hcju/isabelleps huggingface dataset:
lea_refl: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B" shows "A B C LeA A B C"
CD_on_range: fixes ds :: "'b set"   and A :: "'a set" shows "Y.CD_on ds A \<subseteq> A \<inter> (\<Union>d\<in>ds. Field (Pd_singletons_for_ds Y ds d))"
effect__strips_iii_a: fixes s' :: "'a \<Rightarrow> bool option"   and s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and v :: "'a" assumes "s' = s \<then> op"   and "v \<in> set (add_effects_of op)" shows "s' v = Some True"
smap_pseudominimal: fixes w :: "'w"   and s :: "'w" assumes "w \<in> W"   and "s \<in> S" shows "dual_order.poset_simplex_map \<P> (w +o \<langle>S - {s}\<rangle>) = {w +o \<langle>S - {s}\<rangle>}"
r_dummy_recfn: fixes f :: "recf"   and a :: "nat"   and n :: "nat" assumes "recfn a f" shows "recfn (a + n) (r_dummy n f)"
futures_def: fixes P :: "'a process"   and xs :: "'a list" shows "futures P xs \<equiv> {(ys, Y). (xs @ ys, Y) \<in> failures P}"
intros_6: fixes s :: "'a"   and b :: "'a set"   and c :: "'a com"   and s'' :: "'a"   and x :: "'a set"   and y :: "'a \<Rightarrow> nat"   and s' :: "'a" assumes "s \<in> b"   and "Sem c s s''"   and "Sem (While b x y c) s'' s'" shows "Sem (While b x y c) s s'"
scale_right_imp_eq: fixes x :: "'a"   and a :: "complex"   and b :: "complex" assumes "x \<noteq> (0::'a)"   and "a *\<^sub>C x = b *\<^sub>C x" shows "a = b"
intros: fixes s :: "'s1"   and tl :: "'tl1"   and s' :: "'s1" assumes "s -1-tl\<rightarrow> s'"   and "\<tau>move1 s tl s'" shows "s -\<tau>1\<rightarrow> s'"
nn_integral_cmult_indicator: fixes A :: "'b set"   and M :: "'b measure"   and c :: "ennreal" assumes "A \<in> sets M" shows "\<integral>\<^sup>+x\<in>A. c\<partial>M = c * emeasure M A"
Gromov_extension_onto: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d Gromov_completion" assumes "lambda C -quasi_isometry_between UNIV UNIV f"   and "y \<in> Gromov_boundary" shows "\<exists>x\<in>Gromov_boundary. Gromov_extension f x = y"
oppOrdering_simps_1: shows "oppOrdering\<cdot>LT = GT"
cos_lt_zero_on_pi2_pi: fixes x :: "real" assumes "pi / 2 < x"   and "x \<le> pi" shows "cos x < 0"
hyps_2: assumes "K \<in> multisets A" shows "mulex_on P A (K + L) (K + M)"
pathVertices_fwd_def: fixes u :: "nat"   and p :: "(nat \<times> nat) list" shows "pathVertices_fwd u p = u # map snd p"
not_integrable_integral: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'b set" assumes "\<not> f integrable_on i" shows "integral i f = (0::'c)"
Inf_lower2: fixes Inf :: "'a::type set \<Rightarrow> 'a::type"   and Sup :: "'a::type set \<Rightarrow> 'a::type"   and inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and bot :: "'a::type"   and top :: "'a::type"   and u :: "'a::type"   and A :: "'a::type set"   and v :: "'a::type" assumes "class.complete_lattice (Inf::'a::type set \<Rightarrow> 'a::type) (Sup::'a::type set \<Rightarrow> 'a::type) (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (bot::'a::type) (top::'a::type)"   and "(u::'a::type) \<in> (A::'a::type set)"   and "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (u::'a::type) (v::'a::type)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) ((Inf::'a::type set \<Rightarrow> 'a::type) (A::'a::type set)) (v::'a::type)"
simps_8: fixes v :: "nat" shows "unpower v TrueF = TrueF"
f: fixes i :: "nat" assumes "i \<le> degree p" shows "coeff p i = of_rat (f i)"
