Rightarrow> if n = 1 then showsp_int p d else showsp_int p d \<circ> shows_string ''/'' \<circ> showsp_int p n)"
process_prop_4_def: fixes P :: "('e list \<times> 'e set) set \<times> 'e list set" shows "process_prop_4 P \<equiv> \<forall>xs x X. (xs, X) \<in> fst P \<longrightarrow> (xs @ [x], {}) \<in> fst P \<or> (xs, insert x X) \<in> fst P"
less_oSucE: fixes x :: "ordinal"   and y :: "ordinal" assumes "x < oSuc y" obtains "x < y"   | "x = y"
isomorphicE: fixes a :: "'a \<times> 'a"   and a' :: "'a \<times> 'a" assumes "VoV.isomorphic a a'" obtains f :: "'a \<times> 'a" where "VoV.in_hom f a a' \<and> VoV.iso f"
ab_le: fixes x :: "complex" assumes "x \<in> closed_segment a b" shows "d \<bullet> x \<le> k"
field_differentiable_imp_continuous_at: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and S :: "'a set" assumes "f field_differentiable at x within S" shows "continuous (at x within S) f"
Cauchy_Schwarz_ineq: fixes x :: "'a"   and y :: "'a" shows "(inner x y)\<^sup>2 \<le> inner x x * inner y y"
dec_clock_def: fixes s :: "'a state" shows "dec_clock s = update_clock (\<lambda>_. clock s - 1) s"
distinct_5: fixes x2 :: "nat"   and x31 :: "type_scheme"   and x32 :: "type_scheme" shows "BVar x2 \<noteq> x31 =-> x32"
enumerate_gexp_inputs_list: fixes g :: "vname gexp" shows "\<exists>l. enumerate_gexp_inputs g = set l"
edka_partial_def: shows "edka_partial \<equiv> let f = \<lambda>_. 0::'capacity in while\<^bsup>fofu_invar\<^esup> (\<lambda>(f, brk). \<not> brk) (\<lambda>(f, uu_). find_shortest_augmenting_spec f \<bind> (\<lambda>p. case p of None \<Rightarrow> return (f, True) | Some p \<Rightarrow> assert (p \<noteq> []) \<bind> (\<lambda>_. assert (NPreflow.isAugmentingPath c s t f p) \<bind> (\<lambda>_. assert (Graph.isShortestPath (cf_of f) s p t) \<bind> (\<lambda>_. let f = NFlow.augment_with_path c f p in assert (NFlow c s t f) \<bind> (\<lambda>_. return (f, False))))))) (f, False) \<bind> (\<lambda>(f, uu_). assert (NFlow c s t f) \<bind> (\<lambda>_. return f))"
check_instr'_pc: fixes ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list"   and pc' :: "nat" assumes "check_instr' ins P h stk loc C M pc frs"   and "ins_jump_ok ins pc'" shows "check_instr' ins P h stk loc C M pc' frs"
c: fixes x :: "'a vec" assumes "x \<in> X" shows "\<bar>c x\<bar> \<le> (1::'a)"
subst_apply_pairs_nil: fixes \<delta> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "[] \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta> = []"
lens_quotient_def: fixes X :: "'d \<Longrightarrow> 'f"   and Y :: "'e \<Longrightarrow> 'f" shows "X /\<^sub>L Y = \<lparr>lens_get = \<lambda>\<sigma>. get\<^bsub>X\<^esub> (create\<^bsub>Y\<^esub> \<sigma>), lens_put = \<lambda>\<sigma> v. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> (create\<^bsub>Y\<^esub> \<sigma>) v)\<rparr>"
power_strict_decreasing_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero b"   and "less b one" shows "less (power.power one times b m) (power.power one times b n) = (n < m)"
sub: fixes i :: "name"   and t1 :: "trm"   and t2 :: "trm"   and t3 :: "trm"   and t4 :: "trm"   and x :: "trm" assumes "atom i \<sharp> tms" shows "(P t1 t2 t3 t4)(i::=x) = P (subst i x t1) (subst i x t2) (subst i x t3) (subst i x t4)"
assms_2: shows "m2 \<in> carrier_mat d2 d2"
list_emb_append_mono: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and xs' :: "'a list"   and ys :: "'a list"   and ys' :: "'a list" assumes "list_emb P xs xs'"   and "list_emb P ys ys'" shows "list_emb P (xs @ ys) (xs' @ ys')"
Span_code_def: fixes S :: "'a ell2 set" shows "Span_code S = ccspan S"
A1: shows "\<forall>S. (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel"
invimage_iff2: fixes a :: "V"   and r :: "V"   and A :: "V" shows "(a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A) = (a \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ> `\<^sub>\<circ> A)"
C_gets_correct_PG: fixes A :: "agent"   and M :: "agent"   and LID_M :: "nat"   and xid :: "msg"   and cc :: "msg"   and cm :: "msg"   and P :: "agent"   and EKj :: "nat"   and evs :: "event list" assumes "Gets A (sign (priSK M) \<lbrace>Number LID_M, xid, cc, cm, cert P EKj onlyEnc (priSK RCA)\<rbrace>) \<in> set evs"   and "evs \<in> set_pur"   and "M \<notin> bad" shows "\<exists>j trans. P = PG j \<and> Notes M \<lbrace>Number LID_M, Agent P, trans\<rbrace> \<in> set evs \<and> EKj = pubEK P"
