hcju/isabelleps huggingface dataset:
weakCongruence_def: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "env.weakCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' P Q \<equiv> \<forall>\<Psi> \<sigma>. wellFormedSubst \<sigma> \<longrightarrow> env.weakPsiCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> (substPsi.seqSubs substTerm substAssert substCond P \<sigma>) (substPsi.seqSubs substTerm substAssert substCond Q \<sigma>)"
5: shows "t \<sqinter> v \<squnion> u \<sqinter> w = y"
vc_sound: fixes C :: "acom"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "vc C Q" shows "\<turnstile>\<^sub>3\<^sub>' {pre C Q} strip C { Q}"
TL: fixes h :: "'a"   and t :: "'a list" shows "tl (h # t) = t"
mod_ring_rel64_def: fixes pp :: "uint64"   and x :: "uint64"   and y :: "'a mod_ring" assumes "p = int_of_uint64 pp"   and "p \<le> 4294967295" shows "mod_ring_rel64 x y = (\<exists>z. urel64 x z \<and> mod_ring_rel z y)"
simps: fixes i :: "nat"   and xs :: "'a list" shows "fisher_yates_alt_aux i xs = (if i = 0 then return_pmf xs else pmf_of_set {..i} \<bind> (\<lambda>j. fisher_yates_alt_aux (i - 1) (swap xs i j)))"
rOpen: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P M (\<lparr>\<nu>*(xvec @ yvec)\<rparr>N \<prec>' P') A\<^sub>P \<Psi>\<^sub>P"   and "x \<in> supp N"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> A\<^sub>P"   and "x \<sharp> xvec"   and "x \<sharp> yvec"   and "A\<^sub>P \<sharp>* \<Psi>"   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>* xvec"   and "A\<^sub>P \<sharp>* yvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* M"   and "A\<^sub>P \<sharp>* C"   and "x \<sharp> C"   and "xvec \<sharp>* C"   and "yvec \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) M (\<lparr>\<nu>*(xvec @ x # yvec)\<rparr>N \<prec>' P') (x # A\<^sub>P) \<Psi>\<^sub>P"
delAround_def: fixes g :: "graph"   and u :: "nat"   and ps :: "(nat \<times> nat) list" shows "deleteAround g u ps = filter (\<lambda>p. \<not> close g u (fst p)) ps"
rotate2_dot: fixes u :: "(real, 2) vec"   and v :: "(real, 2) vec" shows "rotate2 u \<bullet> rotate2 v = u \<bullet> v"
st_eq_approx_iff: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "(x \<approx> y) = (st x = st y)"
l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf set_mode set_mode_locs"
case_eq_if: fixes f :: "'d \<Rightarrow> real \<Rightarrow> 'c"   and sctn :: "'d sctn" shows "(case sctn of Sctn x xa \<Rightarrow> f x xa) = f (normal sctn) (pstn sctn)"
that: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set ys" shows "compare cmp x y \<noteq> Greater"
index_middle_element: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set" assumes "[f[a .. b ..  c]X]" shows "\<exists>n>0. n < card X - 1 \<and> f n = b"
is_SNF_Smith_mxn_less_2: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m n"   and "n \<le> 2 \<or> m \<le> 2" shows "is_SNF A (Smith_mxn A)"
quasi_isometry_on_change_params: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and mu :: "real"   and D :: "real" assumes "lambda C -quasi_isometry_on X f"   and "lambda \<le> mu"   and "C \<le> D" shows "mu D -quasi_isometry_on X f"