fresh_star_bij_2: fixes pi :: "(name \<times> name) list"   and b :: "name list"   and x :: "'a" shows "(pi \<bullet> b) \<sharp>* (pi \<bullet> x) = b \<sharp>* x"
ne_free_theorem: shows "map (h \<circ> n2l) (candidate (\<lambda>x y. l2n (f (n2l x) (n2l y))) (map l2n zs)) = candidate g (map (h \<circ> n2l) (map l2n zs))"
imp_list_def: fixes is_list :: "'a list \<Rightarrow> 'b \<Rightarrow> assn" shows "imp_list is_list \<equiv> precise is_list"
fps_integral_nth_0_Suc_2: fixes a :: "'a fps"   and a0 :: "'a"   and n :: "nat" shows "fps_nth (fps_integral a a0) (Suc n) = inverse (of_nat (Suc n)) * fps_nth a n"
i_take_i_join: fixes I :: "nat set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "infinite I" shows "f \<Join>\<^sub> I \<Down> n = f \<Down> (I \<rightarrow> n) \<Join>\<^sub> I"
subst_cls_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AA :: "'b literal multiset"   and \<sigma> :: "'c" shows "substitution_ops.subst_cls subst_atm AA \<sigma> = {#substitution_ops.subst_lit subst_atm A \<sigma>. A \<in># AA#}"
iso_inv_iso: fixes f :: "'s" assumes "local.iso f" shows "local.iso (local.inv f)"
select_convs_1: fixes mReturn :: "nat"   and mName :: "nat"   and mParams :: "varDef list"   and mBody :: "exp"   and more :: "'a" shows "mReturn \<lparr>mReturn = mReturn, mName = mName, mParams = mParams, mBody = mBody, \<dots> = more\<rparr> = mReturn"
fls_X_times_conv_shift_1: fixes f :: "'a fls" shows "fls_X * f = fls_shift (- 1) f"
berlekamp_basis_eq_8: fixes v :: "'b mod_ring poly"   and u :: "'b mod_ring poly" assumes "v \<in> set (berlekamp_basis u)" shows "[v ^ CARD('b) = v] (mod u)"
hyps: assumes "list_all (\<lambda>i. i < length signs) I"   and "list_all (\<lambda>s. s = 1 \<or> s = 0 \<or> s = - 1) signs" shows "prod_list (map ((!) signs) I) = 1 \<or> prod_list (map ((!) signs) I) = 0 \<or> prod_list (map ((!) signs) I) = - 1"
mono_image_least: fixes f :: "'g \<Rightarrow> 'h"   and m :: "'g"   and n :: "'g"   and m' :: "'h"   and n' :: "'h" assumes "mono f"   and "f ` {m..<n} = {m'..<n'}"   and "m < n" shows "f m = m'"
wordinterval_eq_set_eq: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_eq r1 r2 = (wordinterval_to_set r1 = wordinterval_to_set r2)"
inverse_unit: shows "comm.multiplicative.inverse \<one> = \<one>"
transitions_invariant: fixes Q :: "(att \<times> char list, att, nat) env \<Rightarrow> bool"   and P :: "operation \<Rightarrow> bool"   and root :: "(att \<times> char list, att, nat) env"   and xs :: "operation list"   and root' :: "(att \<times> char list, att, nat) env" assumes "\<And>r x r'. \<lbrakk>r \<midarrow>x\<rightarrow> r'; Q r; P x\<rbrakk> \<Longrightarrow> Q r'"   and "root \<Midarrow>xs\<Rightarrow> root'"   and "Q root"   and "\<forall>x\<in>set xs. P x" shows "Q root'"
simps_1: fixes f :: "'c" shows "Can \<^bold>\<langle>f\<^bold>\<rangle> = C.ide f"
Fvars_S: shows "Fvars S = {xx, yy}"
emeasure_pair_measure: fixes M :: "'b measure"   and X :: "('c \<times> 'b) set"   and N :: "'c measure" assumes "sigma_finite_measure M"   and "X \<in> sets (N \<Otimes>\<^sub>M M)" shows "emeasure (N \<Otimes>\<^sub>M M) X = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>M \<partial>N"
g_def: fixes t :: "'t" shows "g t = (SOME s. R s t)"
harm_expansion_bigo_simple1: shows "(\<lambda>n. harm n - (ln (real n) + euler_mascheroni + 1 / real (2 * n))) \<in> O(\<lambda>n. 1 / real (n\<^sup>2))"
app_vfield_def: fixes r :: "V" shows "\<F>\<^sub>\<circ> r = \<D>\<^sub>\<circ> r \<union>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
