hcju/isabelleps huggingface dataset:
push_precond_eq_sat_or_nonsat: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "Network c s t" shows "Network.push_precond c f l e = (Network.sat_push_precond c f l e \<or> Network.nonsat_push_precond c f l e)"
rep_eq: fixes x :: "('h, 'i) \<I>" shows "responses_\<I> x = Rep_\<I> x"
arrE: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" obtains "f \<noteq> Null" and "Dom f \<in> {False, True}" and "Cod f \<in> {False, True}" and "Map f \<in> Collect (G.path (Dom f) (Cod f))"
monotoneI: fixes orda :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ordb :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)" shows "monotone orda ordb f"
algo_rel: shows "((f, l), fo, lo) \<in> pr_algo_rel\<^sup>*"
distinct_236: fixes x11 :: "Syntax.term"   and x101 :: "Syntax.term"   and x102 :: "Syntax.term"   and x103 :: "Syntax.term" shows "Prj1 x11 \<noteq> Case x101 x102 x103"
bi_unique_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "bi_unique R = ((\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y))"
drop_nonLNil: fixes t :: "'b llist"   and i :: "nat" assumes "t \<up> i \<noteq> LNil" shows "t \<noteq> LNil"
approx_reorient: fixes x :: "'b star"   and y :: "'b star" shows "(x \<approx> y) = (y \<approx> x)"
absCase_Abs_vsubst: fixes xs :: "'varSort"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and f :: "'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and x :: "'var" assumes "isInBar (xs, s)"   and "wls s X"   and "compatAbsVsubst xs s f" shows "absCase xs s f (Abs xs x X) = f x X"
order_iff: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (a = max a b)"
induct: fixes P :: "nat set \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat set"   and a1 :: "nat"   and a2 :: "nat" assumes "\<And>xs x y. (\<lbrakk>\<not> (x \<notin> xs \<or> infinite xs); \<not> y \<notin> xs\<rbrakk> \<Longrightarrow> P xs (x - 1) (y + 1)) \<Longrightarrow> P xs x y" shows "P a0 a1 a2"
w5: fixes x :: "'b \<Rightarrow> real" shows "(\<Sum>b | (b \<in> B \<longrightarrow> x b \<noteq> 0) \<and> b \<in> B. x b *\<^sub>R b) = (\<Sum>b\<in>B. x b *\<^sub>R b)"
perm_1: fixes pi :: "(name \<times> name) list"   and x2 :: "name"   and x1 :: "name" shows "pi \<bullet> x2[x1] = (pi \<bullet> x2)[(pi \<bullet> x1)]"
ground_se_fm_induction: fixes \<alpha> :: "fm"   and n :: "nat"   and e :: "(name, hf) finfun" assumes "ground_fm \<alpha>"   and "size \<alpha> < n"   and "se_fm \<alpha>"   and "eval_fm e \<alpha>" shows "{} \<turnstile> \<alpha>"
append_assoc: fixes xs :: "'f list"   and ys :: "'f list"   and zs :: "'f list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
lr_eq_ab: shows "(\<Union>i. {l i<..r i}) = {a<..b}"
bit_numeral_int_simps_6: fixes w :: "num"   and n :: "num" shows "bit (- numeral (num.Bit1 w)) (numeral n) = (\<not> bit (numeral w) (pred_numeral n))"
ordering_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Orderings.ordering less_eq less \<equiv> ((\<forall>a b. less a b = (less_eq a b \<and> a \<noteq> b)) \<and> (\<forall>a. less_eq a a)) \<and> (\<forall>a b. less_eq a b \<longrightarrow> less_eq b a \<longrightarrow> a = b) \<and> (\<forall>a b c. less_eq a b \<longrightarrow> less_eq b c \<longrightarrow> less_eq a c)"
hyps_2: fixes x :: "name"   and M :: "trm"   and M' :: "trm" assumes "x:M \<in> \<parallel>(A)\<parallel>"   and "M \<longrightarrow>\<^sub>a* M'" shows "x:M' \<in> \<parallel>(A)\<parallel>"