wellBehavedComposition_def: fixes ES1 :: "'a ES_rec"   and ES2 :: "'a ES_rec"   and \<V> :: "'a V_rec"   and \<V>1 :: "'a V_rec"   and \<V>2 :: "'a V_rec" shows "wellBehavedComposition ES1 ES2 \<V> \<V>1 \<V>2 \<equiv> N\<^bsub>\<V>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {} \<and> N\<^bsub>\<V>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {} \<or> (\<exists>\<rho>1. N\<^bsub>\<V>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {} \<and> EventSystems.total ES1 (C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub>) \<and> BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub>) \<or> (\<exists>\<rho>2. N\<^bsub>\<V>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {} \<and> EventSystems.total ES2 (C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub>) \<and> BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub>) \<or> (\<exists>\<rho>1 \<rho>2 \<Gamma>1 \<Gamma>2. \<nabla>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub> \<and> \<Delta>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub> \<and> \<Upsilon>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub> \<and> \<nabla>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<Delta>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<Upsilon>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub> \<and> BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub> \<and> BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub> \<and> EventSystems.total ES1 (C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub>) \<and> EventSystems.total ES2 (C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub>) \<and> FCIA \<rho>1 \<Gamma>1 \<V>1 Tr\<^bsub>ES1\<^esub> \<and> FCIA \<rho>2 \<Gamma>2 \<V>2 Tr\<^bsub>ES2\<^esub> \<and> V\<^bsub>\<V>1\<^esub> \<inter> V\<^bsub>\<V>2\<^esub> \<subseteq> \<nabla>\<^bsub>\<Gamma>1\<^esub> \<union> \<nabla>\<^bsub>\<Gamma>2\<^esub> \<and> C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub> \<subseteq> \<Upsilon>\<^bsub>\<Gamma>1\<^esub> \<and> C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub> \<subseteq> \<Upsilon>\<^bsub>\<Gamma>2\<^esub> \<and> N\<^bsub>\<V>1\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {} \<and> N\<^bsub>\<V>2\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {})"
FG_fddg_closed: fixes g :: "'g"   and a :: "'f" assumes "g \<in> G" shows "a \<delta>\<delta> g \<in> FG"
dom_cod: fixes f :: "'a1" assumes "A1.arr f" shows "A1.dom (A1.cod f) = A1.cod f"
zero_not_in_nonzero: fixes R :: "('c, 'd) ring_scheme" shows "\<zero>\<^bsub>R\<^esub> \<notin> nonzero R"
induct: fixes P :: "'e list \<Rightarrow> ('d option \<Rightarrow> 'd) \<Rightarrow> ('e, 'd) trie \<Rightarrow> bool"   and a0 :: "'e list"   and a1 :: "'d option \<Rightarrow> 'd"   and a2 :: "('e, 'd) trie" assumes "\<And>f v ps. P [] f (Trie v ps)"   and "\<And>k ks f v ps. (\<And>x. P ks f x) \<Longrightarrow> P (k # ks) f (Trie v ps)" shows "P a0 a1 a2"
relcycle_group_def: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "relcycle_group p X S \<equiv> subgroup_generated (chain_group p X) (singular_relcycle_set p X S)"
echelon_form_upt_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" shows "echelon_form_upt_row A k = ((\<forall>i. mod_type_class.to_nat i < k \<and> is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. mod_type_class.to_nat j < k \<and> \<not> is_zero_row j A)) \<and> (\<forall>i j. i < j \<and> mod_type_class.to_nat j < k \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row j A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ j $ n \<noteq> (0::'d))))"
permLength: fixes p :: "(name \<times> name) list"   and xvec :: "'d list" shows "length (p \<bullet> xvec) = length xvec"