a: shows "map (\<lambda>i. f ((x1 # a # l) ! i) ((a # l) ! i)) [0..<length l] = f x1 a # map (\<lambda>i. f ((a # l) ! i) (l ! i)) [0..<length l - 1]"
crsp_step_goto: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires" shows "\<exists>stp>0. crsp ly (abc_step_l (as, lm) (Some (Goto n))) (steps (s, l, r) (ci ly (start_of ly as) (Goto n), start_of ly as - Suc 0) stp) ires"
create_gba__old_fin: shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. \<forall>nd\<in>g_V \<A>. finite (old nd))"
shconf_upd_obj: fixes P :: "'a prog"   and h :: "heap"   and sh :: "sheap"   and C :: "char list"   and sfs' :: "sfields"   and i' :: "init_state" assumes "P,h \<turnstile>\<^sub>s sh \<surd>"   and "P,h,C \<turnstile>\<^sub>s sfs' \<surd>" shows "P,h \<turnstile>\<^sub>s sh(C \<mapsto> (sfs', i')) \<surd>"
final_no_red: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "final x" shows "False"
up: shows "limsup (\<lambda>n. u n + v n) \<le> a + limsup v"
P2_cong: fixes tms :: "tm list"   and P :: "tm \<Rightarrow> tm \<Rightarrow> fm"   and H :: "fm set"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" assumes "\<And>i t u x. atom i \<sharp> tms \<Longrightarrow> (P t u)(i::=x) = P (subst i x t) (subst i x u)"   and "H \<turnstile> x EQ x'"   and "H \<turnstile> y EQ y'" shows "H \<turnstile> P x y IFF P x' y'"
1: shows "S.adjoint_pair (S.UP f) (S.UP g) \<and> S.adjoint_pair (S.UP f) (S.UP g')"
Sup_atMost: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and y :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (ord.atMost less_eq y) = y"
Dijkstra_Impl_Defs_axioms: shows "Dijkstra_Impl_Defs G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin G_\<alpha>"
seq_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1 ;; c\<^sub>2, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"   and "c\<^sub>1 \<noteq> Stop" shows "\<exists>c\<^sub>1''. \<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>1'', mds', mem'\<rangle> \<and> c' = c\<^sub>1'' ;; c\<^sub>2"
concl_of_def: fixes inference :: "'a inference" shows "concl_of inference = (case inference of Infer x1 x2 \<Rightarrow> x2)"
oprod_minus_Id1: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "r \<subseteq> Id" shows "r *o r' - Id \<subseteq> {((x, y1), x, y2). x \<in> Field r \<and> (y1, y2) \<in> r' - Id}"
semiring_normalization_rules_19: fixes lx :: "'a"   and rx :: "'a"   and ry :: "'a" shows "lx * (rx * ry) = rx * (lx * ry)"
replicate_eq_replicate: fixes m :: "nat"   and x :: "'c"   and n :: "nat"   and y :: "'c" shows "(replicate m x = replicate n y) = (m = n \<and> (m \<noteq> 0 \<longrightarrow> x = y))"
simps_6: fixes F :: "'a formula"   and G :: "'a formula" shows "to_mini_formula (F \<^bold>\<or> G) = (to_mini_formula F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> to_mini_formula G"
atLeastLessThan_def: fixes l :: "'b word"   and u :: "'b word" shows "signed.atLeastLessThan l u \<equiv> signed.atLeast l \<inter> signed.lessThan u"
sin_times_sin: fixes z :: "'a" shows "sin w * sin z = (cos (w - z) - cos (w + z)) / (2::'a)"
preImage_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set" shows "preImage f A = {x. f x \<in> A}"
not_bot_below_trans: fixes a :: "'c"   and b :: "'c" assumes "a \<noteq> \<bottom>"   and "a \<sqsubseteq> b" shows "b \<noteq> \<bottom>"