submodule_def: fixes H :: "'e set"   and R :: "('f, 'g) ring_scheme"   and M :: "('f, 'e, 'h) module_scheme" shows "Module.submodule H R M \<equiv> subgroup H (add_monoid M) \<and> submodule_axioms H R M"
ctr_1: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lnull xs"   and "lnull ys" shows "lappend xs ys = LNil"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "a \<le> sup a b"
simps_59: fixes x9 :: "Term.var"   and x21 :: "ty"   and x22 :: "expr" shows "Acc x9 \<noteq> New x21[x22]"
prob_space_axioms_def: fixes M :: "'a measure" shows "prob_space_axioms M \<equiv> emeasure M (space M) = 1"
assms: shows "(P, D, Q) = diagonalize_2x2 A"
sum_over_zero: fixes mp :: "real mpoly"   and x :: "nat" shows "mp = (\<Sum>i\<le>MPoly_Type.degree mp x. isolate_variable_sparse mp x i * Var x ^ i)"
nat_to_ce_set_srj: fixes A :: "nat set" assumes "A \<in> ce_sets" shows "\<exists>n. A = nat_to_ce_set n"
atMostTwoRes: fixes ts :: "traffic"   and c :: "cars" shows "|res ts c| \<le> 2"
finite_lprefix_nitpick_simps_1: fixes xs :: "'c llist" shows "Coinductive_List.finite_lprefix xs LNil = (xs = LNil)"
R5: fixes x :: "'b"   and xt :: "'b bintree"   and y :: "'b"   and yl :: "'b bintree"   and yr :: "'b bintree"   and ts :: "'b bintree list"   and X :: "'b t_type" assumes "\<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr> \<in> t_ins_set X"   and "\<not> x \<le> y" shows "\<lparr>t_type.folding = True, item = x, subtrees = Branch y yl xt # ts\<rparr> \<in> t_ins_set X"
pure_rel_eq_false_iff: fixes R :: "('a \<times> 'b) set"   and x :: "'b"   and y :: "'a" shows "(pure R x y = false) = ((y, x) \<notin> R)"
intro: fixes M :: "'c \<Rightarrow> 'd measure" assumes "product_sigma_finite M"   and "product_prob_space_axioms M" shows "product_prob_space M"
poly_compare_0_code: fixes f :: "'a poly" shows "(f = 0) = (case coeffs f of [] \<Rightarrow> True | a # list \<Rightarrow> False)"
scene_indep_override: fixes X :: "'a scene"   and Y :: "'a scene" shows "(X \<bowtie>\<^sub>S Y) = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on X \<oplus>\<^sub>S s\<^sub>3 on Y = s\<^sub>1 \<oplus>\<^sub>S s\<^sub>3 on Y \<oplus>\<^sub>S s\<^sub>2 on X)"
subset_fSuc: fixes f :: "nat \<Rightarrow> 'a set"   and u :: "nat"   and v :: "nat" assumes "\<And>u. f u \<subseteq> f (Suc u)"   and "u \<le> v" shows "f u \<subseteq> f v"
game_equiv_subst_eq: fixes \<alpha> :: "game"   and \<beta> :: "game"   and P :: "(variable \<Rightarrow> real) set \<Rightarrow> 'a"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set" assumes "game_equiv \<alpha> \<beta>" shows "P (game_sem I \<alpha> X) \<equiv> P (game_sem I \<beta> X)"
map_component_term_of_pair: fixes f :: "nat \<Rightarrow> nat"   and t :: "('a, 'b) pp"   and k :: "nat" shows "map_component_pprod f (t, k) = (t, f k)"
max_Var_floatariths_lessI: fixes i :: "nat"   and fas :: "floatarith list"   and j :: "nat" assumes "i < max_Var_floatarith (fas ! j)"   and "j < length fas" shows "i < max_Var_floatariths fas"
that: fixes xs' :: "'a list" assumes "xs' \<in> ys \<otimes> zs"   and "xs = filter P xs'" shows "thesis"
runit_hcomp_4: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<a>[f, g, src g] \<cdot> \<r>\<^sup>-\<^sup>1[f \<star> g] = f \<star> \<r>\<^sup>-\<^sup>1[g]"
simps_2: fixes y :: "Syntax.var"   and t' :: "Syntax.term"   and x :: "Syntax.var" shows "Var y[t' / x] = (if x = y then t' else Var y)"