map_def: fixes f :: "'a arrow_of_spans_data" shows "local.map f = (if arr f then f else null)"
derived_set_of_finite: fixes X :: "'c topology"   and S :: "'c set" assumes "Hausdorff_space X"   and "finite S" shows "X derived_set_of S = {}"
assms_1: shows "equivalentFormulae (F @ val2form M) F0"
discr_coind: fixes phi :: "('test, 'atom) com \<Rightarrow> bool"   and c :: "('test, 'atom) com" assumes "phi c"   and "\<And>c s c' s'. \<lbrakk>phi c; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> s \<approx> s' \<and> (phi c' \<or> discr c')"   and "\<And>c s s'. \<lbrakk>phi c; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> s \<approx> s'" shows "discr c"
A_2: fixes i :: "'a" shows "a $ i \<in> A i"
commute: fixes a :: "'g"   and b :: "'g" shows "a \<sqinter> b = b \<sqinter> a"
nhdsin_def: fixes X :: "'b topology"   and a :: "'b" shows "nhdsin X a = (if a \<in> topspace X then Inf (principal ` {S. openin X S \<and> a \<in> S}) else bot)"
strand_fv_bvars_disjointD_2: fixes S :: "('b, 'a) strand_step list"   and X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S = {}"   and "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X \<subseteq> fv\<^sub>s\<^sub>t S"
shEx_pbm_right: fixes P :: "('d, 'f) urel"   and M :: "(('d, 'f, 'g) mrg, 'e) urel"   and Q :: "'h \<Rightarrow> ('d, 'g) urel" shows "P \<parallel>\<^bsub>M\<^esub> ushEx Q = (\<^bold>\<exists> x \<bullet> P \<parallel>\<^bsub>M\<^esub> Q x)"
extends_imp_wf_sub_rel: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and sub :: "('c \<times> nat) \<times> 'c \<times> nat"   and g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs' :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "wf_sub_rel subs"   and "extends g subs sub subs'" shows "wf_sub_rel subs'"
pair_pmf_def: fixes A :: "'e pmf"   and B :: "'f pmf" shows "pair_pmf A B = A \<bind> (\<lambda>x. B \<bind> (\<lambda>y. return_pmf (x, y)))"
hyps_3: shows "subst_domain (compose ss) \<inter> range_vars (compose ss) = {}"
Rep_downset: fixes x :: "'a downset" shows "Rep_downset x \<in> range \<down>"
length_nths: fixes xs :: "'c list"   and I :: "nat set" shows "length (nths xs I) = card {i. i < length xs \<and> i \<in> I}"
rev_append: fixes xs :: "'g list"   and ys :: "'g list" shows "rev (xs @ ys) = rev ys @ rev xs"
prv_imp_refl2: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "\<phi> = \<psi>" shows "prv (imp \<phi> \<psi>)"
success_def: fixes f :: "['c]"   and h :: "heap" shows "success f h = (execute f h \<noteq> None)"
not_eq_order_implies_strict: fixes a :: "'t"   and b :: "'t" assumes "a \<noteq> b"   and "b \<preceq>\<^sub>t a" shows "b \<prec>\<^sub>t a"
TermEquivCl_def: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" shows "[A,e,B]\<^bsub>T\<^esub> \<equiv> m2ZF (TermEquivCl' T A e B)"
Ana_keys_fv: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and R :: "('fun, 'var) Term.term list" assumes "Ana t = (K, R)" shows "fv\<^sub>s\<^sub>e\<^sub>t (set K) \<subseteq> fv t"
GOAL_INDICATION_def: fixes uu :: "'b" shows "\<paragraph>uu \<equiv> True"
sets_Collect_countable_Ex': fixes I :: "'c set"   and M :: "'d measure"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> {x \<in> space M. P i x} \<in> sets M"   and "countable I" shows "{x \<in> space M. \<exists>i\<in>I. P i x} \<in> sets M"
alpha_qSkel: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "X #= X'" shows "qSkel X = qSkel X'"
infinite_arbitrarily_large: fixes A :: "'c set"   and n :: "nat" assumes "infinite A" shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
triple_star: fixes x :: "'a" assumes "is_inj x"   and "is_p_fun x" shows "x\<^sup>\<star> ; x\<^sup>T\<^sup>\<star> ; x\<^sup>\<star> = x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>"
completeness_0: fixes p :: "'b pl" assumes "{} |= p" shows "{} |- p"
pivot_funD_5: fixes A :: "'b mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat"   and i' :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr"   and "f i < nc"   and "i' < nr"   and "i' \<noteq> i" shows "A $$ (i', f i) = (0::'b)"