eval_in_carrier: fixes p :: "'a list"   and x :: "'a" assumes "set p \<subseteq> carrier R"   and "x \<in> carrier R" shows "local.eval p x \<in> carrier R"
simps: fixes x :: "'b"   and xl :: "'b bintree"   and xr :: "'b bintree" shows "t_right (Branch x xl xr) = xr"
loop: fixes st :: "'s \<times> ('c \<Rightarrow> real) set" shows "return_pmf st \<in> \<K> st"
s: shows "\<Turnstile> \<Gamma> \<Rightarrow> \<Delta>"
simps_3: fixes b :: "nat"   and t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "Formula.fvi b (formula.Eq t1 t2) = Formula.fvi_trm b t1 \<union> Formula.fvi_trm b t2"
inj_map_strong: fixes x :: "'c option"   and xa :: "'c option"   and f :: "'c \<Rightarrow> 'd"   and fa :: "'c \<Rightarrow> 'd" assumes "\<And>z za. \<lbrakk>z \<in> set_option x; za \<in> set_option xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_option f x = map_option fa xa" shows "x = xa"
\<T>e_ide_1: fixes b :: "'c" assumes "Cop.ide b" shows "S.arr (\<T> e b)"
strong_del_point_incidence_wf: fixes p :: "'a" shows "incidence_system (del_point p) (str_del_point_blocks p)"
env_restr_env_restr_subset: fixes d1 :: "'d set"   and d2 :: "'d set"   and x :: "'d \<Rightarrow> 'e" assumes "d1 \<subseteq> d2" shows "x f|` d2 f|` d1 = x f|` d1"
adjoint_dim_row: fixes A :: "'c mat" shows "dim_row (adjoint A) = dim_col A"
bitlen_bounds: fixes x :: "int" assumes "0 < x" shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
inverse_eq_imp_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"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "inverse a = inverse b" shows "a = b"
simps_6: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "hnf ([a\<noteq>b]P) = False"
carrier_mat_triv: fixes m :: "'c mat" shows "m \<in> carrier_mat (dim_row m) (dim_col m)"
MI_base: fixes p :: "PrIds"   and l :: "Locs" shows "basevars (caller memCh!p, (rtrner memCh!p, caller crCh!p, cst!p), (rtrner crCh!p, caller rmCh!p, rst!p), mm!l, rtrner rmCh!p, ires!p)"
fvi_Suc: fixes x :: "nat"   and b :: "nat"   and \<phi> :: "'b formula" shows "(x \<in> fvi (Suc b) \<phi>) = (Suc x \<in> fvi b \<phi>)"
smalls_seq: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and c' :: "com"   and s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and cx :: "com" assumes "small_steps \<pi> (c, s) (Some (c', s'))" shows "small_steps \<pi> (c;; cx, s) (Some (c';; cx, s'))"
nth_sbintr: fixes m :: "nat"   and w :: "int"   and n :: "nat" shows "bit (signed_take_bit m w) n = (if n < m then bit w n else bit w m)"
hyps_2: shows "\<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id"
positive_rat: fixes a :: "int"   and b :: "int" shows "Rat.positive (Rat.Fract a b) = (0 < a * b)"
igAbsCongSSTR_imp_igAbsCongS: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igAbsCongSSTR MOD" shows "igAbsCongS MOD"
moebius_mu_power': fixes a :: "nat"   and n :: "nat" shows "moebius_mu (a ^ n) = (if a = 1 \<or> n = 0 then 1::'b else if n = 1 then moebius_mu a else (0::'b))"
sup_exists: fixes A :: "'a set" assumes "A \<subseteq> carrier (L\<lparr>carrier := f ` carrier L\<rparr>)" shows "\<exists>s. is_lub (L\<lparr>carrier := f ` carrier L\<rparr>) s A"
Suc: shows "bound (of_nat n * v) \<le> n * bound v"
map_simp: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "identity_functor C"   and "partial_magma.arr C f" shows "identity_functor.map C f = f"
arr_dom: fixes f :: "'b" assumes "C'.arr f" shows "C'.arr (C'.dom f)"