tendsto_lowerbound: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a"   and F :: "'b filter"   and a :: "'a" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq a (f i)"   and "F \<noteq> \<bottom>" shows "less_eq a x"
double_negate: fixes l :: "lit" shows "negate (negate l) = l"
map_default_update_const: fixes f :: "'a \<Rightarrow> 'b option"   and a :: "'a"   and g :: "'a \<Rightarrow> 'b option"   and d :: "'b" assumes "finite (dom f)"   and "a \<notin> dom f"   and "f \<subseteq>\<^sub>m g" shows "upd a d (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)) = Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)"
assms_1: shows "x \<in>\<^sub>\<circ> vinsert y A"
Lt_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Lt C D \<or> B A Lt C D \<or> A B Lt D C \<or> B A Lt D C" shows "A B Lt C D"
cases: fixes a1 :: "nat"   and a2 :: "nat" assumes "residue_primroot a1 a2" obtains n :: "nat" and g :: "nat" where "a1 = n" and "a2 = g" and "0 < n" and "coprime n g" and "ord n g = totient n"
comp_assoc_assoc'_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "\<a>[a, b, c] \<cdot> \<a>\<^sup>-\<^sup>1[a, b, c] = a \<otimes> b \<otimes> c"
axioms_1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "vertical_composite A B F G H \<sigma> \<tau>" shows "category A"
TER_restrict_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current \<Gamma> g" shows "TER\<^bsub>\<Gamma>\<^esub> g \<subseteq> TER\<^bsub>quotient_web \<Gamma> f\<^esub> (g \<upharpoonleft> \<Gamma> / f)"
cupcake_list_preserve: fixes env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and vs :: "v list" assumes "list_all2_shortcircuit (cupcake_evaluate_single env) es rs"   and "is_cupcake_all_env env"   and "list_all is_cupcake_exp es"   and "sequence_result rs = Rval vs" shows "list_all is_cupcake_value vs"
inject_19: fixes x191 :: "char list"   and x192 :: "char list list"   and x193 :: "bool"   and x194 :: "'a exp"   and y191 :: "char list"   and y192 :: "char list list"   and y193 :: "bool"   and y194 :: "'a exp" shows "(INIT x191 (x192,x193) \<leftarrow> x194 = INIT y191 (y192,y193) \<leftarrow> y194) = (x191 = y191 \<and> x192 = y192 \<and> x193 = y193 \<and> x194 = y194)"
b_ge_k: fixes k :: "nat" assumes "0 < k" shows "k - 1 \<le> \<Sqinter> (b k (k, k - 1))"
vt: shows "vars_term_ms (SCF t) = vars_term_ms (SCF s) + u"
vec_eq_iff: fixes x :: "('e, 'f) Finite_Cartesian_Product.vec"   and y :: "('e, 'f) Finite_Cartesian_Product.vec" shows "(x = y) = (\<forall>i. x $h i = y $h i)"
valid_as_valid_as_proj: fixes as :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) list"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" assumes "as \<in> valid_plans PROB" shows "as_proj as vs \<in> valid_plans (prob_proj PROB vs)"
star_back_loop_least_fixpoint: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x * y \<squnion> z = x" shows "z * y\<^sup>\<star> \<le> x"
Tv_2: shows "has_sort (osig (sig \<Theta>)) (Tv idn S') S"
case_1: fixes f1 :: "CTypeId \<Rightarrow> 'a"   and f2 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and x1 :: "CTypeId" shows "(case new_instance x1 of new_instance x \<Rightarrow> f1 x | new_array x xa \<Rightarrow> f2 x xa) = f1 x1"
throwError_def: shows "throwError \<equiv> return \<circ> Inl"
DiffE: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A - B" obtains "c \<in> A" and "c \<notin> B"
is_none_def: fixes x :: "'a option" shows "Option.is_none x = (x = None)"
vec_setE: fixes a :: "'a"   and v :: "'a vec" assumes "a \<in> set\<^sub>v v" obtains i :: "nat" where "v $ i = a" and "i < dim_vec v"