Exit_source: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g"   and a :: "'f" assumes "CFGExit sourcenode targetnode kind valid_edge Entry Exit"   and "valid_edge a"   and "sourcenode a = Exit" shows "False"
assign_loc_reach: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c' :: "('b, 'a, 'c) Stmt"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'd"   and x :: "'b"   and e :: "'a"   and annos :: "'b ModeUpd list"   and mds :: "Mode \<Rightarrow> 'b set"   and mem :: "'b \<Rightarrow> 'd" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "((c', mds'), mem') \<in> sifum_security.loc_reach (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((sifum_modes.annotate (x \<leftarrow> e) annos, mds), mem)" shows "c' = Stop \<and> mds' = sifum_modes.update_annos mds annos \<or> c' = sifum_modes.annotate (x \<leftarrow> e) annos \<and> mds' = mds"
has_limits_of_shape_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "J'J.A.has_limits_of_shape J \<equiv> \<forall>D. diagram J J' D \<longrightarrow> (\<exists>a \<chi>. limit_cone J J' D a \<chi>)"
ANDSep: fixes p :: "('b set set, 'c) Combinators list" assumes "allNetsDistinct p" shows "allNetsDistinct (separate p)"
poly_minus: fixes p :: "'a poly"   and x :: "'a" shows "poly (- p) x = - poly p x"
r_add_def: shows "r_add = real_of_int z2n1 * sqrt (real_of_int (int b))"
jump_poly_coprime: fixes p :: "real poly"   and x :: "real"   and q :: "real poly" assumes "poly p x = 0"   and "coprime p q" shows "jump_poly q p x = jump_poly 1 (q * p) x"
1: fixes x :: "complex" assumes "x \<in> \<real>" shows "cmod (inverse (1 + x\<^sup>2)) \<le> 1"
induct: fixes P1 :: "Term.var \<Rightarrow> bool"   and P2 :: "expr \<Rightarrow> bool"   and P3 :: "stmt \<Rightarrow> bool"   and P4 :: "expr list \<Rightarrow> bool"   and stmt :: "stmt" assumes "\<And>x. P1 (LVar x)"   and "\<And>x1 x2 x3 x4 x5. P2 x4 \<Longrightarrow> P1 ({x1,x2,x3}x4..x5)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P1 (x1.[x2])"   and "\<And>x1 x2. \<lbrakk>P3 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (InsInitV x1 x2)"   and "\<And>x. P2 (NewC x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (New x1[x2])"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Cast x1 x2)"   and "\<And>x1 x2. P2 x1 \<Longrightarrow> P2 (x1 InstOf x2)"   and "\<And>x. P2 (Lit x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (UnOp x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (BinOp x1 x2 x3)"   and "P2 Super"   and "\<And>x. P1 x \<Longrightarrow> P2 (Acc x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1:=x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (x1 ? x2 : x3)"   and "\<And>x1 x2 x3 x4 x5 x6 x7. \<lbrakk>P2 x4; P4 x7\<rbrakk> \<Longrightarrow> P2 ({x1,x2,x3}x4\<cdot>x5( {x6}x7))"   and "\<And>x1 x2. P2 (Methd x1 x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P2 (Body x1 x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (InsInitE x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Callee x1 x2)"   and "P3 Skip"   and "\<And>x. P2 x \<Longrightarrow> P3 (Expr x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (x1\<bullet> x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P3 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (If(x1) x2 Else x3)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (x1\<bullet> While(x2) x3)"   and "\<And>x. P3 (Jmp x)"   and "\<And>x. P2 x \<Longrightarrow> P3 (Throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P3 x1; P3 x4\<rbrakk> \<Longrightarrow> P3 (Try x1 Catch(x2 x3) x4)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 Finally x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (FinA x1 x2)"   and "\<And>x. P3 (Init x)"   and "P4 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P4 x2\<rbrakk> \<Longrightarrow> P4 (x1 # x2)" shows "P3 stmt"
ennreal_neq_top: fixes r :: "real" shows "ennreal r \<noteq> \<top>"
\<L>\<^sub>P_leftderives: fixes a :: "symbol list" assumes "a \<in> \<L>\<^sub>P" shows "\<exists>b. leftderives [\<SS>] (a @ b)"