that: fixes S :: "(real, 'n) vec set" assumes "S \<in> lmeasurable"   and "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> S"   and "Sigma_Algebra.measure lebesgue S \<le> 2 * e * (2 * m) ^ (CARD('n) - 1)" shows "thesis"
P_P: fixes i :: "nat"   and r :: "nat \<Rightarrow> 'a"   and k1 :: "nat \<Rightarrow> nat"   and N :: "nat"   and k2 :: "nat \<Rightarrow> nat" assumes "P i (r \<circ> k1)"   and "\<And>j. N \<le> j \<Longrightarrow> \<exists>j'\<ge>j. k2 j = k1 j'" shows "P i (r \<circ> k2)"
igSwapAllIPresIGWlsAll_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSwapAllIPresIGWlsAll MOD \<equiv> igSwapIPresIGWls MOD \<and> igSwapAbsIPresIGWlsAbs MOD"
children_eq_h: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr" shows "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
arcosh_real_def: fixes x :: "real" assumes "1 \<le> x" shows "arcosh x = ln (x + sqrt (x\<^sup>2 - 1))"
in_obs_valid_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n' :: "'b"   and n :: "'b"   and S :: "'b set" assumes "CFG sourcenode targetnode valid_edge Entry"   and "n' \<in> CFG.obs sourcenode targetnode valid_edge n S" shows "CFG.valid_node sourcenode targetnode valid_edge n"
mult_le_cancel2: fixes m :: "nat"   and k :: "nat"   and n :: "nat" shows "(m * k \<le> n * k) = (0 < k \<longrightarrow> m \<le> n)"
map_simp: fixes f :: "'a" assumes "arr f" shows "local.map f = f"
set_intros: fixes x2 :: "'c" shows "x2 \<in> set_option (Some x2)"
IH: fixes h :: "heap"   and l :: "sfields"   and sh :: "sheap" assumes "s\<^sub>1 = (h, l(V := None), sh)" shows "P \<turnstile> \<langle>{V:T; e\<^sub>1},(h, l, sh),b\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2(V := l V), sh\<^sub>2),b\<^sub>2\<rangle>"
minGraphProps_2: fixes g :: "graph"   and f :: "face" assumes "minGraphProps g"   and "f \<in> \<F> g" shows "distinct (vertices f)"
Seq_sound: fixes P :: "'b com"   and pre :: "'b set"   and rely :: "('b \<times> 'b) set"   and guar :: "('b \<times> 'b) set"   and mid :: "'b set"   and Q :: "'b com"   and post :: "'b set" assumes "\<Turnstile> P sat [pre, rely, guar, mid]"   and "\<Turnstile> Q sat [mid, rely, guar, post]" shows "\<Turnstile> Seq P Q sat [pre, rely, guar, post]"
that: fixes H :: "'a set"   and K :: "'a set" assumes "H \<lhd> B"   and "subgroup K B"   and "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}"   and "H <#>\<^bsub>B\<^esub> K = carrier B"   and "f \<in> Group.iso A (subgroup_generated B H)"   and "g \<in> Group.iso (subgroup_generated B K) C" shows "thesis"
PPs_insert: fixes x :: "'b"   and X :: "'b set" shows ".[insert x X] = (\<Union>e. (+) (monomial e x) ` .[X])"
growth: fixes n :: "nat" shows "prefix (bc (Suc n)) (bc n) \<or> (\<exists>b. bc (Suc n) = bc n @ [b]) \<and> mining (Suc n)"
tt_in_keys: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "tt p \<in> keys p"
one_greatest1: fixes x :: "'a" shows "(1::'a) \<oplus> !x = (1::'a)"
mat2matofpoly_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f" shows "mat2matofpoly_row A i = (\<chi>j. [:A $ i $ j:])"
gA_1: shows "OrderingSetIso (\<^bold>\<le>*) (\<^bold><*) dual_order.greater_eq dual_order.greater (f ` P) g"
clD_2: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "matroid carrier indep" shows "matroid.cl carrier indep X \<subseteq> carrier"
last_strict_2: fixes x :: "'b" shows "last\<cdot>(x : \<bottom>) = \<bottom>"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "B.par f f'"   and "UP f = UP f'" shows "f = f'"