gfh: fixes x :: "'a" assumes "x \<in> S" shows "(g \<circ> f) x = exp (h x)"
stop_doesnt_access: 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 x :: "'b" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_security.doesnt_modify (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) Stop x \<and> sifum_security.doesnt_read (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) Stop x"
simps_1: fixes n :: "nat" shows "pr_index_enumerator n 0 = n"
before_def: fixes f :: "'a \<Rightarrow> 'b" shows "$f \<equiv> %s. f (first s)"
wls_vsubst_eq_swap: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y1 :: "'var"   and y2 :: "'var"   and ys :: "'varSort" assumes "wls s X"   and "y1 = y2 \<or> fresh ys y1 X" shows "X #[y1 // y2]_ys = X #[y1 \<and> y2]_ys"
F_minus_D_seq_def: fixes P :: "'a process"   and Q :: "'a process" shows "F_minus_D_seq P Q \<equiv> {(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q}"
that: fixes j :: "nat"   and s :: "'a list" assumes "j < \<^bold>|ws\<^bold>|"   and "s <s ws ! j"   and "concat (take (Suc j) ws) = u \<cdot> s" shows "thesis"
max_Suc_Suc: fixes m :: "nat"   and n :: "nat" shows "max (Suc m) (Suc n) = Suc (max m n)"
isCont_eq_Lb: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x" shows "\<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x\<ge>a. x \<le> b \<and> f x = M)"
subdegree_minus_commute: fixes f :: "'a fps"   and g :: "'a fps" shows "subdegree (f - g) = subdegree (g - f)"
lasso_v0_def: fixes L :: "('e, 'f) lasso_scheme" shows "lasso_v0 L \<equiv> case lasso_reach L of [] \<Rightarrow> lasso_va L | v0 # x \<Rightarrow> v0"
mod_ring_gen_axioms: shows "mod_ring_gen (finite_field_ops_int p) mod_ring_rel p"
up_mult_closed: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "ring R"   and "p \<in> up R"   and "q \<in> up R" shows "(\<lambda>n. \<Oplus>\<^bsub>R\<^esub>i\<in>{..n}. p i \<otimes>\<^bsub>R\<^esub> q (n - i)) \<in> up R"
l2product_lzero: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" shows "l2product S (\<lambda>x. 0) f = 0"
plays_from_strategy_chain: fixes n :: "nat"   and m :: "nat"   and \<sigma> :: "'c list \<Rightarrow> 'c"   and p :: "'c list" assumes "n \<le> m" shows "prefix (strategy_progression \<sigma> p $ n) (strategy_progression \<sigma> p $ m)"
ppos: fixes s :: "'s" assumes "inv s" shows "0 \<le> \<Phi> s"
injective_comp_right_dist_inf: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "injective z" shows "(x \<sqinter> y) * z = x * z \<sqinter> y * z"
wp_loop_unfold: fixes P :: "'b \<Rightarrow> real"   and body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> bool" assumes "sound P"   and "healthy (wp body)" shows "wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P = (\<lambda>s. \<guillemotleft> \<N> G \<guillemotright> s * P s + \<guillemotleft> G \<guillemotright> s * wp body (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P) s)"
rref_upt_condition2_explicit: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and i :: "'d" assumes "reduced_row_echelon_form_upt_k A k"   and "\<not> is_zero_row_upt_k i k A" shows "A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'b)) = (1::'b)"
collapse_1: fixes sum :: "'c + 'd" assumes "isl sum" shows "Inl (projl sum) = sum"
leg0_composite: shows "doms.\<mu>\<nu>.leg0 = doms.\<nu>.leg0 \<cdot> doms.\<mu>\<nu>.prj\<^sub>0"
ih_13: shows "InvariantVarsM (getM state') F0 Vbl"
rimg_ring: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "Ring (rimg A R f)"