lookup_someD: fixes T :: "(('e \<times> 'd) \<times> 'f) tree"   and x :: "'e"   and y :: "'d" assumes "lookup T x = Some y" shows "\<exists>p. p \<in> set (Tree2.inorder T) \<and> p = (x, y)"
weak_seq_inf_distrib: fixes c :: "'a::refinement_lattice"   and d\<^sub>0 :: "'a::refinement_lattice"   and d\<^sub>1 :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice) ; ((d\<^sub>0::'a::refinement_lattice) \<sqinter> (d\<^sub>1::'a::refinement_lattice)) \<sqsubseteq> c ; d\<^sub>0 \<sqinter> c ; d\<^sub>1"
awalk_verts_arc2: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and e :: "'b" assumes "G.awalk u p v"   and "e \<in> set p" shows "head G e \<in> set (G.awalk_verts u p)"
adopt_node_removes_first_child: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.adopt_node owner_document node \<rightarrow>\<^sub>h h'"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr' \<rightarrow>\<^sub>r node # children" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr' \<rightarrow>\<^sub>r children"
in_set_real_minus_interval: fixes x :: "real"   and X :: "float interval"   and y :: "real"   and Y :: "float interval" assumes "x \<in>\<^sub>r X"   and "y \<in>\<^sub>r Y" shows "x - y \<in>\<^sub>r X - Y"
no_inv_on_left_moving_in_middle_B_Oc: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_on_left_moving_in_middle_B (as, lm) (s, l, Oc # r) ires = False"
cases: fixes x :: "pratt" obtains p :: "nat" where "x = Prime p__"   | p :: "nat" and a :: "nat" and xa :: "nat" where "x = Triple p a xa"
integral_zero: fixes M :: "'e measure" shows "LINT x|M. (0::'d) = (0::'d)"
induct: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'c"   and a1 :: "'b list" assumes "\<And>f. P f []"   and "\<And>f x xs. P f (filter (\<lambda>y. f x \<noteq> f y) xs) \<Longrightarrow> P f (x # xs)" shows "P a0 a1"
state_trans_def: fixes asig :: "'d set \<times> 'd set \<times> 'd set"   and R :: "('e \<times> 'd \<times> 'e) set" shows "state_trans asig R \<equiv> (\<forall>triple. triple \<in> R \<longrightarrow> fst (snd triple) \<in> actions asig) \<and> (\<forall>a. a \<in> inputs asig \<longrightarrow> (\<forall>s1. \<exists>s2. (s1, a, s2) \<in> R))"
nonzero_inverse_minus_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse (uminus a) = uminus (inverse a)"
\<tau>2: shows "typ_of (Abs_fv x \<tau>' t) = Some (\<tau>' \<rightarrow> \<tau>)"
rel_funI: fixes A :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and B :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and f :: "'n \<Rightarrow> 'p"   and g :: "'o \<Rightarrow> 'q" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "(A ===> B) f g"
hoare_weakening: fixes p :: "'a"   and p' :: "'a"   and q' :: "'a"   and q :: "'a"   and c :: "'b" assumes "p \<le> p'"   and "q' \<le> q"   and "\<lbrace>\<iota> p'\<rbrace> c \<lbrace>\<iota> q'\<rbrace>" shows "\<lbrace>\<iota> p\<rbrace> c \<lbrace>\<iota> q\<rbrace>"
in_ST_imp: fixes q :: "hf"   and u :: "'a list" assumes "q \<in> ST.nextl {q0} u" shows "q = q0 \<and> u = [] \<or> (\<exists>u1 u2. u = u1 @ u2 \<and> u1 \<in> star (dfa.language MS) \<and> q \<in> ST.epsclo {dfa.nextl MS (dfa.init MS) u2})"
lex_diff2: fixes c :: "real \<times> real"   and a :: "real \<times> real"   and b :: "real \<times> real" shows "Counterclockwise_2D_Arbitrary.lex c (a - b) = Counterclockwise_2D_Arbitrary.lex (c + b) a"
bind_left_identity: fixes a :: "'c"   and f :: "'c \<Rightarrow> ('a, 'b) state" shows "State_Monad.return a \<bind> f = f a"