leibniz_rule_field_derivative: fixes U :: "'a set"   and a :: "'b"   and b :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and fx :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and x0 :: "'a" assumes "\<And>x t. \<lbrakk>x \<in> U; t \<in> cbox a b\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"   and "\<And>x. x \<in> U \<Longrightarrow> f x integrable_on cbox a b"   and "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"   and "x0 \<in> U"   and "convex U" shows "((\<lambda>x. integral (cbox a b) (f x)) has_field_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
smult_r_minus: fixes a :: "'a"   and x :: "'c" assumes "a \<in> carrier K"   and "x \<in> carrier V" shows "a \<odot>\<^bsub>V\<^esub> \<ominus>\<^bsub>V\<^esub> x = \<ominus>\<^bsub>V\<^esub> (a \<odot>\<^bsub>V\<^esub> x)"
factorize_complex_main: fixes p :: "complex poly"   and c :: "complex"   and xis :: "(complex \<times> nat) list" assumes "factorize_complex_main p = Some (c, xis)" shows "p = smult c (\<Prod>(x, i)\<leftarrow>xis. [:- x, 1:] ^ Suc i)"
qmpt_factor_def: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" shows "qmpt_factor proj M2 T2 = (proj \<in> quasi_measure_preserving M M2 \<and> (AE x in M. proj (T x) = T2 (proj x)) \<and> qmpt M2 T2)"
elims_2: fixes x :: "'c"   and xa :: "'c arctic_delta"   and xb :: "'c arctic_delta" assumes "gt_arctic_delta x xa xb" obtains \<delta> :: "'c" and uu_ :: "'c arctic_delta" where "x = \<delta>'__" and "xa = uu_" and "xb = MinInfty_delta"   | \<delta> :: "'c" and xc :: "'c" and y :: "'c" where "x = \<delta>" and "xa = Num_arc_delta xc" and "xb = Num_arc_delta y" and "delta_gt \<delta> xc y"
I_intP_Fext_imp: fixes p :: "'psym"   and al :: "univ list" assumes "polC p = Fext"   and "\<not> Ik_intP p al" shows "\<not> I_intP p al"
fv: fixes \<phi> :: "'fmla" assumes "set (map snd txs) \<subseteq> var"   and "set (map fst txs) \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla"   and "\<phi> \<in> set [\<phi>]" shows "set (getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)) \<inter> Fvars \<phi> = {}"
less_linear: fixes x :: "'a word"   and y :: "'a word" shows "x <s y \<or> x = y \<or> y <s x"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and B :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A @ B) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A @ dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t B"
prems: fixes x :: "'a list" assumes "x \<in> set (z # zs)" shows "x \<noteq> []"
same_diag_refl: fixes n :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.same_diag n A A"
return_only_one_intra_edge: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and a :: "'g"   and Q' :: "('i \<Rightarrow> 'j option) \<times> 'k \<Rightarrow> bool"   and p :: "'l"   and f' :: "('i \<Rightarrow> 'j option) \<Rightarrow> ('i \<Rightarrow> 'j option) \<Rightarrow> 'i \<Rightarrow> 'j option" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<exists>!a'. valid_edge a' \<and> targetnode a' = targetnode a \<and> intra_kind (kind a')"
sum_list_map_filter': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero (map f (filter P xs)) = monoid_add.sum_list plus zero (map (\<lambda>x. if P x then f x else zero) xs)"
simps_1: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and k :: "'a"   and v :: "'b" assumes "comparator comp" shows "comparator.update_by_pair comp (k, v) [] = (if v = (0::'b) then [] else [(k, v)])"
index_tensor_gate_unitary1: fixes m :: "nat"   and G1 :: "complex Matrix.mat"   and n :: "nat"   and G2 :: "complex Matrix.mat"   and i :: "nat"   and j :: "nat" assumes "gate m G1"   and "gate n G2"   and "i < dim_row (1\<^sub>m (dim_col G1 * dim_col G2))"   and "j < dim_col (1\<^sub>m (dim_col G1 * dim_col G2))" shows "(G1 \<Otimes> G2\<^sup>\<dagger> * (G1 \<Otimes> G2)) $$ (i, j) = 1\<^sub>m (dim_col G1 * dim_col G2) $$ (i, j)"
assms_3: shows "P''' \<Longrightarrow>\<^sub>\<tau> P'"
poly_lift_hom_X_minus: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and a :: "'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "a \<in> carrier R" shows "poly_lift_hom R S \<phi> (X_poly_minus R a) = X_poly_minus S (\<phi> a)"