iarray_append_code: fixes xs :: "'a iarray"   and x :: "'a" shows "iarray_append xs x = IArray (IArray.list_of xs @ [x])"
ac: shows "AE x in M. f x = 0 \<longrightarrow> g x = 0"
n_rw3: shows "ncols_iarray (IArray (map (\<lambda>x. IArray (map (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) [0..<CARD('b)])) [0..<CARD('c)])) = CARD('b)"
adjoint_def_alter: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec"   and m :: "nat"   and A :: "'c mat" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec m"   and "A \<in> carrier_mat n m" shows "inner_prod v (A *\<^sub>v w) = inner_prod (adjoint A *\<^sub>v v) w"
that: fixes y :: "'b" assumes "(x, y) \<in> S" shows "thesis"
nop_bij_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "nop_bij A n f \<equiv> v11 f \<and> nop_bij_axioms A n f"
weakenBisimE_1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx>\<^sub>w Q" shows "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<weakenBisim> Q"
eq: shows "{\<langle>b, a\<rangle> |a b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r} = (\<lambda>\<langle>a, b\<rangle>. \<langle>b, a\<rangle>) ` elts (vpairs r)"
ccField_cc_restr: fixes S :: "Vars.var set"   and G :: "CoCalls" shows "ccField (G G|` S) \<subseteq> ccField G \<inter> S"
inv_inv: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "inv (inv x) = x"
strict_ordered_set_def: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "strict_ordered_set A less \<equiv> irreflexive A less \<and> transitive A less"
2: fixes p_in_bs :: "bool" assumes "(p_in_bs, q, p) \<in> set (np gs bs hs data)" shows "thesis"
CutFreePC_def: shows "CutFreePC = \<Union> {Perms, Axioms, Conjs, Disjs, Alls, Exs, Weaks, Contrs}"
ctype: shows "case_nat REAL \<Gamma> \<turnstile>\<^sub>c IF\<^sub>c CReal 0 <\<^sub>c CVar 0 THEN (f \<circ>\<^sub>c ln\<^sub>c (CVar 0)) /\<^sub>c CVar 0 ELSE CReal 0 : REAL"
nn_integral_monotone_convergence_SUP: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "incseq f"   and "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (\<Squnion>i. f i x) \<partial>M = (\<Squnion>i. integral\<^sup>N M (f i))"
nonempty_has_size: fixes S :: "'a multiset" shows "(S \<noteq> {#}) = (0 < size S)"
Diff_Compl: fixes A :: "'a set"   and B :: "'a set" shows "A - - B = A \<inter> B"
rel_interior_Times: fixes S :: "'a set"   and T :: "'b set" assumes "convex S"   and "convex T" shows "rel_interior (S \<times> T) = rel_interior S \<times> rel_interior T"
array_set_oo'_def: fixes f :: "unit \<Rightarrow> 'a array"   and a :: "'a array" shows "array_set_oo' f a \<equiv> array_set_oo f a \<circ> nat_of_integer"
simps_14: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "sat \<sigma> V v i (Since \<phi> I \<psi>) = (\<exists>j\<le>i. mem (\<tau> \<sigma> i - \<tau> \<sigma> j) I \<and> sat \<sigma> V v j \<psi> \<and> (\<forall>k\<in>{j<..i}. sat \<sigma> V v k \<phi>))"
pcs_subset_1: fixes e :: "nat exp"   and pc :: "nat"   and d :: "nat" shows "pcs (compxE\<^sub>2 e pc d) \<subseteq> {pc..<pc + length (compE\<^sub>2 e)}"
xy_1: shows "x = rat_of_int a / rat_of_int b"