doctor_optimal_match_def: fixes ds :: "'d set"   and Y :: "'x set" shows "doctor_optimal_match ds Y = (stable_on ds Y \<and> (\<forall>X. \<forall>x\<in>X. stable_on ds X \<longrightarrow> (\<exists>y\<in>Y. (x, y) \<in> Pd (Xd x))))"
subspace_orthogonal_to_vector: fixes x :: "'a" shows "subspace {y. orthogonal x y}"
Compl_INT: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "- \<Inter> (B ` A) = (\<Union>x\<in>A. - B x)"
lemma_4_2_iii: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a * c \<le> b * c) = (a \<le> b)"
OrdDom_def: fixes r :: "hf" shows "OrdDom r \<equiv> \<forall>x y. \<langle>x, y\<rangle> \<^bold>\<in> r \<longrightarrow> Ord x"
UTM_pre_def: shows "UTM_pre = t_wcode |+| t_utm"
R_def: shows "R = \<lparr>carrier = carrier_irr, monoid.mult = mult_irr, one = 1, zero = 0, add = plus_irr\<rparr>"
closed_euler2_2: fixes ua :: "'a"   and p :: "'b list"   and u :: "'a" assumes "euler_trail ua p ua"   and "u \<in> verts G" shows "in_degree G u = out_degree G u"
l2_inv2D: fixes x :: "l2_state"   and R :: "fid_t" assumes "in_progress (progress x R) xsk"   and "x \<in> l2_inv2" shows "\<exists>X N. sklvl1.guessed_frame R xsk = Some (Exp X (NonceF (R $ N)))"
NN_unique: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (all yy (all yy' (imp (cnj (NN \<langle>\<phi>\<rangle> (Var yy)) (NN \<langle>\<phi>\<rangle> (Var yy'))) (eql (Var yy) (Var yy')))))"
induct: fixes P :: "'a \<Rightarrow> bool"   and a0 :: "'a" assumes "\<And>t. (\<And>x2 x y. \<lbrakk>unapp t = Some x2; (x, y) = x2\<rbrakk> \<Longrightarrow> P x) \<Longrightarrow> P t" shows "P a0"
constant_term_denominator_nonzero_imp_constant_term_denominator_div_gcd_nonzero: fixes x :: "'b poly fract"   and a :: "'b poly" assumes "constant_term (denominator x) \<noteq> (0::'b)" shows "constant_term (denominator x div gcd a (denominator x)) \<noteq> (0::'b)"
empty_fv_not_var: fixes t :: "('b, 'a) Term.term"   and x :: "'a" assumes "fv t = {}" shows "t \<noteq> Var x"
extremum_strict: fixes a :: "'a word" shows "\<not> a < 0"
last_appendL: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys = []" shows "last (xs @ ys) = last xs"
small_nstep_Suc_nend: fixes \<sigma>' :: "'state"   and P :: "'prog"   and \<sigma> :: "'state"   and n1 :: "nat" assumes "\<sigma>' \<in> small_nstep P \<sigma> (Suc n1)" shows "\<sigma> \<notin> endset"
indifferent_prefD: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and y :: "'a" assumes "x \<^bsub>r\<^esub>\<approx> y" shows "x \<^bsub>r\<^esub>\<preceq> y \<and> y \<^bsub>r\<^esub>\<preceq> x"
prove_KRP: fixes i :: "name"   and A :: "fm" shows "{} \<turnstile> KRP \<guillemotleft>Var i\<guillemotright> \<guillemotleft>A\<guillemotright> \<guillemotleft>A(i::=\<guillemotleft>A\<guillemotright>)\<guillemotright>"
prime_factorization_prime_elem: fixes p :: "'b" assumes "prime_elem p" shows "prime_factorization p = {#normalize p#}"
assu2: shows "\<forall>n\<ge>s. (\<Prod>j. d (n + j)) < A / real_of_int (a n) powr (1 / real_of_int (2 ^ n))"
Bot_def: fixes po :: "'b potype" shows "Bot po \<equiv> least (\<lambda>x. True) po"
power_minus_mult: fixes n :: "nat"   and a :: "'a" assumes "zero_class.zero < n" shows "power.power 1' (;) a (n - one_class.one) ; a = power.power 1' (;) a n"
restrict_map_subdom: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" assumes "dom m \<subseteq> S" shows "m |` S = m"
pat_type_eqvt: fixes pi :: "(name \<times> name) list"   and p :: "pat" shows "pi \<bullet> pat_type p = pat_type (pi \<bullet> p)"