Ord_ordermap: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and x :: "'a" assumes "wf r"   and "trans r" shows "Ord (ordermap A r x)"
verit_sko_ex: fixes P :: "'a \<Rightarrow> bool" shows "(\<exists>x. P x) = P (SOME x. P x)"
left_add_twice: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_numeral one times plus" shows "plus a (plus a b) = plus (times (numeral.numeral one plus (num.Bit0 num.One)) a) b"
AbstFormP_Disj: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{AbstFormP v i x x', AbstFormP v i y y'} \<turnstile> AbstFormP v i (Q_Disj x y) (Q_Disj x' y')"
opeT: shows "openin (top_of_set (S \<union> T)) T"
G_relpow_matpow_pos_ge: fixes i :: "'n"   and j :: "'n"   and m :: "nat"   and n :: "nat" assumes "(i, j) \<in> G ^^ m"   and "m \<le> n" shows "0 < matpow (A + mat 1) n $h i $h j"
exhaust_sel: fixes state :: "('a, 'b) state" obtains "state = State (runState state)"
convergent: fixes f :: "'a \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'a" assumes "bounded_linear f"   and "convergent X" shows "convergent (\<lambda>n. f (X n))"
select_convs_1: fixes root :: "'c"   and edges :: "'c edge set"   and more :: "'d" shows "root \<lparr>root = root, edges = edges, \<dots> = more\<rparr> = root"
ra_2: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x ; (z \<cdot> y ; 1) = (x \<cdot> (y ; 1)\<^sup>T) ; z"
left_commute: fixes b :: "'k"   and a :: "'k"   and c :: "'k" shows "b * (a * c) = a * (b * c)"
elims_3: fixes x :: "'b list \<times> 'b list" assumes "\<not> invar x" obtains fs :: "'b list" and rs :: "'b list" where "x = (fs, rs)" and "\<not> (fs = [] \<longrightarrow> rs = [])"
point_not_equal_1: fixes p :: "'a"   and q :: "'a" assumes "is_point p"   and "is_point q" shows "(p \<noteq> q) = (p \<le> - q)"
special_axioms_into_Special_ax: fixes A :: "fm"   and e :: "(name, hf) finfun" assumes "A \<in> special_axioms" shows "\<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<in> Special_ax"
that: fixes r' :: "nat \<Rightarrow> 'Q \<times> nat" assumes "b_graph.is_acc_run (degeneralize_ext m) r'"   and "r = fst \<circ> r'" shows "thesis"
2: shows "convergent_subseq s n l = residue (p ^ l) (convergent_subseq s l l)"
fg: shows "homotopic_with (\<lambda>k. k ` T \<subseteq> T) Y Y (f \<circ> g) id"
pdivmod_monic_2: fixes g :: "'a poly"   and f :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" assumes "monic g"   and "pdivmod_monic f g = (q, r)" shows "r = 0 \<or> degree r < degree g"
closed_singleton: fixes "open" :: "'d set \<Rightarrow> bool"   and a :: "'d" assumes "class.t1_space open" shows "topological_space.closed open {a}"
divmod_numeral_simps_16: fixes m :: "num" shows "(num.Bit1 m \<le> num.One) = False"
all_not_in_conv: fixes A :: "'b set" shows "(\<forall>x. x \<notin> A) = (A = {})"
qe_FMo_def: shows "qe_FMo = R\<^sub>e.lift_dnfeq_qe qe_FMo\<^sub>1"
concurrent_expunge_delete_independent: fixes i :: "'a"   and e1 :: "'b"   and mo :: "'a"   and ir :: "'a"   and "is" :: "'a set"   and e2 :: "'b"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat" assumes "\<not> hb (i, Expunge e1 mo i) (ir, Delete is e2)"   and "\<not> hb (ir, Delete is e2) (i, Expunge e1 mo i)"   and "xs prefix of j"   and "(i, Expunge e1 mo i) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e2) \<in> set (node_deliver_messages xs)" shows "i \<notin> is"
collapse_6: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" assumes "is_InSet stateful_strand_step" shows "\<langle>the_check stateful_strand_step: the_elem_term stateful_strand_step \<in> the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