tauInduct: fixes P :: "pi"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto> \<tau> \<prec> P'"   and "\<And>P C. F C (\<tau>.P) P"   and "\<And>P P' a C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C ([a\<frown>a]P) P'"   and "\<And>P P' a b C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C P P'; a \<noteq> b\<rbrakk> \<Longrightarrow> F C ([a\<noteq>b]P) P'"   and "\<And>P P' Q C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) P'"   and "\<And>Q Q' P C. \<lbrakk>Q \<longmapsto> \<tau> \<prec> Q'; \<And>C. F C Q Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) Q'"   and "\<And>P P' Q C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q)"   and "\<And>Q Q' P C. \<lbrakk>Q \<longmapsto> \<tau> \<prec> Q'; \<And>C. F C Q Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P \<parallel> Q')"   and "\<And>P a b P' Q Q' C. \<lbrakk>P \<longmapsto> a<b> \<prec> P'; Q \<longmapsto> a[b] \<prec> Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q')"   and "\<And>P a b P' Q Q' C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; Q \<longmapsto> a<b> \<prec> Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q')"   and "\<And>P a x P' Q Q' C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; Q \<longmapsto> a<\<nu>x> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (<\<nu>x>(P' \<parallel> Q'))"   and "\<And>P a x P' Q Q' C. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; Q \<longmapsto> a<x> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (<\<nu>x>(P' \<parallel> Q'))"   and "\<And>P P' x C. \<lbrakk>P \<longmapsto> \<tau> \<prec> P'; x \<sharp> C; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) (<\<nu>x>P')"   and "\<And>P P' C. \<lbrakk>P \<parallel> !P \<longmapsto> \<tau> \<prec> P'; \<And>C. F C (P \<parallel> !P) P'\<rbrakk> \<Longrightarrow> F C (!P) P'" shows "F C P P'"
finite_positions_in_row: fixes \<Sigma> :: "'b list"   and n :: "nat"   and x :: "('b \<times> bool list) list" assumes "formula \<Sigma>"   and "0 < n" shows "finite (positions_in_row (x @- sconst (formula.any \<Sigma>, replicate n False)) 0)"
p_has_vector_derivative: fixes t :: "real"   and s :: "real set" shows "(p has_vector_derivative p' t) (at t within s)"
gt_imp_chksubs_gt: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t >\<^sub>t s" shows "chksubs (>\<^sub>t) t s"
PCon: fixes r :: "'a pfm"   and p :: "'a pfm"   and q :: "'a pfm" shows "\<turnstile>\<^sub>! ([r]\<^sub>! (p \<^bold>\<and>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<and>\<^sub>! [r]\<^sub>! q)"
intro: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "\<And>a. a \<in> S \<Longrightarrow> pls z a = a"   and "z \<in> S" shows "comm_monoid_add_ow_axioms S pls z"
simps_2: fixes a :: "'c"   and x :: "('c, 'd) bintree option"   and xs :: "('c, 'd) bintree option list" shows "Binomial_Queue.find a (x # xs) = (case x of None \<Rightarrow> Binomial_Queue.find a xs | Some t \<Rightarrow> if priority t = a then Some t else Binomial_Queue.find a xs)"
represents_def: fixes T :: "'s Trace set" shows "represents T \<equiv> jkbpT T = T"
set_filter_option: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option" shows "set_option (filter_option P x) = {y \<in> set_option x. P y}"
fundchamber_S_image_inj_on: shows "inj_on (\<lambda>s. s `\<rightarrow> C0) (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
FL_valid_Tree_eqvt: fixes P :: "'state"   and t :: "('b, 'pred, 'act, 'effect) Tree"   and p :: "perm" assumes "FL_valid_Tree P t" shows "FL_valid_Tree (p \<bullet> P) (p \<bullet> t)"
insert_subset: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "order.greater_eq B (insert x A) = (x \<in> B \<and> order.greater_eq B A)"
length_stake: fixes n :: "nat"   and s :: "'d stream" shows "length (stake n s) = n"
valid_cfg_def: shows "MDP.valid_cfg = \<Union> (MDP.cfg_on ` S)"