cut_ge_greater_conv_if: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<ge> t = (if t \<in> I then insert t (I \<down>> t) else I \<down>> t)"
Gram_Schmidt_column_k_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" shows "Gram_Schmidt_column_k A k = (\<chi>a b. (if b = mod_type_class.from_nat k then column b A - proj_onto (column b A) {column i A |i. i < b} else column b A) $ a)"
delete_edges_set_nodes: fixes G :: "'a graph"   and E :: "('a \<times> 'a) set" shows "nodes (delete_edges G E) = nodes G"
fac_simp: fixes i :: "nat" assumes "0 < i" shows "fac i = i * fac (i - 1)"
codomains_def: fixes f :: "'a2" shows "C2.codomains f \<equiv> {b. C2.ide b \<and> b \<cdot>\<^sub>2 f \<noteq> C2.null}"
Hermite_reduce_above_Least_eq_le: fixes i :: "'a"   and ia :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "i < ia"   and "\<not> is_zero_row ia (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'b)) res)" shows "(LEAST n. A $ ia $ n \<noteq> (0::'b)) = (LEAST n. Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'b)) res $ ia $ n \<noteq> (0::'b))"
paths_le_def: fixes k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "paths_le k P = {p \<in> P. charslength p \<le> k}"
flowderiv_eq: fixes x0 :: "'a"   and t :: "real"   and \<xi>\<^sub>1 :: "'a"   and \<xi>\<^sub>2 :: "real" shows "blinfun_apply (flowderiv x0 t) (\<xi>\<^sub>1, \<xi>\<^sub>2) = blinfun_apply (Dflow x0 t) \<xi>\<^sub>1 + \<xi>\<^sub>2 *\<^sub>R f (flow0 x0 t)"
g_S1: shows "{Fun.swap j k id \<circ> p |p. p \<in> {p \<in> {p. p permutes UNIV}. evenperm p}} = (\<circ>) (Fun.swap j k id) ` {p \<in> {p. p permutes UNIV}. evenperm p}"
index_eq_index_conv: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "x \<in> set xs \<or> y \<in> set xs" shows "(index xs x = index xs y) = (x = y)"
drop_bit_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit 0 = id"
deriv: fixes x :: "real" assumes "x \<in> {a..b} - s" shows "((\<lambda>x. integral {c..x} f) \<circ> g has_vector_derivative g' x *\<^sub>R f (g x)) (at x within {a..b} - s)"
And_correct: fixes M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and v :: "'e \<Rightarrow> nat"   and n :: "nat"   and M2 :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry" shows "[M1]\<^bsub>v,n\<^esub> \<inter> [M2]\<^bsub>v,n\<^esub> = [And M1 M2]\<^bsub>v,n\<^esub>"
ind_cpa_game_eq_game\<^sub>0: shows "game \<A> = game\<^sub>0"
wellFormedSubst_def: fixes \<sigma> :: "('e list \<times> 'f list) list" shows "wellFormedSubst \<sigma> = (filter (\<lambda>(xvec, Tvec). \<not> (length xvec = length Tvec \<and> distinct xvec)) \<sigma> = [])"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D I\<^sub>D.map I\<^sub>D.cmp"
pred_lemma: fixes x :: "nat" shows "rec_exec UF.rec_pred [x] = x - 1"
mapping_of_cases: fixes y :: "'a \<Rightarrow>\<^sub>0 'b" assumes "y \<in> UNIV" obtains x :: "('a, 'b) pp" where "y = pp.mapping_of x"
exe_check_proof_def: fixes e :: "exetheory"   and P :: "proofterm"   and res :: "Core.term" shows "exe_check_proof e P res \<equiv> exe_wf_theory' e \<and> exereplay e P = Some res"
configuration_finite: fixes q :: "'b"   and n :: "nat" shows "finite (configuration q n)"
of_int_eq: fixes z :: "int" shows "of_int z = Abs (z mod n)"
lookupEvalHeap': fixes \<Gamma> :: "('e \<times> 'd) list"   and v :: "'e"   and e :: "'d"   and f :: "'d \<Rightarrow> 'f" assumes "map_of \<Gamma> v = Some e" shows "evalHeap \<Gamma> f v = f e"