that: fixes t' :: "nat" assumes "t' < N"   and "has_ticket (ts t') c" shows "thesis"
int: shows "integrable lborel (\<lambda>\<phi>. indicat_real {0..pi} \<phi> * min (d / 2) (sin \<phi> * l / 2))"
simps_9: fixes c :: "int"   and t :: "MIR.num"   and s :: "MIR.num" shows "simpnum (CF c t s) = simpnum (Add (Mul c (Floor t)) s)"
poly_divides_conv0: fixes q :: "'a poly"   and p :: "'a poly" assumes "degree q < degree p"   and "p \<noteq> 0" shows "(p dvd q) = (q = 0)"
span_add_eq: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "module scale"   and "x \<in> module.span scale S" shows "(x + y \<in> module.span scale S) = (y \<in> module.span scale S)"
proper_it_parE: fixes it :: "'a \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and it' :: "'a \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<forall>x. proper_it (it x) (it' x)" obtains f :: "'a \<Rightarrow> 'c list" where "it = (\<lambda>x. foldli (f x))" and "it' = (\<lambda>x. foldli (f x))"
preserves: fixes M :: "moebius" assumes "unit_disc_fix M"   and "P (moebius_circline M H)" shows "P H"
assms: shows "(SubnetS s t, a, st') \<in> pnet_sos (automaton.trans (pnet np p1)) (automaton.trans (pnet np p2))"
leading_monomial_tail: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "p = monomial (lcf p) (lpp p) + punit.tail p"
singular_relboundary_alt: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_relboundary p X S c = (\<exists>d e. singular_chain (Suc p) X d \<and> singular_chain p (subtopology X S) e \<and> chain_boundary (Suc p) d = c + e)"
eqvt_cases: fixes \<pi> :: "perm"   and f :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>x. \<pi> \<bullet> f x = f (\<pi> \<bullet> x)" obtains "f x" and "f (\<pi> \<bullet> x)"   | "\<not> f x" and "\<not> f (\<pi> \<bullet> x)"
ih: fixes s :: "('s, 'v) tm" assumes "size s < size sa__"   and "wary s" shows "\<not> s >\<^sub>t s"
left_total_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "left_total R = (\<forall>x. \<exists>y. R x y)"
case_eq_if: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and strand_step :: "('d, 'e) strand_step" shows "(case strand_step of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = (if is_Send strand_step then f1 (un_Send strand_step) else if is_Receive strand_step then f2 (un_Receive strand_step) else if is_Equality strand_step then f3 (un_Equality1 strand_step) (un_Equality2 strand_step) (un_Equality3 strand_step) else f4 (bvars\<^sub>s\<^sub>t\<^sub>p strand_step) (un_Inequality2 strand_step))"
norm_signed01: fixes xs :: "bit list" shows "norm_signed (\<zero> # \<one> # xs) = \<zero> # \<one> # xs"
cons: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> U" shows "\<forall>k\<le>n. \<psi> (the (s (f \<triangleright> n))) k = f k"
DK_fresh_not_KeyCryptKey: fixes DK :: "nat"   and evs :: "event list"   and K :: "nat" assumes "Key DK \<notin> used evs"   and "evs \<in> set_cr" shows "\<not> KeyCryptKey DK K evs"
exit_def: fixes s :: "'state"   and v :: "'value" shows "exit s v \<equiv> \<forall>tr trn. validFrom s (tr ## trn) \<and> never T (tr ## trn) \<and> \<phi> trn \<longrightarrow> f trn \<noteq> v"
FSwImorph_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'l, 'h, 'j, 'm) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'l, 'i, 'k, 'n) model_scheme" shows "FSwImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD'"